\college

Lady Margaret Hall \degreeMSc Mathematical Sciences \degreedateTrinity 2024

Synthetic Homotopy Theory

Yuhang Wei

Abstract

The goal of this dissertation is to present results from synthetic homotopy theory based on homotopy type theory (HoTT). After an introduction to Martin-Löf’s dependent type theory and homotopy type theory, key results include a synthetic construction of the Hopf fibration, a proof of the Blakers–Massey theorem, and a derivation of the Freudenthal suspension theorem, with calculations of some homotopy groups of $n$ -spheres.

Keywords: homotopy type theory, homotopy theory, algebraic topology, type theory, constructive mathematics

Acknowledgements

I would like to thank my dissertation supervisor, Prof. Kobi Kremnitzer, for his invaluable guidance, inspirations and encouragement during this journey. His expertise and support have been instrumental in the completion of this work.

My gratitude also goes to my college supervisor, Dr. Rolf Suabedissen, for providing insightful advice on the structure and direction of this dissertation.

I am profoundly grateful to my parents for their unconditional and unwavering support in my academic pursuits and, indeed, all aspects of my life. Their efforts in their own field of research and teaching have been a constant source of motivation for me.

Finally, special thanks are due to all of my friends who helped me decide my academic path in pure mathematics or supported me in different ways during the writing of this dissertation. An incomplete list of their names follows: Qixuan Fang, Franz Miltz, Samuel Knutsen, Quanwen Chen, Sifei Li, Phemmatad Tansoontorn, Hao Chen and Haoyang Liu. This dissertation would not have been possible without each and every one of them.

Note

A slightly abridged version of this dissertation, originally written in Typst, was submitted in fulfillment of the requirements for the degree of MSc Mathematical Sciences at University of Oxford. That version was awarded the Departmental Dissertation Prize by the Mathematical Institute, University of Oxford. The current version has been extended and reformatted in LaTeX.

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Chapter 1 Introduction

The goal of this dissertation is to present results in synthetic homotopy theory based on homotopy type theory (HoTT, also known as univalent foundations), including the Hopf fibration and Blakers–Massey theorem.

Compared to set theory, type theory is often considered as a better foundation for constructive mathematics and mechanised reasoning [Hou17, Section 1.2]. Martin-Löf’s intuitionistic type theory (ITT, also known as dependent type theory) was first proposed in the 1970s [ML75]. In ITT, any mathematical object is seen as inherently a term of some type. In particular, ITT follows the Curry–Howard correspondence, or the paradigm of propositions-as-types, where any proposition is regarded as a type and proving it is equivalent to constructing a term of that type. Since all terms of types can be seen as computer programmes and be executed, proof assistants, e.g. Coq, Agda and Lean, have been developed to ensure the correctness of proofs through type-checking [Bru16, Introduction].

HoTT is based on the intensional version of ITT, where a term of any identity type (i.e., the proposition that two terms of a type are equal) is not necessarily unique. This seemingly counterintuitive situation is addressed by interpreting types as topological spaces and identities as paths between points. The link between homotopy theory and type theory was first discovered in 2006 independently by Voevodsky and by Awodey and Warren [AW09]. In 2009, Voevodsky proposed the univalence axiom (UA), which formalises the idea that ‘isomorphic structures have the same properties’. In 2011, the concept of higher inductive types (HIT) emerged [Lum11], which allows the use of path constructors in addition to point constructors, so that many fundamental topological spaces can be elegantly constructed. In summary, the components of HoTT are

\text{ HoTT }=\text{ ITT }+\text{ UA }+\text{ HIT}.

HoTT and $\infty$ -topoi form a syntax-semantics duality, and all results proven in HoTT hold in the $\infty$ -category of homotopy types [Car22]. This lays the foundation for our synthetic approach to homotopy theory, where all constructions are invariant under homotopy equivalences. Unlike in classical (or analytical) homotopy theory, notions such as spaces, points, continuous functions, and continuous paths are not based on set-theoretic constructions but are axiomatised as primitive ones, showing the formal abstraction provided by type theory [Hou17, Section 1.2].

Many results of synthetic homotopy theory, including the Hopf fibration, Freudenthal suspension theorem, and the van Kampen theorem, can be found in [Uni13, Section 8]. More recent efforts in this field include $\pi_{4}\left({\mathbb{S}}^{3}\right)\simeq{\mathbb{Z}}/2{\mathbb{Z}}$ [Bru16], the Blakers–Massey theorem [HFLL16], the Cayley–Dickson construction [BR16], the construction of the real projective spaces [BR17], and the long exact sequence of homotopy $n$ -groups [BR21]. Additionally, synthetic (co)homology theory has been investigated by [Cav15], [BH18] and [Gra18].

This dissertation begins with a brief summary of dependent type theory in Chapter 2 and an overview of HoTT in Chapter 3. Chapter 4 mainly discusses homotopy $n$ -types and $n$ -connected types. Chapter 5 develops the fibre sequence, Hopf construction and Hopf fibration. Chapter 6 proves the Blakers–Massey theorem. Along the journey, we will calculate the homotopy groups of $n$ -spheres in Table 1.1.

Table 1.1: Homotopy groups of

n

-spheres calculated in this dissertation.

	$\mathbb{S}^{1}$	$\mathbb{S}^{2}$	$\mathbb{S}^{3}$	$\mathbb{S}^{n\geq 4}$
$\pi_{1}$	$\mathbb{Z}$	$0$	$0$	$\pi_{k<n}(\mathbb{S}^{n})\simeq 0$
$\pi_{2}$	$0$	$\mathbb{Z}$	$0$
$\pi_{3}$	$0$	$\mathbb{Z}$	$\mathbb{Z}$
$\pi_{k\geq 4}$	$0$	$\pi_{k}(\mathbb{S}^{2})\simeq\pi_{k}(\mathbb{S}^{3})$		$\pi_{n}(\mathbb{S}^{n})\simeq\mathbb{Z}$

Although no result is essentially new, calculations and discussions original to this dissertation are marked with a red square ( $\blacksquare$ ). For instance, Chapter 4 contains discussions on path inductions, Corollary 4.4 and Corollary 4.6, and novel examples, Example 4.2 and Example 4.25. Most noticeably, Chapter 6 ‘informalises’ a mechanised Blakers–Massey theorem proof [HFLL16] by integrating details from a Freudenthal suspension theorem proof [Uni13, Section 8.6], before explicitly deriving the Freudenthal suspension theorem as a corollary. Additionally, visualisation diagrams are plotted for certain definitions and theorems to aid understanding.

Chapter 2 Dependent type theory

Dependent type theory is a formal system that organises all mathematical objects, structures and knowledge by expressing them as types or terms of types [Rij22, p. 1]. A type is defined by rules specifying its own formation and the construction of its terms. Types resemble sets; however, a major difference is that membership in a type is not a predicate but is part of the inherent nature of its term.

We write $a:A$ if $a$ is a term of type $A$ , where we also say that type $A$ is inhabited and $a$ is an inhabitant of $A$ . All types are considered terms of a special type, $\mathtt{Type}$ , and we denote $A:\mathtt{Type}$ if $A$ is a type¹¹1We suppress the underlying universe level (see [Uni13, Section 1.3]).. If $A:\mathtt{Type}$ and $a,b:A$ , we write $a\equiv b$ to indicate that $a$ and $b$ are judgementally equal, meaning that $a$ and $b$ are only symbolically different. When defining a new term $b:A$ from $a:A$ , we use the notation $b:\equiv a$ . Judgemental equalities are to be contrasted with propositional equalities introduced in Section 2.3.

We shall briefly sketch the fundamentals of dependent type theory, based on [Rij22, Chapter I] and [Uni13, Chapter 1]. Importantly, these constructions all bear interpretations in logic and topology in Table 2.1.

Table 2.1: Interpretations of type theory in logic and topology.

Type theory	Logic	Topology
$A:\mathtt{Type}$	a proposition $A$	a topological space $A$
$a:A$	a proof $a$ of the proposition $A$	a point $a$ in the space $A$
$A\to B$	if $A$ then $B$	a continuous map $A\to B$
$P:A\to\mathtt{Type}$	a predicate $P(x)$ for each $x:A$	a fibration $P$ over $A$ with fibre $P(x)$ for each $x:A$
$(x:A)\to P(x)$	for all $x:A$ , $P(x)$	a continuous section of fibration $P$
$\sum_{x:A}P(x)$	there exists $x:A$ , $P(x)$ ²²2This type carries more information than merely a ’true or false’ judgement of the existence, since a term of this type provides a concrete $a:A$ as well as a term $b:P(a)$ , i.e., a proof for $P(a)$ .	the total space of fibration $P$
$A\times B$	$A$ and $B$	the product space $A\times B$
$A+B$	$A$ or $B$ ³³3A term of $A+B$ also carries the information ’which one of $A$ and $B$ is true’.	the disjoint union $A\sqcup B$
$\mathbb{1}$	True	the contractible space
$\mathbb{0}$	False	the empty space
$p:u=_{A}v$	$p$ is a proof of the equality of $u$ and $v$	$p$ is a continuous path from $u$ to $v$
$\mathtt{refl}_{a}:a=_{A}a$	the reflexivity of equality at $a$	the constant path at $a$

2.1 Function types

Let $A:\mathtt{Type}$ . A dependent type (or a family of types) $P:A\rightarrow\mathtt{Type}$ over $A$ assigns a type $P(x):\mathtt{Type}$ to each term $x:A$ .

A dependent function is a function whose output types may vary depending on its input. The dependent function type, also known as $\Pi$ -type, is specified by:

(1)

The formation rule: Given a type $A$ and a dependent type $P:A\rightarrow\mathtt{Type}$ , the dependent function type $(x:A)\rightarrow P(x)$ can be formed.
(2)

The introduction rule: To introduce a dependent function of type $(x:A)\rightarrow P(x)$ , a term $b(x):P(x)$ is needed for each $x:A$ . This dependent function is then denoted by $\lambda x.b(x)$ .
(3)

The elimination rule: Given a dependent function $f:(x:A)\rightarrow P(x)$ and a term $a:A$ , a term $f(a):P(a)$ is obtained by evaluating $f$ at $a$ .
(4)

The computation rule: Combining the introduction and elimination rules, $\left(\lambda x.b(x)\right)(a):\equiv b(a)$ .

In particular, when $P$ is a constant type family assigning the same type $B$ to every $a:A$ , we obtain the (non-dependent) function type $A\rightarrow B$ .

2.2 Inductive types

A wide range of constructions are based on inductive types. An inductive type $A$ is specified by the following:

(1)

The constructors (or construction rules) of $A$ state the ways to construct a term of $A$ .
(2)

The induction principle states the necessary data to construct a dependent function $(x:A)\rightarrow P(x)$ for a given dependent type $P:A\rightarrow\mathtt{Type}$ . Particularly, the recursion principle is the induction principle when $P$ is a constant type.
(3)

The computation rule states how the dependent function produced by the induction principle acts on the constructors.

2.2.1 $\Sigma$ -types

Given a type $A$ and a dependent type $P:A\rightarrow\mathtt{Type}$ , we can form the $\Sigma$ -type, written as $\sum_{x:A}P(x),$ whose terms are called dependent pairs.

(1)

The construction rule states that given $a:A$ and $b:P(a)$ , we can construct the dependent pair $(a,b):\sum_{x:A}P(x)$ .
(2)

The induction principle states that given
- •
  
  a dependent type $C:\sum_{x:A}P(x)\rightarrow\mathtt{Type}$ and
- •
  
  a function $g:(a:A)\rightarrow\left(b:P(a)\right)\rightarrow C(a,b)$ ⁴⁴4Arrows $\rightarrow$ are right-associative.,
there is a dependent function $f:\left(p:\sum_{x:A}P(x)\right)\rightarrow C(p)$ .
(3)

The computation rule states that the above $f$ satisfies $f(a,b):\equiv g(a)(b)$ .

The function $g$ above is the curried form of $f$ and $f$ is the uncurried form of $g$ , and they can be used interchangeably. Unlike the convention in [Uni13], we use the notation that reflects the actual curriedness of a function, i.e., $g(a)(b)$ is not written as $g(a,b)$ .

Using the induction principle, we define the first projection function $\mathtt{pr}_{1}:\sum_{x:A}P(x)\rightarrow A$ by $\lambda a.\lambda b.a:(a:A)\rightarrow\left(b:P(a)\right)\rightarrow A$ and the second projection function $\mathtt{pr}_{2}:\left(p:\sum_{x:A}P(x)\right)\rightarrow P\left(\mathtt{pr}_{1}(p)\right)$ by $\lambda a.\lambda b.b:(a:A)\rightarrow\left(b:P(a)\right)\rightarrow P(a)$ .

The case when $P$ is a constant type $B$ gives the cartesian product $A\times B$ . In this case, we write $\mathtt{fst}$ and $\mathtt{snd}$ for the projection maps onto $A$ and $B$ , respectively.

2.2.2 Other inductive types

Some other inductive types are specified as follows based on [Esc22, Section 2].

•

The empty type $\mathbb{0}$ has no construction rule. The induction principle states that given $(P:\mathbb{0}\to\mathtt{Type})$ , there is a dependent function $((x:\mathbb{0})\to P(x))$ . In particular, the recursion principle states that for any $A:\mathtt{Type}$ , we can define a function $f:\mathbb{0}\to A$ , which can be logically interpreted as the principle of explosion, or ex falso quodlibet. There is also no computation rule for $\mathbb{0}$ . For any type $A$ , $\neg A$ is defined as the function type $A\to\mathbb{0}$ .
•

The unit type $\mathbb{1}$ has a single constructor $\star:\mathbb{1}$ . The induction principle for $\mathbb{1}$ states that given a dependent type $P:\mathbb{1}\to\mathtt{Type}$ and a term $p:P(\star)$ , there is a dependent function $f:(x:\mathbb{1})\to P(x)$ . The computation rule⁵⁵5Henceforth, we may concisely state the computation rule as a part of the induction principle. states that such constructed $f$ satisfies $f(\star):\equiv p$ .
•
Let $A,B:\mathtt{Type}$ . The disjoint sum $A+B$ of $A$ and $B$ has construction rules $\mathtt{inl}:A\to A+B$ and $\mathtt{inr}:B\to A+B$ . The induction principle states that given
- –
  
  a dependent type $P:(A+B)\to\mathtt{Type}$ ,
- –
  
  two dependent functions $f_{\mathtt{inl}}:(a:A)\to P(\mathtt{inl}(a))$ and
- –
  
  $f_{\mathtt{inr}}:(b:B)\to P(\mathtt{inr}(b))$ ,
there is a dependent function $f:((x:A+B)\to P(x))$ such that $f(\mathtt{inl}(a)):\equiv f_{\mathtt{inl}}(a)$ and $f(\mathtt{inr}(b)):\equiv f_{\mathtt{inr}}(b)$ .
•

The boolean type $\mathbb{2}$ is defined as $\mathbb{2}\equiv\mathbb{1}+\mathbb{1}$ , and its two constructors are written as $\mathtt{True}$ and $\mathtt{False}$ .
•
The type of natural numbers $\mathbb{N}$ has construction rules $0:\mathbb{N}$ and $\mathtt{succ}:\mathbb{N}\to\mathbb{N}$ . The induction principle states that given
- –
  
  a dependent type $P:\mathbb{N}\to\mathtt{Type}$ ,
- –
  
  a term $f_{0}:P(0)$ , and
- –
  
  a dependent function $f_{\mathtt{succ}}:(n:\mathbb{N})\to P(n)\to P(\mathtt{succ}(n))$ ,
there is a dependent function $f:((n:\mathbb{N})\to P(n))$ such that $f(0):\equiv f_{0}$ and $f(\mathtt{succ}(n)):\equiv f_{\mathtt{succ}}(n)(f(n))$ .
•

The type of integers $\mathbb{Z}$ has three constructors, representing zero, the positive and negative numbers, and its induction principle and computation rules are similar to $\mathbb{N}$ [Bru16, Section 1.3].

2.3 Identity types

Given a type $A$ and two terms $u:A$ and $v:A$ , we can form the identity type $u=_{A}v$ , whose terms are called (propositional) equalities or paths. The constructor for a family of identity types $\left(a=_{A}a\right)$ indexed by $a:A$ is given by a dependent function

\mathtt{refl}:(a:A)\rightarrow\left(a=_{A}a\right).

However, Axiom K, the assumption that $p=\mathtt{refl}_{a}$ for any $p:\left(a=_{A}a\right)$ , does not hold in general.

For a fixed $a:A$ , the induction principle (called the path induction) for a family of identity types $\left(a=_{A}x\right)$ indexed by $x:A$ states that given

•

a dependent type $P:(x:A)\rightarrow\left(a=_{A}x\right)\rightarrow\mathtt{Type}$ and
•

a term $d:P(a)\left(\mathtt{refl}_{a}\right)$ ,

there is a dependent function

J_{P}(d):(x:A)\rightarrow\left(p:\left(a=_{A}x\right)\right)\rightarrow P(x)(p),

such that $J_{P}(d)(a)\left(\mathtt{refl}_{a}\right):\equiv d$ . The path induction implies that to construct a dependent function for a path $p$ from a fixed endpoint $a$ to a free $x$ , it suffices to assume that $x$ is $a$ and $p$ is $\mathtt{refl}_{a}$ . Importantly, one endpoint $x$ should be free, since otherwise we would have Axiom K. Intuitively, the path induction reflects the fact that the space of all paths starting from a fixed point is contractible. We shall revisit this idea in Proposition 4.3.

The reader is advised to review Table 2.1 at this point.

Chapter 3 Homotopy type theory

The implications of the identity type are profound, especially when it interacts with other types. In Section 3.1, the identity type equips each type with an $\infty$ -groupoid structure. In Section 3.2, we explore how equalities can be passed along by non-dependent and dependent functions. The notions of ‘equivalence’ between functions and between types are subtle ones treated in Section 3.3, and their relationship with paths in function types or in $\mathtt{Type}$ is handled in Section 3.4 by function extensionality and univalence axiom. In Section 3.5, we introduce higher inductive types that allow path constructors. Together, these form the basic components of HoTT.

3.1 The $\infty$ -groupoid structure of types

In classical mathematics, equalities are reflexive, symmetric and transitive. While reflexivity is expressed as the $\mathtt{refl}$ constructor for identity types, symmetry and transitivity appear in the form of path inversion and concatenation.

Lemma 3.1 ([Uni13, Lemma 2.1.1]).

For every type $A$ and $x,y:A$ , there is a function $\left(x=_{A}y\right)\rightarrow\left(y=_{A}x\right)$ , denoted $p\mapsto p^{-1}$ , such that $\left(\mathtt{refl}_{x}\right)^{-1}:\equiv\mathtt{refl}_{x}$ , where $p^{-1}$ is called the inverse of $p$ .

Remark 3.2.

By the paradigm of propositions-as-types, this lemma is equivalent to the type

\left(A:\mathtt{Type}\right)\rightarrow(x:A)\rightarrow(y:A)\rightarrow\left(p:\left(x=_{A}y\right)\right)\rightarrow\left(y=_{A}x\right).

The proof shows how we construct a term of this type.

Proof.

Fix $A:\mathtt{Type}$ and $x:A$ . By path induction, it suffices to assume $y$ is $x$ and $p:\left(x=_{A}y\right)$ is $\mathtt{refl}_{x}$ . Then it suffices to give a term of $x=_{A}x$ , which is fulfilled by $\mathtt{refl}_{x}$ . We thus have $\left(\mathtt{refl}_{x}\right)^{-1}:\equiv\mathtt{refl}_{x}$ . ∎

Lemma 3.3 ([Uni13, Lemma 2.1.2]).

For every type $A$ and $x,y,z:A$ , there is a function $\left(x=_{A}y\right)\rightarrow\left(y=_{A}z\right)\rightarrow\left(z=_{A}x\right)$ , denoted $p\mapsto q\mapsto p{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ q$ , such that $\mathtt{refl}_{x}{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \mathtt{refl}_{x}:\equiv\mathtt{refl}_{x}$ , where $p{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ q$ is called the concatenation of $p$ and $q$ .

Proof.

By path induction twice.∎

Lemma 3.4 ([Uni13, Lemma 2.1.4]).

Path concatenation is associative, i.e., for any $x,y,z,w:A$ , $p:x=_{A}y$ , $q:y=_{A}z$ , and $r:z=_{A}w$ , there is a $2$ -dimensional path $p{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \left(q{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ r\right)=\left(p{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ q\right){\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ r$ .

Proof.

By path induction, we assume $p,q,r$ are all $\mathtt{refl}_{x}$ . Then $\mathtt{refl}_{x}{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \left(\mathtt{refl}_{x}{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \mathtt{refl}_{x}\right)\equiv\mathtt{refl}_{x}\equiv\left(\mathtt{refl}_{x}{\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \mathtt{refl}_{x}\right){\ \mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\ }\ \mathtt{refl}_{x}$ by Lemma 3.3, so we give $\mathtt{refl}_{\mathtt{refl}_{x}}$ . ∎

We similarly can show that path concatenation satisfies inverse and identity laws [Uni13, Lemma 2.1.4]. The above are all examples of coherence operations on paths or higher-dimensional paths. We omit further discussions and direct to [Uni13, Section 2.1] or [Bru16, Section 1.4] for details. Henceforth, we shall omit proofs which only require path inductions.

We then observe that the identity types can be constructed recursively: for any type $A$ , we can form $x=_{A}y$ , $p=_{x=_{A}y}q$ , and $r=_{p=_{x=_{A}y}q}s$ , etc. This observation, combined with the next definition, pointed types, gives rise to the notion of loop spaces.

Definition 3.5.

The type of pointed types is defined as $\sum_{A:\mathtt{Type}}A$ . Alternatively, a pointed type is a dependent pair $\left(A,\star_{A}\right)$ where $A:\mathtt{Type}$ and $\star_{A}:A$ , and $\star_{A}$ is called the basepoint of this pointed type. We often leave the basepoint implicit.

Example 3.6.

Let $A:\mathtt{Type}$ , then define a pointed type $A_{+}:\equiv\left(A+\mathbb{1},\mathtt{inr}(\star)\right)$ . This is called adjoining a disjoint basepoint.

Definition 3.7.

The loop space of a pointed type $\left(A,\star_{A}\right)$ is defined as the pointed type

\Omega\left(A,\star_{A}\right):\equiv\left(\left(\star_{A}=_{A}\star_{A}\right),\mathtt{refl}_{\star_{A}}\right).

Then for $n:{\mathbb{N}}$ , define the $n$ -th loop space of $\left(A,\star_{A}\right)$ by $\Omega^{0}A:\equiv A$ and $\Omega^{n+1}A:\equiv\Omega\left(\Omega^{n}A\right)$ (basepoints implicit).

In summary, coherence operations and loop spaces demonstrate that the identity type equips each pointed type with a (weak) $\infty$ -groupoid structure (see [Bru16, Appendix A] for a formal definition).

3.2 Dependent paths

In classical mathematics, for any function between sets $f:A\rightarrow B$ and $x,y\in A$ , $x=y$ implies $f(x)=f(y)$ . The analogue in HoTT is expressed as a path $f(x)=f(y)$ induced by the path $x=y$ , demonstrating the proof relevance feature of type theory.

Lemma 3.8 ([Uni13, Lemma 2.2.1]).

Let $A,B:\mathtt{Type}$ , $f:A\rightarrow B$ , $x,y:A$ and $p:x=_{A}y$ . Then there is a path $\mathtt{ap}_{f}(p):f(x)=_{B}f(y)$ , such that $\mathtt{ap}_{f}\left(\mathtt{refl}_{x}\right):\equiv\mathtt{refl}_{f(x)}$ .

However, for a dependent function $f:(x:A)\rightarrow P(x)$ , there is not necessarily a path from $f(x)$ to $f(y)$ , which generally belong to different types $P(x)$ and $P(y)$ , even when there is $p:x=_{A}y$ . We have to ‘transport’ $f(x)$ from $P(x)$ to $P(y)$ ‘along’ $p$ first, shown by the following two lemmas, visualised in Figure 3.1.

Lemma 3.9 ([Uni13, Lemma 2.3.1]).

Let $A:\mathtt{Type}$ , $P:A\rightarrow\mathtt{Type}$ , $x,y:A$ and $p:x=_{A}y$ . Then there is a function $p_{\ast}\equiv\mathtt{transport}^{P}(p):P(x)\rightarrow P(y)$ , such that $\left(\mathtt{refl}_{x}\right)_{\ast}:\equiv\operatorname{id}\limits_{P(x)}$ .

Lemma 3.10 ([Uni13, Lemma 2.3.4]).

Let $A:\mathtt{Type}$ , $P:A\rightarrow\mathtt{Type}$ , $f:(a:A)\rightarrow P(a)$ , $x,y:A$ and $p:x=_{A}y$ . Then there is a dependent path $\mathtt{apd}_{f}(p):f(x)=_{p}^{P}f(y)$ such that $\mathtt{apd}_{f}\left(\mathtt{refl}_{x}\right):\equiv\mathtt{refl}_{f(x)}$ , where we define the type

\left(u=_{p}^{P}v\right):\equiv\left(\mathtt{transport}^{P}(p)(u)=_{P(y)}v\right)

for any $u:P(x)$ and $v:P(y)$ .

Figure 3.1: Diagram for Lemma 3.9 and Lemma 3.10.

In this and following diagrams, a type is represented as a grey solid circle or ellipse, a term as a black solid dot, a path as a arrow, and a non-dependent function as a arrow.

The next lemma relates the dependent and non-dependent cases, visualised in Figure 3.2.

Lemma 3.11 ([Uni13, Lemma 2.3.5, Lemma 2.3.8]).

Let $A,B:\mathtt{Type}$ , $x,y:A$ , and $p:x=_{A}y$ . Let $P:A\rightarrow\mathtt{Type}$ be a constant family of types given by $\lambda a.B$ . Then for any $b:B$ , there is a path

\mathtt{tr\text{-}const}_{p}^{B}(b):\mathtt{transport}^{P}(p)(b)=_{B}b.

Let $f:A\rightarrow B$ , then $\mathtt{apd}_{f}(p)=\mathtt{tr\text{-}const}_{p}^{B}\left(f(x)\right)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{f}(p)$ .

Figure 3.2: Diagram for Lemma 3.11.

The next result shows the functoriality of $\mathtt{transport}$ .

Proposition 3.12 ([Uni13, Lemma 2.3.9]).

Let $P:A\rightarrow\mathtt{Type}$ , $p:x=_{A}y$ and $q:y=_{A}z$ . Then $\mathtt{transport}^{P}(p\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}q)=\mathtt{transport}^{P}(q)\circ\mathtt{transport}^{P}(p).$

It is helpful to characterise $\mathtt{transport}$ when the dependent type $P:A\rightarrow\mathtt{Type}$ is of particular forms, as in the following lemmas, visualised in Figure 3.3, Figure 3.4 and Figure 3.5.

Lemma 3.13.

Let $A,B:\mathtt{Type}$ , $f,g:A\rightarrow B$ , $x,y:A$ , $p:x=_{A}y$ and $u:f(x)=_{B}g(x)$ . Let $P:\equiv\lambda a.\left(f(a)=_{B}g(a)\right):A\rightarrow\mathtt{Type}$ . Then

\mathtt{transport}^{P}(p)(u)=_{f(y)=_{B}g(y)}\left(\mathtt{ap}_{f}(p)\right)^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}u\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{g}(p).

Figure 3.3: Diagram for Lemma 3.13.

The two dashed ellipses represent identity types

f(x)=g(x)

and

f(y)=g(y)

Lemma 3.14 ([Uni13, (2.9.4)]).

Let $X:\mathtt{Type}$ and $A,B:X\rightarrow\mathtt{Type}$ . Let $P:\equiv\lambda x.A(x)\rightarrow B(x):X\rightarrow\mathtt{Type}$ . Let $x_{1},x_{2}:X$ , $p:x_{1}=_{X}x_{2}$ . Let $f:P\left(x_{1}\right)\equiv\left(A\left(x_{1}\right)\rightarrow B\left(x_{1}\right)\right)$ . Then

\mathtt{transport}^{P}(p)(f)=_{P\left(x_{2}\right)}\mathtt{transport}^{B}(p)\circ f\circ\mathtt{transport}^{A}\left(p^{-1}\right).

Figure 3.4: Diagram for Lemma 3.14.

The two dashed ellipses represent the function types

A\left(x_{1}\right)\rightarrow B\left(x_{1}\right)

and

A\left(x_{2}\right)\rightarrow B\left(x_{2}\right)

Lemma 3.15 ([Uni13, Lemma 2.11.2]).

Let $A:\mathtt{Type}$ , $a,x_{1},x_{2}:A$ and $p:x_{1}=_{A}x_{2}$ . Then

$\displaystyle\mathtt{transport}^{x\mapsto(a=x)}(p)(q_{1})$	$\displaystyle=q_{1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,p\quad$	$\displaystyle\text{for }q_{1}:a=x_{1},$
$\displaystyle\mathtt{transport}^{x\mapsto(x=a)}(p)(q_{2})$	$\displaystyle=p^{-1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,q_{2}\quad$	$\displaystyle\text{for }q_{2}:x_{1}=a,$
$\displaystyle\mathtt{transport}^{x\mapsto(x=x)}(p)(q_{3})$	$\displaystyle=p^{-1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,q_{3}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,p\quad$	$\displaystyle\text{for }q_{3}:x_{1}=x_{1}.$

Figure 3.5: Diagram for the first case of Lemma 3.15.

The two dashed ellipses represent identity types

a=x_{1}

and

a=x_{2}

3.3 Homotopies and equivalences

The next problem is how to construct equalities between functions and between types. In this section, we shall present two constructions, function homotopy and type equivalence, based on the identity types.

Definition 3.16.

Let $A:\mathtt{Type}$ , $B:A\rightarrow\mathtt{Type}$ and $f,g:(a:A)\rightarrow B(a)$ . A (function) homotopy from $f$ to $g$ is a dependent function of type

(f\sim g):\equiv(a:A)\rightarrow\left(f(a)=_{B(a)}g(a)\right),

i.e., a family of pointwise equalities.

Definition 3.17.

Let $A,B:\mathtt{Type}$ , $f:A\rightarrow B$ . A quasi-inverse of $f$ is a triple $(g,\alpha,\beta)$ , where $g:B\rightarrow A$ , $\alpha:f\circ g\sim\operatorname{id}\limits_{B}$ and $\beta:g\circ f\sim\operatorname{id}\limits_{A}$ .

Proposition 3.18 ([Uni13, Section 2.4]).

Let $A,B:\mathtt{Type}$ and $f:A\rightarrow B$ . Define type

\mathtt{isequiv}(f):\equiv\left(\sum_{g:B\rightarrow A}\left(g\circ f\sim\operatorname{id}\limits_{A}\right)\right)\times\left(\sum_{h:B\rightarrow A}\left(f\circ h\sim\operatorname{id}\limits_{B}\right)\right).

We say ‘ $f$ is an equivalence’ if $\mathtt{isequiv}(f)$ is inhabited. Then

(1)

$f$ is an equivalence if and only if $f$ has a quasi-inverse;
(2)

Given any $p,q:\mathtt{isequiv}(f)$ , there is a path $p=_{\mathtt{isequiv}(f)}q$ .

In the language of Section 4.2, (2) states that $\mathtt{isequiv}(f)$ is a mere proposition. In practice, to prove that a function $f$ is an equivalence, by (1), we only have to find a quasi-equivalence of $f$ . An example follows.

Lemma 3.19 ([Uni13, Example 2.4.9]).

Let $A:\mathtt{Type}$ , $P:A\rightarrow\mathtt{Type}$ , $x,y:A$ and $p:x=_{A}y$ . Then $\mathtt{transport}^{P}(p):P(x)\rightarrow P(y)$ is an equivalence.

Proof.

By Proposition 3.12, $\mathtt{transport}^{P}\left(p^{-1}\right)$ forms a quasi-inverse with $\mathtt{transport}^{P}(p)$ .∎

Definition 3.20.

Let $A,B:\mathtt{Type}$ . Then we define the type of equivalences between $A$ and $B$ as

(A\simeq B):\equiv\sum_{f:A\rightarrow B}\mathtt{isequiv}(f).

We now characterise the identity type in certain types using the language of equivalence:

Proposition 3.21 ([Bru16, Proposition 1.6.8]).

Let $A:\mathtt{Type}$ , $B:A\rightarrow\mathtt{Type}$ and two terms $(a,b),(a^{\prime},b^{\prime}):\sum_{x:A}B(x)$ , then there is an equivalence of types

\left((a,b)=_{\sum_{x:A}B(x)}(a^{\prime},b^{\prime})\right)\simeq\left(\sum_{p:a=_{A}a^{\prime}}b=_{p}^{B}b^{\prime}\right).

The function from right to left is written as $\mathtt{pair}^{=}$ , i.e., given $p:a=_{A}a^{\prime}$ and $\ell:b=_{p}^{B}b^{\prime}$ , there is a path $\mathtt{pair}^{=}(p,\ell):(a,b)=(a^{\prime},b^{\prime})$ .

3.4 Univalence axiom and function extensionality

Given $A,B:\mathtt{Type}$ , it is natural to wonder what is the relationship between type equality $A=_{\mathtt{Type}}B$ and type equivalence $A\simeq B$ . It turns out that we can derive the latter from the former:

Proposition 3.22 ([Uni13, Lemma 2.10.1]).

Let $A,B:\mathtt{Type}$ . Then there is a function $\mathtt{idtoeqv}:\left(A=_{\mathtt{Type}}B\right)\rightarrow(A\simeq B)$ .

Proof.

Let $p:A=_{\mathtt{Type}}B$ . The identity function $\operatorname{id}\limits_{\mathtt{Type}}\equiv\lambda X.X:\mathtt{Type}\rightarrow\mathtt{Type}$ can be regarded as a dependent type, so by Lemma 3.9, we have a function $\mathtt{idtoeqv}(p):\equiv\mathtt{transport}^{\operatorname{id}\limits_{\mathtt{Type}}}(p):A\rightarrow B$ , which is an equivalence by Lemma 3.19. ∎

Desirable as it is, we however cannot go the other way around with what we currently have. Our wish is instead formulated as an axiom:

Axiom 3.23 (Univalence).

Let $A,B:\mathtt{Type}$ . Then $\mathtt{idtoeqv}:\left(A=_{\mathtt{Type}}B\right)\rightarrow(A\simeq B)$ is an equivalence.

Remark 3.24.

The axiom means that we assume the type $\mathtt{isequiv}(\mathtt{idtoeqv})$ is inhabited without constructing its term.

In particular, this implies that there is a function

\mathtt{ua}:(A\simeq B)\rightarrow\left(A=_{\mathtt{Type}}B\right),

which forms a quasi-inverse of $\mathtt{idtoeqv}$ . By abuse of notation, we write $\mathtt{ua}(f):\left(A=_{\mathtt{Type}}B\right)$ if $f:A\rightarrow B$ is an equivalence. In practice, to find a path between two types $A$ and $B$ , it suffices to give a function $f:A\rightarrow B$ and show that $f$ is an equivalence.

A similar situation arises with function equality and function homotopy.

Proposition 3.25.

Let $A:\mathtt{Type}$ , $B:A\rightarrow\mathtt{Type}$ and $f,g:(a:A)\rightarrow B(a)$ . Then there is a function $\mathtt{happly}:\left(f=_{(a:A)\rightarrow B(a)}g\right)\rightarrow(f\sim g)$ .

Proof.

Let $p:\left(f=_{(a:A)\rightarrow B(a)}g\right)$ . For any $a:A$ , the evaluation function $\lambda h.h(a):\left((a:A)\rightarrow B(a)\right)\rightarrow B(a)$ is a non-dependent one, so we can apply Lemma 3.8 and get $\mathtt{happly}(p)(a):\equiv\mathtt{ap}_{\lambda h.h(a)}(p):\left(f(a)=_{B(a)}g(a)\right).$ ∎

The next theorem can be proved with univalence [Uni13, Section 4.9].

Theorem 3.26 (Function extensionality).

The function $\mathtt{happly}$ is an equivalence.

This implies that there is a function

\mathtt{funext}:\left((a:A)\rightarrow\left(f(a)=_{B(a)}g(a)\right)\right)\rightarrow\left(f=_{(a:A)\rightarrow B(a)}g\right),

which forms a quasi-inverse of $\mathtt{happly}$ . In practice, to give an equality between two functions $f$ and $g$ , it suffices to give a family of pointwise equalities.

3.5 Higher inductive types

In classical dependent type theory, any constructor for an inductive type $A$ should produce a term of $A$ . This is not the case for higher inductive types (HIT), where path constructors are allowed which produce paths, or ‘customised equalities’, between points of $A$ . We now introduce two HITs, the circle and pushouts.

3.5.1 Circle

The circle ${\mathbb{S}}^{1}$ is the HIT generated by a (point) constructor $\mathtt{base}:{\mathbb{S}}^{1}$ and a path constructor $\mathtt{loop}:\mathtt{base}=_{{\mathbb{S}}^{1}}\mathtt{base}$ . The wording here is ‘generated’, in the sense that by coherence operations, from the constructor $\mathtt{loop}$ we automatically obtain paths such as $\mathtt{loop}^{-1}$ and $\mathtt{loop}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}$ . Note that ${\mathbb{S}}^{1}$ is non-trivial, in the sense that $\mathtt{loop}\neq\mathtt{refl}_{\mathtt{base}}$ ¹¹1Note that this means the type $(\mathtt{loop}=\mathtt{refl}_{\mathtt{base}})\to\mathbb{0}$ is inhabited.[Uni13, Lemma 6.4.1].

The induction principle states that given

•

a dependent type $P:{\mathbb{S}}^{1}\rightarrow\mathtt{Type}$ ,
•

a term $b:P\left(\mathtt{base}\right)$ and
•

a dependent path $\ell:b=_{\mathtt{loop}}^{P}b$ ,

there is a dependent function $f:\left(x:{\mathbb{S}}^{1}\right)\rightarrow P(x)$ such that $f\left(\mathtt{base}\right):\equiv b$ and $\mathtt{apd}_{f}\left(\mathtt{loop}\right):=\ell$ ²²2Here the computation rule for the path is a propositional equality instead of a judgemental one; see [Uni13, Section 6.2]..

Remark 3.27 ([Uni13, Remark 6.2.4]).

When informally applying circle induction on $x:{\mathbb{S}}^{1}$ given $P:{\mathbb{S}}^{1}\rightarrow\mathtt{Type}$ , we may use expressions ‘when $x$ is $\mathtt{base}$ ’ and ‘when $x$ varies along $\mathtt{loop}$ ’ to introduce the constructions of $b$ and $\ell$ as above.

The recursion principle states that given

•

$B:\mathtt{Type}$ ,
•

$b:B$ and
•

$p:b=_{B}b$ ,

there is a function $f:{\mathbb{S}}^{1}\rightarrow B$ such that $f\left(\mathtt{base}\right):\equiv b$ and $\mathtt{ap}_{f}\left(\mathtt{loop}\right):=p$ [Uni13, Lemma 6.2.5].

3.5.2 Pushouts

Let $A,B,C:\mathtt{Type}$ , $f:C\rightarrow A$ and $g:C\rightarrow B$ . Then we get a diagram $A\overset{f}{\leftarrow}C\overset{g}{\rightarrow}B$ , which we call a span. The pushout $A\sqcup^{C}B$ of this span is the HIT generated by two point constructors $\mathtt{inl}:A\rightarrow A\sqcup^{C}B$ and $\mathtt{inr}:B\rightarrow A\sqcup^{C}B$ and a path constructor

\mathtt{glue}:(c:C)\rightarrow\left(\mathtt{inl}(f(c))=_{A\sqcup^{C}B}\mathtt{inr}(g(c))\right),

as visualised by Figure 3.6.

The induction principle states that given

•

a dependent type $P:A\sqcup^{C}B\rightarrow\mathtt{Type}$ ,
•

two dependent functions $h_{\mathtt{inl}}:(a:A)\rightarrow P\left(\mathtt{inl}(a)\right)$ and
•

$h_{\mathtt{inr}}:(b:B)\rightarrow P\left(\mathtt{inr}(b)\right)$ , and

•

a family of dependent paths

h_{\mathtt{glue}}:(c:C)\rightarrow\left(h_{\mathtt{inl}}\left(f(c)\right)=_{\mathtt{glue}(c)}^{P}h_{\mathtt{inr}}\left(g(c)\right)\right),

there is a dependent function $h:\left(x:A\sqcup^{C}B\right)\rightarrow P(x)$ such that $h\left(\mathtt{inl}(a)\right):\equiv h_{\mathtt{inl}}(a)$ , $h\left(\mathtt{inr}(b)\right):\equiv h_{\mathtt{inr}}(b)$ and $\mathtt{apd}_{h}\left(\mathtt{glue}(c)\right):=h_{\mathtt{glue}}(c)$ .

Figure 3.6: Diagram for the pushout

A\sqcup^{C}B

The two dashed circles represent the ‘image’ of

A

and

B

under

\mathtt{inl}

and

\mathtt{inr}

, respectively.

By specialising the span, we can obtain many interesting constructions from the pushout.

(1)

Let $A:\mathtt{Type}$ . The suspension $\Sigma A$ of $A$ is the pushout of

$\mathbb{1}_{N}\leftarrow A\rightarrow\mathbb{1}_{S}.$

We denote $\mathtt{north}:\equiv\mathtt{inl}(\star_{N})$ , $\mathtt{south}:\equiv\mathtt{inr}(\star_{S})$ , and for any $a:A$ , $\mathtt{merid}(a):\equiv\mathtt{glue}(a):\mathtt{north}=_{\Sigma A}\mathtt{south}$ . $\Sigma A$ can be seen as two points (’north and south poles’) connected by a family of paths (’meridians’) indexed by the terms of $A$ .
(2)

Let $A,B:\mathtt{Type}$ . The join $A\ast B$ of $A$ and $B$ is the pushout of

$A\overset{\mathtt{fst}}{\leftarrow}A\times B\overset{\mathtt{snd}}{\rightarrow}B.$
(3)

Let $\left(A,\star_{A}\right)$ and $\left(B,\star_{B}\right)$ be two pointed types. The wedge sum $A\vee B$ of $A$ and $B$ is the pushout of

$A\overset{\star_{A}}{\leftarrow}\mathbb{1}\overset{\star_{B}}{\rightarrow}B.$

With suspension, we define the $n$ -sphere ${\mathbb{S}}^{n}$ by induction on $n\geq 0$ : ${\mathbb{S}}^{0}:\equiv\mathbb{2}$ and ${\mathbb{S}}^{n+1}:\equiv\Sigma{\mathbb{S}}^{n}$ . Note that ${\mathbb{S}}^{1}\simeq\Sigma\mathbb{2}$ [Uni13, Lemma 6.5.1], where ${\mathbb{S}}^{1}$ is in the sense of Section 3.5.1.

For any type $X$ , we define its pointed suspension as $\left(\Sigma X,\mathtt{north}\right).$ This turns ${\mathbb{S}}^{n}$ into pointed types for $n\geq 1$ . For $n=0$ , we view ${\mathbb{S}}^{0}\equiv\mathbb{2}$ as $\mathbb{1}_{+}$ . We write the basepoint of ${\mathbb{S}}^{n}$ as $\mathtt{base}$ .

Chapter 4 Homotopy $n$ -types and $n$ -connected types

Section 4.1 discusses contractible types, ‘trivial types’ that are equivalent to $\mathbb{1}$ . Section 4.2 generalises contractible types to (homotopy) $n$ -types: types with trivial ‘homotopy information’ in any dimension greater than $n$ (as in Theorem 4.8). Section 4.3 discusses how to truncate any given type into an $n$ -type. Then our tools at hand will allow us to define and calculate the homotopy groups of the circle in Section 4.4. Finally, Section 4.5 focuses on the ‘dual’ of $n$ -types, $n$ -connected types, which have trivial ‘homotopy information’ in any dimension less than or equal to $n$ .

4.1 Contractible types

Definition 4.1.

For $A:\mathtt{Type}$ , we define the predicate

\mathtt{is\text{-}contr}(A):\equiv\sum_{a:A}\left((x:A)\rightarrow\left(a=_{A}x\right)\right)

which reads ‘ $A$ is contractible’, and we say $a$ is the centre of contraction.

Clearly, $\mathbb{1}$ is a contractible type. In fact, $A$ is contractible if and only if $A\simeq\mathbb{1}$ [Uni13, Lemma 3.11.3]. Informally, contractibility means having trivial ‘homotopy information’ in all dimensions (observe that $\Omega^{n}\left(\mathbb{1}\right)\simeq\mathbb{1}$ for all $n:{\mathbb{N}}$ ). We first give a non-example.

Example 4.2.

$\blacksquare$ ${\mathbb{S}}^{1}$ is not contractible. Otherwise, taking $\mathtt{base}$ as the centre of contraction, we have to prove $\left(x:{\mathbb{S}}^{1}\right)\rightarrow\left(\mathtt{base}=x\right)$ . By circle induction, when $x$ is $\mathtt{base}$ , we need $p:\mathtt{base}=\mathtt{base}$ . When $x$ ‘varies along’ $\mathtt{loop}$ , we need $\mathtt{loop}_{\ast}(p)=p$ . By Lemma 3.15, $\mathtt{loop}_{\ast}(p)=p\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}$ , so by coherence operations, we need $\mathtt{loop}=\mathtt{refl}_{\mathtt{base}}$ , but this would give us absurdity by the non-triviality of ${\mathbb{S}}^{1}$ [Uni13, Lemma 6.4.1].

As mentioned earlier, the space of all paths from a fixed point is contractible:

Proposition 4.3.

Let $X:\mathtt{Type}$ and $a:X$ . Then the type $\sum_{x:X}\left(a=_{X}x\right)$ is contractible.

Proof.

The centre of contraction is given by $\left(a,\mathtt{refl}_{a}\right)$ . For any $(x,p):\sum_{x:X}\left(a=_{X}x\right)$ , we have $p:a=x$ and $\ell:p_{\ast}\left(\mathtt{refl}_{a}\right)=p$ by Lemma 3.15. Then we have $\mathtt{pair}^{=}(p,\ell):\left(a,\mathtt{refl}_{a}\right)=(x,p)$ by Proposition 3.21. ∎

$\blacksquare$ This lends us some further insights into path induction, as in the two following corollaries.

Corollary 4.4 (Path induction, strengthened).

Let $A:\mathtt{Type}$ , $a:A$ , and $P:(x:A)\rightarrow\left(a=_{A}x\right)\rightarrow\mathtt{Type}$ . Then for any $y:A$ , $q:a=_{A}y$ and $d:P(y)(q)$ , there is a function $f:(x:A)\rightarrow\left(p:\left(a=_{A}x\right)\right)\rightarrow P(x)(p)$ .

Proof.

Let $\hat{P}:\sum_{x:A}\left(a=_{A}x\right)\rightarrow\mathtt{Type}$ be the uncurried form of $P$ . Then by the induction principle of $\mathbb{1}$ , it suffices to take any $(y,q):\sum_{x:A}\left(a=_{A}x\right)$ and $d:\hat{P}(y,q)\equiv P(y)(q)$ to obtain $\hat{f}:\left((x,p):\sum_{x:A}\left(a=_{A}x\right)\right)\rightarrow\hat{P}(x,p)$ . The desired $f$ is the curried form of $\hat{f}$ . ∎

Remark 4.5.

In the usual path induction, $y$ and $q$ should be $a$ and $\mathtt{refl}_{a}$ respectively.

We may also use $\mathtt{transport}$ to express functions defined using path induction:

Corollary 4.6.

Let $A:\mathtt{Type}$ , $a,x:A$ , $p:\left(a=_{A}x\right)$ , $P:(x:A)\rightarrow\left(a=_{A}x\right)\rightarrow\mathtt{Type}$ and $d:P(a)\left(\mathtt{refl}_{a}\right)$ . Then

J_{P}(d)(x)(p)=_{P(x)(p)}\mathtt{transport}^{\hat{P}}\left(\mathtt{pair}^{=}(p,\ell)\right)(d),

where $\ell:p_{\ast}\left(\mathtt{refl}_{a}\right)=p$ by Lemma 3.15, $J$ is defined in Section 2.3 and $\hat{P}:\sum_{x:A}\left(a=_{A}x\right)\rightarrow\mathtt{Type}$ is the uncurried form of $P$ .

Proof.

By path induction, assume $x$ is $a$ and $p$ is $\mathtt{refl}_{a}$ , then both sides are $d$ .∎

4.2 Homotopy $n$ -types

Definition 4.7.

We recursively define the predicate

\mathtt{is\text{-}}n\mathtt{\text{-}Type}(A):\equiv\begin{cases}\mathtt{is\text{-}contr}(A)\quad&\text{if }n=-2,\\ (x:A)\rightarrow(y:A)\rightarrow\mathtt{is\text{-}}(n-1)\mathtt{\text{-}Type}(x=_{A}y)\quad&\text{if }n\geq-1,\end{cases}

for any $n\geq-2$ , which reads ‘ $A$ is an $n$ -type’. We also define

n\mathtt{\text{-}Type}:\equiv\sum_{X:\mathtt{Type}}\mathtt{is\text{-}}n\mathtt{\text{-}Type}(X)

and in particular, we write $\mathtt{Prop}:\equiv(-1)\mathtt{\text{-}Type}$ .

The type $\mathtt{is\text{-}}n\mathtt{\text{-}Type}(A)$ is a $(-1)$ -type [Uni13, Theorem 7.1.10], and the type $n\mathtt{\text{-}Type}$ is an $(n+1)$ -type [Uni13, Theorem 7.1.11].

A $(-2)$ -type is by definition a contractible type; we also define a mere proposition as a $(-1)$ -type and a set as a $0$ -type. They are characterised as follows:

•

$A:\mathtt{Type}$ is contractible, if and only if $A$ is a pointed mere proposition, if and only if $A\simeq\mathbb{1}$ [Uni13, Lemma 3.11.3].
•

$A:\mathtt{Type}$ is a mere proposition if and only if for any $x,y:A$ , we have $x=_{A}y$ [Uni13, Definition 3.3.1 and Lemma 3.11.10]. This aligns with the ‘proof-irrelevant’ propositions in classical logic. For example, $\mathbb{0}$ is a mere proposition [Bru16, Proposition 1.10.5].
•

$A:\mathtt{Type}$ is a set, if and only if for any $x,y:A$ and $p,q:x=_{A}y$ we have $p=_{x=_{A}y}q$ , if and only if $A$ satisfies Axiom K, i.e., $a=_{A}a$ is contractible for any $a:A$ [Uni13, Theorem 7.2.1]. This aligns with the ‘equality-irrelevant’ sets in classical mathematics and allows us to talk about set-theoretic notions such as subsets. The types $\mathbb{2}$ , $\mathbb{N}$ and $\mathbb{Z}$ are sets [Bru16, Proposition 1.10.7].

As a generalisation of Axiom K, we can characterise $n$ -types in terms of loop spaces:

Theorem 4.8 ([Uni13, Theorem 7.2.9]).

For any $n\geq-1$ , a type $A$ is an $n$ -type if and only if $\Omega^{n+1}(A,a)$ is contractible for all $a:A$ .

Noticeably, $n$ -types have the cumulativity property:

Proposition 4.9 ([Uni13, Theorem 7.1.7]).

Let $m\geq n\geq-2$ . An $n$ -type is also an $m$ -type.

4.3 $n$ -truncations

Any type $A$ can be turned into an $n$ -type $\left\|A\right\|_{n}$ , called the $n$ -truncation of $A$ , by a map $|-|_{n}:A\rightarrow\left\|A\right\|_{n}$ . The construction of $\left\|A\right\|_{n}$ as an HIT is elaborated in Appendix A.

The induction principle (called the truncation induction) for $\left\|A\right\|_{n}$ states that given

•

a dependent type $P:\left\|A\right\|_{n}\rightarrow n\mathtt{\text{-}Type}$ and
•

a dependent function $g:(a:A)\rightarrow P\left(|a|_{n}\right)$ ,

there is a dependent function $f:\left(x:\left\|A\right\|_{n}\right)\rightarrow P(x)$ , such that $f\left(|a|_{n}\right):\equiv g(a)$ for all $a:A$ [Uni13, Theorem 7.3.2]. The recursion principle states that given

•

an $n$ -type $E$ and
•

a function $g:A\rightarrow E$ ,

there is a function $f:\left\|A\right\|_{n}\rightarrow E$ such that $f\left(|a|_{n}\right):\equiv g(a)$ .

Particularly, the $(-1)$ -truncation turns any type into a mere proposition. This allows us to get rid of the ‘superfluous’ information carried by the $\Sigma$ -type and disjoint union, as footnoted in Table 2.1, and perform logic in the classical sense. For $A:\mathtt{Type}$ and $P:A\rightarrow\mathtt{Type}$ , we shall use the expression ‘there merely exists $a:A$ such that $P(a)$ ’ to represent the type $\left\|{\sum_{a:A}P(a)}\right\|_{-1}$ (see [Uni13, Sections 3.6, 3.7, 3.10]).

The next result shows that truncation operation is a functor:

Proposition 4.10.

Given $n\geq-2$ and $f:A\rightarrow B$ , there is a map $\left\|f\right\|_{n}:\left\|A\right\|_{n}\rightarrow\left\|B\right\|_{n}$ such that $\left\|f\right\|_{n}\left(|a|_{n}\right):\equiv|f(a)|_{n}$ .

Proof.

Since $\left\|B\right\|_{n}$ is an $n$ -type, define $\left\|f\right\|_{n}:\left\|A\right\|_{n}\rightarrow\left\|B\right\|_{n}$ by $\left\|f\right\|_{n}\left(|a|_{n}\right):\equiv|f(a)|_{n}$ by the recursion principle. ∎

Unsurprisingly, $n$ -truncating an $n$ -type gives an equivalence:

Proposition 4.11 ([Uni13, Corollary 7.3.7]).

Let $A:\mathtt{Type}$ . $A$ is an $n$ -type if and only if the map $|-|_{n}:A\rightarrow\left\|A\right\|_{n}$ is an equivalence.

Proof.

Suppose $A$ is an $n$ -type. Then we are allowed to define $f:\left\|A\right\|_{n}\rightarrow A$ by $f\left(|a|_{n}\right):\equiv a$ . By definition, $f\circ|-|_{n}\equiv\operatorname{id}\limits_{A}$ . Then we would like to define a dependent function $\left(x:\left\|A\right\|_{n}\right)\rightarrow\left(\left|{f(x)}\right|_{n}=_{\left\|A\right\|_{n}}x\right)$ . Notice that $\left(\left|{f(x)}\right|_{n}=_{\left\|A\right\|_{n}}x\right)$ is an $n$ -type (since it is by definition an $(n-1)$ -type), so by its induction principle it suffices to give a dependent function $(a:A)\rightarrow\left(\left|{f\left(|a|_{n}\right)}\right|_{n}=_{\left\|A\right\|_{n}}|a|_{n}\right)$ , which can be given by $\lambda a.\mathtt{refl}_{|a|_{n}}$ since $f\left(|a|_{n}\right):\equiv a$ . Therefore, $f$ forms a quasi-inverse for $|-|_{n}$ and $|-|_{n}$ is an equivalence.

Conversely, if $|-|_{n}$ is an equivalence, then since $\left\|A\right\|_{n}$ is an $n$ -type, so is $A$ . ∎

The following two results deal with $n$ -truncations of identity types and loop spaces:

Proposition 4.12 ([Uni13, Theorem 7.3.12]).

Let $n\geq-2$ , $A:\mathtt{Type}$ and $x,y:A$ . Then

\left\|{x=_{A}y}\right\|_{n}\simeq\left(|x|_{n+1}=_{\left\|A\right\|_{n+1}}|y|_{n+1}\right).

Proposition 4.13 ([Uni13, Corollary 7.3.14]).

Let $n\geq-2$ , $k\geq 0$ , and $A$ be a pointed type. Then $\left\|{\Omega^{k}(A)}\right\|_{n}\simeq\Omega^{k}\left(\left\|A\right\|_{n+k}\right).$

Finally, $n$ -truncations have the cumulativity property:

Lemma 4.14 ([Uni13, Lemma 7.3.15]).

For $A:\mathtt{Type}$ and $-2\leq m\leq n$ , $\left\|\left\|A\right\|_{n}\right\|_{m}\simeq\left\|A\right\|_{m}.$

4.4 Interlude: homotopy groups of the circle

In this interlude, we first take groups as an example to demonstrate how to construct algebraic structures. Then we define homotopy groups and calculate them for ${\mathbb{S}}^{1}$ .

Definition 4.15.

A group is a set $G$ with functions $m:G\rightarrow G\rightarrow G$ (called the multiplication function), $i:G\rightarrow G$ (called the inversion function) and a term $e:G$ (called the neutral element), such that the equalities hold for all $x,y,z:G$ :

•

$m\left(m(x)(y)\right)(z)=_{G}m(x)\left(m(y)(z)\right)$ ;
•

$m(x)\left(i(x)\right)=_{G}e$ ;
•

$m\left(i(x)\right)(x)=_{G}e$ ;
•

$m(x)(e)=_{G}x$ ;
•

$m(e)(x)=_{G}x$ .

The group is abelian if further $m(x)(y)=_{G}m(y)(x)$ hold for all $x,y:G$ .

Remark 4.16.

Since a group is by definition a set, any of these equalities has at most one inhabitant. Hence, the predicate ‘ $(G,m,i,e)$ is a group’ is a mere proposition. This allows the groups we define to behave identically as in classical mathematics.

Definition 4.17.

Let $(G,m,i,e)$ and $(G^{\prime},m^{\prime},i^{\prime},e^{\prime})$ be two groups. Then a function $f:G\rightarrow G^{\prime}$ is called a group homomorphism if $m^{\prime}\left(f(x)\right)\left(f(y)\right)=_{G^{\prime}}f\left(m(x)\right)\left(m(y)\right)$ for all $x,y:G$ . A group homomorphism $f$ is a group isomorphism if it is an equivalence.

Example 4.18.

$\mathbb{1}$ is trivially a group, written as $0$ .

Importantly, set-truncation enables us to turn a loop space into a group:

Definition 4.19.

Let $A$ be a pointed type and $n\geq 1$ . Define the $n$ -th homotopy group of $A$ as $\pi_{n}(A):\equiv\left\|{\Omega^{n}(A)}\right\|_{0}$ .

When $n\geq 1$ , the set $\pi_{n}(A)$ is a group with path concatenation, path inversion and reflexivity. When $n\geq 2$ , by the Eckmann-Hilton argument [Uni13, Theorem 2.1.6], $\pi_{n}(A)$ is an abelian group. We may also write $\pi_{0}(A)\equiv\left\|A\right\|_{0}$ , although it is not a group in general. Finally, $\pi_{n}$ is a functor, as truncation and loop space are both functors by Lemma 5.6 and Proposition 4.10.

We know that ${\mathbb{S}}^{1}$ is not contractible by Example 4.2. For the rest of this section, we briefly sketch the calculation of $\pi_{k}\left({\mathbb{S}}^{1}\right)$ for all $k\geq 1$ . We proceed by first showing $\Omega^{1}\left({\mathbb{S}}^{1}\right)\simeq{\mathbb{Z}}$ with the encode-decode method based on [Uni13, Section 8.1.4]. This technique will be used again in Chapter 6.

The initial idea would be to give two functions $\Omega^{1}\left({\mathbb{S}}^{1}\right)\leftrightarrows{\mathbb{Z}}$ and show that they are quasi-inverses. The function ${\mathbb{Z}}\rightarrow\Omega^{1}\left({\mathbb{S}}^{1}\right)$ is given by $\varphi:\equiv\lambda n.\mathtt{loop}^{n}$ , where $\mathtt{loop}^{n}$ denotes the $|n|$ -time concatenation of $\mathtt{loop}$ if $n>0$ or $\mathtt{loop}^{-1}$ if $n<0$ and $\mathtt{refl}_{\mathtt{base}}$ if $n=0$ . However, we cannot easily define a function $\left(\mathtt{base}=_{{\mathbb{S}}^{1}}\mathtt{base}\right)\rightarrow{\mathbb{Z}}$ , as path induction is unavailable when the two endpoints are both fixed. We thus generalise the claim $\Omega^{1}\left({\mathbb{S}}^{1}\right)\simeq{\mathbb{Z}}$ to the following.

Lemma 4.20.

There exists a dependent type $\mathtt{code}:{\mathbb{S}}^{1}\rightarrow\mathtt{Type}$ , such that

\left(x:{\mathbb{S}}^{1}\right)\rightarrow\left(\left(\mathtt{base}=_{{\mathbb{S}}^{1}}x\right)\simeq\mathtt{code}(x)\right)

and $\mathtt{code}(\mathtt{base}):\equiv{\mathbb{Z}}$ .

Proof sketch.

To finish the definition of $\mathtt{code}$ , as $\mathtt{succ}:{\mathbb{Z}}\rightarrow{\mathbb{Z}}$ is an equivalence, $\mathtt{ua}(\mathtt{succ})$ is a path ${\mathbb{Z}}={\mathbb{Z}}$ . We then set $\mathtt{ap}_{\mathtt{code}}\left(\mathtt{loop}\right):=\mathtt{ua}(\mathtt{succ})$ . $\mathtt{code}$ represents the universal cover of the circle: the fibre over $\mathtt{base}$ is $\mathbb{Z}$ and going around $\mathtt{loop}$ once corresponds to going up by one in $\mathbb{Z}$ .

From this definition, we can prove that

\mathtt{transport}^{\mathtt{code}}\left(\mathtt{loop}^{c}\right)(x)=(x+c)

(4.1)

for any $c:{\mathbb{Z}}$ (by [Uni13, Lemma 8.1.2] and Proposition 3.12).

It then remains to define four functions of the types:

(1)

$\mathtt{encode}:\left(x:{\mathbb{S}}^{1}\right)\rightarrow\left(\mathtt{base}=_{{\mathbb{S}}^{1}}x\right)\rightarrow\mathtt{code}(x)$ ,
(2)

$\mathtt{decode}:\left(x:{\mathbb{S}}^{1}\right)\rightarrow\mathtt{code}(x)\rightarrow\left(\mathtt{base}=_{{\mathbb{S}}^{1}}x\right)$ ,
(3)

$\left(x:{\mathbb{S}}^{1}\right)\rightarrow\left(p:\mathtt{base}=_{{\mathbb{S}}^{1}}x\right)\rightarrow\left(\mathtt{decode}(x)(\mathtt{encode}(x)(p))=p\right)$ , and
(4)

$\left(x:{\mathbb{S}}^{1}\right)\rightarrow\left(c:\mathtt{code}(x)\right)\rightarrow\left(\mathtt{encode}(x)(\mathtt{decode}(x)(c))=c\right)$ ,

where (3) and (4) state that for each $x:{\mathbb{S}}^{1}$ , $\mathtt{encode}(x)$ and $\mathtt{decode}(x)$ are quasi-inverses.

To proceed, (1) is given by

\mathtt{encode}(x)(p):\equiv\mathtt{transport}^{\mathtt{code}}(p)(0);

(2) uses circle induction, where $\mathtt{decode}(\mathtt{base})$ is given by the function $\varphi:{\mathbb{Z}}\rightarrow\left(\mathtt{base}=\mathtt{base}\right)$ defined earlier; (3) uses path induction and (4) circle induction, and they both apply Equality 4.1. We omit the details. ∎

Now setting $x$ to $\mathtt{base}$ in Lemma 4.20 yields $\Omega^{1}\left({\mathbb{S}}^{1}\right)\simeq{\mathbb{Z}}$ .

Corollary 4.21.

$\pi_{1}\left({\mathbb{S}}^{1}\right)\simeq{\mathbb{Z}}$ and $\pi_{k}\left({\mathbb{S}}^{1}\right)\simeq 0$ for $k\geq 2$ .

Proof.

Since $\mathbb{Z}$ is a set, $\left\|{\mathbb{Z}}\right\|_{0}={\mathbb{Z}}$ . Thus, $\pi_{1}\left({\mathbb{S}}^{1}\right)\equiv\left\|{\Omega^{1}\left({\mathbb{S}}^{1}\right)}\right\|_{0}\simeq\left\|{\mathbb{Z}}\right\|_{0}={\mathbb{Z}}$ . Further, since $\mathbb{Z}$ is a set, $\Omega^{1}\left({\mathbb{S}}^{1}\right)$ is also a set, which implies that $\Omega^{k}\left({\mathbb{S}}^{1}\right)$ is trivial for any $k\geq 2$ , and therefore so is $\pi_{k}\left({\mathbb{S}}^{1}\right)$ . ∎

4.5 $n$ -connected types

Definition 4.22.

Let $A,B:\mathtt{Type}$ and $f:A\rightarrow B$ . The fibre of $f$ over $y:B$ is defined as the type

\mathtt{fib}_{f}(y):\equiv\sum_{x:A}\left(f(x)=_{B}y\right).

Definition 4.23.

A type $A$ is $n$ -connected if $\left\|A\right\|_{n}$ is contractible, and a function $f:A\rightarrow B$ is $n$ -connected if for all $y:B$ , $\left\|{\mathtt{fib}_{f}(y)}\right\|_{n}$ is contractible. In particular, a function $f:A\rightarrow B$ is surjective, connected, or simply-connected if it is $(-1)$ -, $0$ -, or $1$ -connected, respectively.

Remark 4.24.

An $n$ -connected function $f$ can be thought of as ‘being surjective up to dimension $(n+1)$ ’. In particular, $f$ preserves ‘homotopy information’ up to dimension $n$ . This is formalised in Lemma 4.32 and Corollary 5.14. We also remark that different conventions in defining the $n$ -connectedness of functions exist.

Easily seen from the definitions are:

(1)

A type $A$ is $n$ -connected if and only if the unique function $A\rightarrow\mathbb{1}$ is $n$ -connected;
(2)

Every type is $(-2)$ -connected and so is every function;
(3)

Every pointed type is $(-1)$ -connected.

Example 4.25 ( $\blacksquare$ Inspired by [Hou17, Figure 3.4]).

Consider an HIT $G$ generated by a non-empty graph $\overline{G}$ : each vertex $\overline{v}\in\overline{G}$ gives a term $v:G$ and each edge $\overline{e}$ connecting vertices $\overline{v_{1}}$ and $\overline{v_{2}}$ gives a path $e:v_{1}=_{G}v_{2}$ , where the order of $v_{1}$ and $v_{2}$ is arbitrary thanks to path inversion.

If $\overline{G}$ has only one connected component, then $G$ is $0$ -connected. Indeed, as $\left\|G\right\|_{0}$ is pointed, it suffices to show that $x^{\prime}=y^{\prime}$ is contractible for any $x^{\prime},y^{\prime}:\left\|G\right\|_{0}$ . The goal is a mere proposition, so we assume $x^{\prime}$ is $|x|_{0}$ and $y^{\prime}$ is $|y|_{0}$ for $x,y:G$ . By Proposition 4.12, $\left(|x|_{0}=|y|_{0}\right)\simeq\left\|{x=y}\right\|_{-1}$ , but $\left\|{x=y}\right\|_{-1}$ is pointed by assumption and is thus contractible.

If further $\overline{G}$ is a tree, then $G$ is $1$ -connected. Indeed, using the same reasoning twice, it suffices to show that $\left\|{p=q}\right\|_{-1}$ is contractible for any $x,y:G$ and $p,q:x=y$ , which holds since any two vertices on a tree has a unique path between them up to homotopy.

Example 4.26.

${\mathbb{S}}^{0}\equiv\mathbb{2}$ is pointed and is $(-1)$ -connected. Example 4.25 indicates that ${\mathbb{S}}^{1}$ is $0$ -connected. In fact, for all $n:{\mathbb{N}}$ , the sphere ${\mathbb{S}}^{n}$ is $(n-1)$ -connected [Uni13, Corollary 8.2.2].

We observe that $n$ -connected types and functions have the (downward) cumulativity property:

Proposition 4.27 ([Bru16, Proposition 2.3.7]).

Let $-2\leq m\leq n$ . Any $n$ -connected type (resp. function) is also an $m$ -connected type (resp. function).

Proof.

Let $A:\mathtt{Type}$ be $n$ -connected. Then $\left\|A\right\|_{m}\simeq\left\|\left\|A\right\|_{n}\right\|_{m}$ by Lemma 4.14, and $\left\|\left\|A\right\|_{n}\right\|_{m}\simeq\left\|A\right\|_{n}$ by Proposition 4.11 since $\left\|A\right\|_{n}$ is contractible and a fortiori an $m$ -type. So $\left\|A\right\|_{m}$ is contractible. The case for $n$ -connected functions is similar. ∎

We can also say something about the lower homotopy groups of an $n$ -connected type:

Lemma 4.28 ([Uni13, Lemma 8.3.2]).

If $A$ is a pointed $n$ -connected type, then $\pi_{k}(A)\simeq 0$ for all $k\leq n$ .

Proof.

We calculate

\pi_{k}(A)\equiv\left\|{\Omega^{k}(A)}\right\|_{0}\simeq\Omega^{k}\left(\left\|A\right\|_{k}\right)\simeq\Omega^{k}\left(\left\|\left\|A\right\|_{n}\right\|_{k}\right)\simeq\Omega^{k}\left(\left\|\mathbb{1}\right\|_{k}\right)\simeq\Omega^{k}\left(\mathbb{1}\right)\simeq\mathbb{1},

by Proposition 4.13 and Lemma 4.14. ∎

Corollary 4.29.

$\pi_{k}\left({\mathbb{S}}^{n}\right)\simeq 0$ for $k<n$ .

Proof.

By Example 4.26 and Lemma 4.28. ∎

We have ‘induction principles’ for $n$ -connected types and functions:

Proposition 4.30 (Induction principle for $n$ -connected types).

Suppose $n\geq-2$ and $A$ is an $(n+1)$ -connected type. Given

•

a dependent type $P:A\rightarrow n\mathtt{\text{-}Type}$ ,
•

$a_{0}:A$ and
•

$p:P\left(a_{0}\right)$ ,

there is a dependent function $f:(x:A)\rightarrow P(x)$ such that $f\left(a_{0}\right):\equiv p$ .

Proof.

$\blacksquare$ Since $n\mathtt{\text{-}Type}$ is an $(n+1)$ -type, by the recursion principle of $\left\|A\right\|_{n+1}$ , there is a dependent type $P^{\prime}:\left\|A\right\|_{n+1}\rightarrow n\mathtt{\text{-}Type}$ such that $P^{\prime}\left(|a|_{n+1}\right):\equiv P(a)$ . Since $\left\|A\right\|_{n+1}$ is contractible, there is a function $f^{\prime}:\left(x:\left\|A\right\|_{n+1}\right)\rightarrow P^{\prime}(x)$ such that $f^{\prime}\left(\left|a_{0}\right|_{n+1}\right):\equiv p:P\left(a_{0}\right)\equiv P^{\prime}\left(\left|a_{0}\right|_{n+1}\right)$ . Therefore, we obtain $f:\equiv\left(f^{\prime}\circ|-|_{n+1}\right):(a:A)\rightarrow P^{\prime}\left(|a|_{n+1}\right)\equiv P(a)$ . ∎

Proposition 4.31 (‘Induction principle’ for $n$ -connected maps, [Uni13, Lemma 7.5.7]).

Suppose $n\geq-2$ , $A,B:\mathtt{Type}$ , and $f:A\rightarrow B$ . Then $f$ is $n$ -connected, if and only if for any $P:B\rightarrow n\mathtt{\text{-}Type}$ , the map

\lambda s.s\circ f:\left((b:B)\rightarrow P(b)\right)\rightarrow\left((a:A)\rightarrow P\left(f(a)\right)\right)

is an equivalence.

Intuitively, Proposition 4.30 says that an $(n+1)$ -connected type ‘looks like’ a contractible type to any family of $n$ -types, and Proposition 4.31 says that in the presence of an $n$ -connected map $f:A\rightarrow B$ , a family of $n$ -types over $B$ sees $B$ as being ‘generated’ by $A$ [HFLL16, Section 2].

Finally, we mention that an $n$ -connected function induces an equivalence on $n$ -truncations:

Lemma 4.32 ([Uni13, Lemma 7.5.14]).

If $f:A\rightarrow B$ is $n$ -connected, then it induces an equivalence $\left\|A\right\|_{n}\simeq\left\|B\right\|_{n}$ .

Chapter 5 Hopf fibration

The centre of this chapter is the following definition:

Definition 5.1.

Let $X$ , $Y$ be pointed types. Define the type of pointed maps from $X$ to $Y$ as

\mathtt{Map}_{\star}(X,Y):\equiv\sum_{f:X\rightarrow Y}\left(f\left(\star_{X}\right)=_{Y}\star_{Y}\right).

We may say that $f:X\rightarrow_{\star}Y$ is a pointed map witnessed by path $f_{\star}:f\left(\star_{X}\right)=_{Y}\star_{Y}$ .

For any pointed map $f:X\rightarrow_{\star}Y$ , the homotopy groups of $F\equiv\mathtt{fib}_{f}\left(\star_{Y}\right)$ , $X$ and $Y$ are related by a long exact sequence, obtained from a fibre sequence, shown in Section 5.1. In particular, given a ‘pointed fibration’ (in the sense of Corollary 5.3), $F$ , $X$ , $Y$ correspond respectively to the fibre, total space and base space of the fibration. In Section 5.2, we work to build the Hopf construction, a fibration based on H-spaces. In Section 5.3, we demonstrate an H-space structure on ${\mathbb{S}}^{1}$ and derive the Hopf fibration, whose long exact sequence fascinatingly connects the homotopy groups of ${\mathbb{S}}^{1}$ , ${\mathbb{S}}^{2}$ and ${\mathbb{S}}^{3}$ .

5.1 Fibre sequences

We begin with a simple yet powerful observation.

Lemma 5.2.

Let $X$ and $Y$ be pointed types and let $f:X\rightarrow_{\star}Y$ be witnessed by $f_{\star}$ . Let $F:\equiv\mathtt{fib}_{f}\left(\star_{Y}\right)$ and $i\equiv\mathtt{pr}_{1}:F\rightarrow X$ . Then we have pointed types and pointed maps:

F\overset{i}{\rightarrow}X\overset{f}{\rightarrow}Y.

Proof.

$F$ is pointed by $\left(\star_{X},f_{\star}\right)$ , and since $i\left(\star_{X},f_{\star}\right)\equiv\star_{X}$ , $i$ is a pointed map witnessed by $\mathtt{refl}_{\star_{X}}$ . ∎

A special case is the following, a fibration over $Y$ with fibre $F$ and total space $X$ .

Corollary 5.3.

Let $Y$ be a pointed type and $P:Y\rightarrow\mathtt{Type}$ be a dependent type such that $P\left(\star_{Y}\right)$ is a pointed type. Then we have pointed types and pointed maps:

P\left(\star_{Y}\right)\overset{i}{\rightarrow}\sum_{y:Y}P(y)\overset{f}{\rightarrow}Y,

called the fibration sequence, where $f\equiv\mathtt{pr}_{1}$ and $i\equiv\lambda z.\left(\star_{Y},z\right)$ .

Proof.

$\blacksquare$ $X\equiv\sum_{y:Y}P(y)$ has basepoint $\left(\star_{Y},\star_{P\left(\star_{Y}\right)}\right)$ and $f$ is thus a pointed map. Then we can apply Lemma 5.2, but $F\equiv\mathtt{fib}_{f}\left(\star_{Y}\right)\simeq P\left(\star_{Y}\right)$ by [Uni13, Lemma 4.8.1] (which uses Proposition 4.3). The map $i$ is as above under this equivalence. ∎

The rest of this section is based on the more general Lemma 5.2, although later in practice we only use the case Corollary 5.3. Since $i:F\rightarrow X$ is again a pointed map, we apply Lemma 5.2 recursively to yield the following construction:

Definition 5.4.

Let $X$ and $Y$ be pointed types. The fibre sequence of a pointed map $f:X\rightarrow_{\star}Y$ witnessed by $f_{\star}$ is the infinite sequence of pointed types and pointed maps

\ldots\overset{f^{(n)}}{\rightarrow}X^{(n)}\overset{f^{(n-1)}}{\rightarrow}X^{(n-1)}\overset{f^{(n-2)}}{\rightarrow}X^{(n-2)}\rightarrow\ldots\overset{f^{(0)}}{\rightarrow}X^{(0)}.

We denote $\star^{(n)}:X^{(n)}$ as the basepoint of $X^{(n)}$ and $f_{\star}^{(n)}:f^{(n)}\left(\star^{(n+1)}\right)=_{X^{(n)}}\star^{(n)}$ as the witnessing path of $f^{(n)}$ . The sequence is then defined recursively by

•

$X^{(0)}:\equiv Y$ , $X^{(1)}:\equiv X$ , $f^{(0)}:\equiv f$ , $\star^{(0)}:\equiv\star_{Y}$ , $\star^{(1)}:\equiv\star_{X}$ , $f_{\star}^{(0)}:\equiv f_{\star}$ ;

•

If $n\geq 2$ ,

	$\displaystyle X^{(n)}:\equiv\mathtt{fib}_{f^{(n-2)}}\left(\star^{(n-2)}\right)$	$\displaystyle,\quad f^{(n-1)}:\equiv\mathtt{pr}_{1}:X^{(n)}\rightarrow X^{(n-1)},$
	$\displaystyle\star^{(n)}:\equiv\left(\star^{(n-1)},f_{\star}^{(n-2)}\right)$	$\displaystyle,\quad f_{\star}^{(n-1)}:\equiv\mathtt{refl}_{\star^{(n-1)}}.$

We denote $F:\equiv\mathtt{fib}_{f}\left(\star_{Y}\right)\equiv X^{(2)}$ .

This fibre sequence can be (magically) turned into an equivalent form where the loop spaces of $F$ , $X$ and $Y$ show up in a periodic fashion. The following two ingredients are needed before we move on.

Definition 5.5.

If $f:A\rightarrow_{\star}B$ is a type equivalence, then $f$ is a pointed equivalence and in this case $A$ and $B$ are pointed equivalent.

Lemma 5.6.

The looping of $f:A\rightarrow_{\star}B$ defined as $\Omega f:\equiv\left(\lambda q.f_{\star}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{f}(q)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}f_{\star}\right):\Omega A\rightarrow\Omega B$ is also a pointed map.

Proof.

$(\Omega f)\left(\mathtt{refl}_{\star_{A}}\right)\equiv f_{\star}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{refl}_{f\left(\star_{A}\right)}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}f_{\star}=\mathtt{refl}_{\star_{B}}$ . ∎

We now work to reveal the first step of the alleged equivalence.

Lemma 5.7 ([Uni13, Lemma 8.4.4]).

Let $X$ and $Y$ be pointed types and let $f:X\rightarrow_{\star}Y$ . Then $X^{(3)}\simeq\Omega Y$ in the fibre sequence of $f$ . Further, under the equivalence, $f^{(2)}$ is identified with

\partial:\equiv\lambda r.\left(\star_{X},f_{\star}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r\right):\Omega Y\rightarrow F.

Figure 5.1: Diagram for Lemma 5.7.

Proof.

See Figure 5.1. Using the definition of fibres twice, a term of $X^{(3)}$ is of the form $\left((x,p),q\right)$ , where $x:X$ , $p:f(x)=_{Y}\star_{Y}$ and $q:x=_{X}\star_{X}$ . We first define a function $\varphi:X^{(3)}\rightarrow\Omega(Y)$ . Take any $\left((x,p),q\right):X^{(3)}$ . Then $\mathtt{ap}_{f}\left(q^{-1}\right):f\left(\star_{X}\right)=_{Y}f(x)$ . Thus since $f_{\star}:f\left(\star_{X}\right)=_{Y}\star_{Y}$ , we can produce $f_{\star}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{f}\left(q^{-1}\right)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}p:\Omega(Y)$ . Conversely, we define $\psi:\Omega(Y)\rightarrow X^{(3)}$ by

\lambda r.\left(\left(\star_{X},f_{\star}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r\right),\mathtt{refl}_{\star_{X}}\right).

Now take any $\left((x,p),q\right):X^{(3)}$ , then

	$\displaystyle\psi(\varphi((x,p),q))$	$\displaystyle\equiv\psi\left(f_{\star}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{f}\left(q^{-1}\right)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}p\right)$
		$\displaystyle=\left(\left(\star_{X},\mathtt{ap}_{f}\left(q^{-1}\right)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}p\right),\mathtt{refl}_{\star_{X}}\right)$
		$\displaystyle=\left((x,p),q\right),$

where the last equality is by Proposition 3.21 and Lemma 3.13. Conversely, for any $r:\Omega(Y,\star_{Y})$ ,

\varphi(\psi(r))\equiv\varphi(\left(\star_{X},f_{\star}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r\right),\mathtt{refl}_{\star_{X}})=f_{\star}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{ap}_{f}\left(\mathtt{refl}_{\star_{X}}^{-1}\right)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}f_{\star}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r=r.

Thus $\psi$ forms a quasi-inverse of $\varphi$ . Also $\psi$ and $\varphi$ are both pointed maps, so $X^{(3)}$ and $\Omega Y$ are pointed equivalent. We therefore identify $f^{(2)}:X^{(3)}\rightarrow F$ with $\partial:\equiv f^{(2)}\circ\psi:\Omega Y\rightarrow F$ , which can be simplified as $\lambda r.\left(\star_{X},f_{\star}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r\right)$ , easily a pointed map. ∎

By applying Lemma 5.7 recursively to the fibre sequence, we have:

Corollary 5.8 ([Uni13, Lemma 8.4.4]).

The fibre sequence of $f:X\rightarrow_{\star}Y$ can be equivalently written as

\rightarrow\Omega^{2}X\overset{\Omega^{2}f}{\rightarrow}\Omega^{2}Y\overset{-\Omega\partial}{\rightarrow}\Omega F\overset{-\Omega i}{\rightarrow}\Omega X\overset{-\Omega f}{\rightarrow}\Omega Y\overset{\partial}{\rightarrow}F\overset{i}{\rightarrow}X\overset{f}{\rightarrow}Y.

where $i:\equiv\mathtt{pr}_{1}$ and the minus sign represents composition with path inversion.

We will then show that this fibre sequence, when set-truncated, becomes an exact sequence, defined below.

Definition 5.9.

Let $A,B$ be sets and let $f:A\rightarrow B$ , then the image of $f$ is defined as $\operatorname{im}(f):\equiv\sum_{b:B}\left\|{\mathtt{fib}_{f}(b)}\right\|_{-1}.$ If $A,B$ are pointed sets and let $f:A\rightarrow_{\star}B$ , then the kernel of $f$ is defined as $\ker(f):\equiv\mathtt{fib}_{f}\left(\star_{B}\right)$ .

Definition 5.10.

An exact sequence of pointed sets is a sequence of pointed sets and pointed maps:

\ldots\rightarrow A^{(n+1)}\overset{f^{(n)}}{\rightarrow}A^{(n)}\overset{f^{(n-1)}}{\rightarrow}A^{(n-1)}\rightarrow\ldots

such that $\operatorname{im}(f^{(n)})=\ker(f^{(n-1)})$ for every $n$ . The sequence is an exact sequence of groups if further each $A^{(n)}$ is a group and each $f^{(n)}$ is a group homomorphism.

Lemma 5.11 ([Uni13, Theorem 8.4.6] ¹¹1 $\blacksquare$ A few typos in [Uni13, Theorem 8.4.6] are fixed in this proof.).

For any $F\overset{i}{\rightarrow}X\overset{f}{\rightarrow}Y$ given by Lemma 5.2,

\left\|F\right\|_{0}\overset{\left\|i\right\|_{0}}{\rightarrow}\left\|X\right\|_{0}\overset{\left\|f\right\|_{0}}{\rightarrow}\left\|Y\right\|_{0}

is an exact sequence of pointed sets.

Proof.

$\operatorname{im}(\left\|i\right\|_{0})\subset\ker(\left\|f\right\|_{0})$ . Take any $(x,p):F$ , then $f\left(i(x,p)\right)\equiv f(x)=_{Y}\star_{Y}$ as witnessed by $p$ . Thus $\left\|{f\circ i}\right\|_{0}$ sends everything to $\left|\star_{Y}\right|_{0}$ , and so does $\left\|f\right\|_{0}\circ\left\|i\right\|_{0}$ by the functoriality of $\left\|-\right\|_{0}$ .

$\ker(\left\|f\right\|_{0})\subset\operatorname{im}(\left\|i\right\|_{0})$ . For any $x^{\prime}:\left\|X\right\|_{0}$ and $p^{\prime}:\left\|f\right\|_{0}(x^{\prime})=_{\left\|Y\right\|_{0}}\left|\star_{Y}\right|_{0}$ , we claim that there merely exists $t:F$ such that $\left|{i(t)}\right|_{0}=_{\left\|X\right\|_{0}}x^{\prime}$ . As the goal is a mere proposition and a fortiori a $0$ -type, by truncation induction, it suffices to assume $x^{\prime}$ is $|x|_{0}$ for some $x:X$ . Then since $\left\|f\right\|_{0}\left(|x|_{0}\right)\equiv\left|{f(x)}\right|_{0}$ , we have $p^{\prime}:\left|{f(x)}\right|_{0}=_{\left\|Y\right\|_{0}}\left|\star_{Y}\right|_{0}$ , which by Proposition 4.12, corresponds to some $p^{\prime\prime}:\left\|{f(x)=_{Y}\star_{Y}}\right\|_{-1}$ . Again by truncation induction, it suffices to assume $p^{\prime\prime}$ is $|p|_{-1}$ for some $p:f(x)=_{Y}\star_{Y}$ . Thus we take $t:\equiv(x,p):F$ , and then $\left|{i(t)}\right|_{0}\equiv|x|_{0}\equiv x^{\prime}$ . ∎

Theorem 5.12 ([Uni13, Theorem 8.4.6]).

Let $X,Y$ be pointed types and let $f:X\rightarrow_{\star}Y$ . Let $F:\equiv\mathtt{fib}_{f}\left(\star_{Y}\right)$ . Then we have an exact sequence of pointed sets

	$\displaystyle\ldots\rightarrow$	$\displaystyle\pi_{k}(F)\rightarrow\pi_{k}(X)\rightarrow\pi_{k}(Y)\rightarrow$
	$\displaystyle\ldots\rightarrow$	$\displaystyle\pi_{2}(F)\rightarrow\pi_{2}(X)\rightarrow\pi_{2}(Y)$
	$\displaystyle\rightarrow$	$\displaystyle\pi_{1}(F)\rightarrow\pi_{1}(X)\rightarrow\pi_{1}(Y)$
	$\displaystyle\rightarrow$	$\displaystyle\pi_{0}(F)\rightarrow\pi_{0}(X)\rightarrow\pi_{0}(Y).$

This is further an exact sequence of groups except the $\pi_{0}$ terms, and an exact sequence of abelian groups except the $\pi_{0}$ and $\pi_{1}$ terms.

Proof.

By recursively applying Lemma 5.11 to the fibre sequence of $f$ , the above is an exact sequence of pointed sets. The rest of the proof, including some technicalities involving path inversion, is omitted. ∎

The following lemma is commonly known in classical algebraic topology.

Lemma 5.13 ([Uni13, Lemma 8.4.7]).

Let $C\rightarrow A\overset{f}{\rightarrow}B\rightarrow D$ be an exact sequence of abelian groups. Then (1) $f$ is injective if $C$ is $0$ ; (2) $f$ is surjective if $D$ is $0$ ; (3) $f$ is an isomorphism if $C$ and $D$ are both $0$ .

From the next result, which characterises an $n$ -connected function in terms of homotopy groups, we formalise the idea that an $n$ -connected function is a generalisation of a surjective function.

Corollary 5.14.

Let $X,Y$ be pointed types and let $f:X\rightarrow_{\star}Y$ be $n$ -connected.

(1)

If $k\leq n$ , then $\pi_{k}(f):\pi_{k}(X)\rightarrow\pi_{k}(Y)$ is an isomorphism;
(2)

If $k=n+1$ , then $\pi_{k}(f):\pi_{k}(X)\rightarrow\pi_{k}(Y)$ is surjective.

Proof.

We omit the case when $k=0$ and suppose $k>0$ . We have an exact sequence

\pi_{k}(F)\rightarrow\pi_{k}(X)\rightarrow\pi_{k}(Y)\rightarrow\pi_{k-1}(F).

Since $f$ is $n$ -connected, by Proposition 4.27, $f$ is also $k$ -connected when $k\leq n$ . Thus $\left\|F\right\|_{k}\equiv\left\|{\mathtt{fib}_{f}\left(\star_{Y}\right)}\right\|_{k}$ is contractible, and $\pi_{k}(F)\equiv\left\|{\Omega^{k}(F)}\right\|_{0}\simeq\Omega^{k}\left(\left\|F\right\|_{k}\right)$ (by Proposition 4.13) is also contractible for $k\leq n$ . Applying Lemma 5.13 yields the result. ∎

Remark 5.15.

Set $Y$ to $\mathbb{1}$ and Corollary 5.14 would imply Lemma 4.28.

5.2 Hopf construction

To trigger all the machinery in the previous section, we only need a fibration sequence as in Corollary 5.3. We now build the Hopf construction, a fibration sequence derived from a $0$ -connected H-space.

Definition 5.16.

A type $A$ is an H-space if it is equipped with a base point $e:A$ , a binary operation $\mu:A\times A\rightarrow A$ and equalities $\mu(e,a)=a$ and $\mu(a,e)=a$ for every $a:A$ .

Lemma 5.17.

Let $A$ be a $0$ -connected H-space. Then $\mu(a,-):A\rightarrow A$ and $\mu(-,a):A\rightarrow A$ are equivalences for every $a:A$ .

Proof.

Let $P:A\rightarrow\mathtt{Prop}$ be the dependent proposition which sends $a:A$ to the predicate that $\mu(a,-)$ (resp. $\mu(-,a)$ ) is an equivalence. By Proposition 4.30, it suffices to prove $P(e)$ , which holds by definition. ∎

The flattening lemma, used in the next proof, is a result concerning the total space of a fibration over an HIT constructed via univalence [Uni13, Section 6.12].

Theorem 5.18 ([Uni13, Lemma 8.5.7]).

Let $A$ be a $0$ -connected $H$ -space. Then there is a dependent type (fibration) $P:\Sigma A\rightarrow\mathtt{Type}$ with fibre $A$ and total space $\sum_{x:\Sigma A}P(x)\simeq A\ast A$ , called the Hopf construction.

Proof.

We define a dependent type $P:\Sigma A\rightarrow\mathtt{Type}$ by $P\left(\mathtt{north}\right):\equiv A$ , $P\left(\mathtt{south}\right):\equiv A$ , and $\mathtt{ap}_{P}\left(\mathtt{merid}(a)\right):=\mathtt{ua}(\mu(a,-))$ since each $\mu(a,-)$ is an equivalence $A\rightarrow A$ . By the flattening lemma (see [Uni13, Lemma 8.5.3] or [Bru16, Proposition 1.9.2]), $\sum_{x:\Sigma A}P(x)$ is equivalent to the pushout of the span $A\overset{\mathtt{snd}}{\leftarrow}A\times A\overset{\mu}{\rightarrow}A.$ We define a function $\alpha:A\times A\rightarrow A\times A$ by $\alpha(x,y):\equiv\left(\mu(x,y),y\right)$ , and $\alpha$ is an equivalence because it has quasi-inverse $\lambda(u,v).\left(\left(\mu(-,v)\right)^{-1}(u),v\right)$ . Therefore, we have an equivalence of spans

since both squares commute and each vertical line is an equivalence. Thus, the pushout of the two spans are also equivalent, but the pushout of the lower span is by definition $A\ast A$ . ∎

5.3 Hopf fibration

We finally present the Hopf construction induced by the H-space structure on ${\mathbb{S}}^{1}$ .

Lemma 5.19 ([Uni13, Lemma 8.5.8]).

There is an $H$ -space structure on ${\mathbb{S}}^{1}$ .

Proof.

Take $e:\equiv\mathtt{base}:{\mathbb{S}}^{1}$ . Now we define $\mu:{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}$ by circle induction on the first argument. We define $\mu(\mathtt{base}):\equiv\operatorname{id}\limits_{{\mathbb{S}}^{1}}$ ; then we need a path $p:\operatorname{id}\limits_{{\mathbb{S}}^{1}}=_{{\mathbb{S}}^{1}\rightarrow{\mathbb{S}}^{1}}\operatorname{id}\limits_{{\mathbb{S}}^{1}}$ such that $\mathtt{ap}_{\mu}\left(\mathtt{loop}\right):=p$ . By function extensionality, this is equivalent to a path $p_{x}:x=_{{\mathbb{S}}^{1}}x$ for each $x:{\mathbb{S}}^{1}$ . By induction on $x:{\mathbb{S}}^{1}$ again, we define $p_{\mathtt{base}}:\equiv\mathtt{loop}$ . It then remains to give a dependent path $\mathtt{loop}=_{\mathtt{loop}}^{\lambda x.x=_{{\mathbb{S}}^{1}}x}\mathtt{loop}$ , which is equivalent to $\mathtt{loop}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}=\mathtt{loop}$ by Lemma 3.13.

Now $\mu(\mathtt{base})(x)\equiv x$ since $\mu(\mathtt{base})\equiv\operatorname{id}\limits_{{\mathbb{S}}^{1}}$ . We show that $\mu(x)(\mathtt{base})=_{{\mathbb{S}}^{1}}x$ by circle induction on $x:{\mathbb{S}}^{1}$ . When $x$ is $\mathtt{base}$ , $\mu(\mathtt{base})(\mathtt{base})\equiv\mathtt{base}$ , so we take $\mathtt{refl}_{\mathtt{base}}$ ; when $x$ ‘varies along’ $\mathtt{loop}$ , we need a dependent path

\mathtt{transport}^{\lambda x.\mu(x)(\mathtt{base})=_{{\mathbb{S}}^{1}}x}\left(\mathtt{loop}\right)\left(\mathtt{refl}_{\mathtt{base}}\right)=\mathtt{refl}_{\mathtt{base}},

which is equivalent to $\mathtt{ap}_{\lambda x.\mu(x)(\mathtt{base})}\left(\mathtt{loop}\right)^{-1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,\mathtt{refl}_{\mathtt{base}}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{loop}=\mathtt{refl}_{\mathtt{base}}$ by Lemma 3.13. By function extensionality,

\mathtt{ap}_{\lambda x.\mu(x)(\mathtt{base})}\left(\mathtt{loop}\right)=\mathtt{happly}(\mathtt{ap}_{\mu}\left(\mathtt{loop}\right))\left(\mathtt{base}\right)=p_{\mathtt{base}}\equiv\mathtt{loop}.

Finally, we change $\mu$ to its uncurried form. ∎

The next result follows from the $3\times 3$ -lemma [Bru16, Section 1.8].

Lemma 5.20 ([Bru16, Proposition 1.8.8]).

For every $m,n:{\mathbb{N}}$ , ${\mathbb{S}}^{m}\ast{\mathbb{S}}^{n}\simeq{\mathbb{S}}^{m+n+1}$ .

Theorem 5.21.

There is a fibration, called the Hopf fibration, over ${\mathbb{S}}^{2}$ with fibre ${\mathbb{S}}^{1}$ and total space ${\mathbb{S}}^{3}$ .

Proof.

${\mathbb{S}}^{1}$ is $0$ -connected by Example 4.26. By Lemma 5.19 and Theorem 5.18, there is a fibration over $\Sigma{\mathbb{S}}^{1}\equiv{\mathbb{S}}^{2}$ with fibre ${\mathbb{S}}^{1}$ and total space ${\mathbb{S}}^{1}\ast{\mathbb{S}}^{1}\simeq{\mathbb{S}}^{3}$ by Lemma 5.20. ∎

We put all the pieces together by writing the long exact sequence induced by the Hopf fibration:

Corollary 5.22 ([Uni13, Corollary 8.5.2]).

$\pi_{2}\left({\mathbb{S}}^{2}\right)\simeq{\mathbb{Z}}$ and $\pi_{k}\left({\mathbb{S}}^{3}\right)\simeq\pi_{k}\left({\mathbb{S}}^{2}\right)$ for $k\geq 3$ .

Proof.

By Corollary 5.3, Theorem 5.12 and Theorem 5.21, we have an exact sequence of groups

	$\displaystyle\ldots$	$\displaystyle\rightarrow\pi_{k}\left({\mathbb{S}}^{1}\right)\rightarrow\pi_{k}\left({\mathbb{S}}^{3}\right)\rightarrow\pi_{k}\left({\mathbb{S}}^{2}\right)\rightarrow$
	$\displaystyle\ldots$	$\displaystyle\rightarrow\pi_{2}\left({\mathbb{S}}^{1}\right)\rightarrow\pi_{2}\left({\mathbb{S}}^{3}\right)\rightarrow\pi_{2}\left({\mathbb{S}}^{2}\right)$
		$\displaystyle\rightarrow\pi_{1}\left({\mathbb{S}}^{1}\right)\rightarrow\pi_{1}\left({\mathbb{S}}^{3}\right)\rightarrow\pi_{1}\left({\mathbb{S}}^{2}\right).$

By Corollary 4.21 and Corollary 4.29, this is reduced to

	$\displaystyle...$	$\displaystyle\to 0\to\pi_{k}(\mathbb{S}^{3})\to\pi_{k}(\mathbb{S}^{2})\to$
	$\displaystyle...$	$\displaystyle\to 0\to 0\to\pi_{2}(\mathbb{S}^{2})$
		$\displaystyle\to\mathbb{Z}\to 0\to 0.$

Applying Lemma 5.13 yields the result. ∎

Chapter 6 Blakers–Massey theorem

In this chapter, we will ‘informalise’ the mechanised Agda proof of the Blakers–Massey theorem in [HFLL16] (or [Hou17, Sections 3.4, 4.3]) by combining [Uni13, Section 8.6]. For this, a generalised version of pushouts is first presented in Section 6.1. Then the theorem is stated, generalised and relaxed in Section 6.2. The proof employs a more sophisticated encode-decode method, where we define $\mathtt{code}$ by the wedge connectivity lemma in Section 6.3 and then show the contractibility of $\mathtt{code}$ in Section 6.4. Some consequences, including the Freudenthal suspension theorem and stability for spheres, are derived in Section 6.5.

Throughout the chapter, we shall omit proofs on coherence operations too technical for presentation, and write a square $\square$ for an unspecified coherence path.

6.1 Generalised pushouts

Lemma 6.1 ([Uni13, Lemma 4.8.2]).

For any function $f:A\rightarrow B$ , $A\simeq\sum_{b:B}\mathtt{fib}_{f}(b)$ .

Proof.

By Proposition 4.3.∎

This result shows that we can interchange function types and families of types, i.e., any function $f:A\rightarrow B$ is equivalent to the projection $\mathtt{pr}_{1}:\sum_{b:B}\mathtt{fib}_{f}(b)\rightarrow B$ .

$\blacksquare$ Recall that Section 3.5.2 defines a pushout for the span $X\overset{f}{\leftarrow}C\overset{g}{\rightarrow}Y$ . By rewriting the functions as families of types and expanding the definition of fibres, this span is equivalent to

X\overset{\mathtt{pr}_{X}}{\leftarrow}\sum_{x:X}\sum_{y:Y}\sum_{c:C}\left(\left(g(c)=_{Y}y\right)\times\left(f(c)=_{X}x\right)\right)\overset{\mathtt{pr}_{Y}}{\rightarrow}Y.

If we define a dependent type $Q:X\rightarrow Y\rightarrow\mathtt{Type}$ , then this span can be generalised as

X\overset{\mathtt{pr}_{X}}{\leftarrow}\sum_{x:X}\sum_{y:Y}Q(x)(y)\overset{\mathtt{pr}_{Y}}{\rightarrow}Y.

Definition 6.2.

Let $X,Y:\mathtt{Type}$ and $Q:X\rightarrow Y\rightarrow\mathtt{Type}$ . The (generalised) pushout $X\sqcup^{Q}Y$ is the HIT generated by two point constructors $\mathtt{inl}:X\rightarrow X\sqcup^{Q}Y$ and $\mathtt{inr}:Y\rightarrow X\sqcup^{Q}Y$ and a path constructor

\mathtt{glue}:(x:X)\rightarrow(y:Y)\rightarrow Q(x)(y)\rightarrow\left(\mathtt{inl}(x)=_{X\sqcup^{Q}Y}\mathtt{inr}(y)\right),

as visualised by Figure 6.1.

Figure 6.1: Diagram for the generalised pushout

X\sqcup^{Q}Y

X

and

Y

are thought of as two ‘axes’ of a ‘coordinate system’, and each pair

(x,y):X\times Y

corresponds to a type

Q(x)(y)

. The two dashed circles represent the ‘image’ of

X

and

Y

under

\mathtt{inl}

and

\mathtt{inr}

, respectively.

By previous discussions, when $Q\equiv\lambda x.\lambda y.\sum_{c:C}\left(\left(g(c)=_{Y}y\right)\times\left(f(c)=_{X}x\right)\right)$ , this definition is equivalent to the one in Section 3.5.2. Henceforth, we only use the generalised pushout.

The induction principle states that given

•

a dependent type $P:X\sqcup^{Q}Y\rightarrow\mathtt{Type}$ ,
•

two dependent functions $h_{\mathtt{inl}}:(x:X)\rightarrow P\left(\mathtt{inl}(x)\right)$ and
•

$h_{\mathtt{inr}}:(y:Y)\rightarrow P\left(\mathtt{inr}(y)\right)$ and
•

a family of dependent paths

h_{\mathtt{glue}}:(x:X)\rightarrow(y:Y)\rightarrow\left(q:Q(x)(y)\right)\rightarrow h_{\mathtt{inl}}(x)=_{\mathtt{glue}_{x,y}(q)}^{P}h_{\mathtt{inr}}(y),

there is a dependent function $h:\left(z:X\sqcup^{Q}Y\right)\rightarrow P(z)$ such that $h\left(\mathtt{inl}(x)\right):\equiv h_{\mathtt{inl}}(x)$ , $h\left(\mathtt{inr}(y)\right):\equiv h_{\mathtt{inr}}(y)$ , and $\mathtt{apd}_{h}\left(\mathtt{glue}_{x,y}(q)\right):=h_{\mathtt{glue}}(x)(y)(q)$ .

Henceforth, we may omit the first two parameters of $\mathtt{glue}$ for conciseness.

6.2 Theorem statement

Theorem 6.3 (Blakers–Massey theorem).

Let $X,Y:\mathtt{Type}$ and $Q:X\rightarrow Y\rightarrow\mathtt{Type}$ . Let $m,n\geq-1$ . Suppose the type $\sum_{y:Y}Q(x)(y)$ is $m$ -connected for each $x:X$ , and the type $\sum_{x:X}Q(x)(y)$ is $n$ -connected for each $y:Y$ . Then for each $x:X$ and each $y:Y$ , the map

\mathtt{glue}_{x,y}:Q(x)(y)\rightarrow\left(\mathtt{inl}(x)=_{X\sqcup^{Q}Y}\mathtt{inr}(y)\right)

is $(m+n)$ -connected.

Let $X,Y,Q,m,n$ be fixed. We also fix $x\equiv x_{0}$ . By the definition of connectedness, the claim is equivalent to

(y:Y)\rightarrow\left(r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}\mathtt{inr}(y)\right)\right)\rightarrow\mathtt{is\text{-}contr}\left(\left\|\mathtt{fib}_{\mathtt{glue}}(r)\right\|_{m+n}\right).

Similar to Section 4.4, in order to apply path induction on $r$ , we need to generalise the right-hand side of $\left(\mathtt{inl}(x_{0})=\mathtt{inr}(y)\right)$ to an arbitrary term $p$ of $X\sqcup^{Q}Y$ . Therefore, we claim the following lemma, which would imply Theorem 6.3 by setting $p$ to $\mathtt{inr}(y)$ :

Lemma 6.4.

There exists a dependent type

\mathtt{code}:\left(p:X\sqcup^{Q}Y\right)\rightarrow\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)\rightarrow\mathtt{Type}

such that

•

$\mathtt{code}(\mathtt{inr}(y))(r)\equiv\left\|\mathtt{fib}_{\mathtt{glue}}(r)\right\|_{m+n}$ , and
•

for any $p:X\sqcup^{Q}Y$ and any $r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)$ , $\mathtt{code}(p)(r)$ is contractible.

Additionally, we need the following relaxation:

Lemma 6.5.

To prove Lemma 6.4, we may assume that there exist some fixed $y_{0}:Y$ and $q_{00}:Q\left(x_{0}\right)\left(y_{0}\right)$ .

Proof.

By assumption, $\sum_{y:Y}Q\left(x_{0}\right)(y)$ is $m$ -connected, so $\left\|{\sum_{y:Y}Q\left(x_{0}\right)(y)}\right\|_{m}$ is contractible and thus inhabited. The proposition that $\mathtt{code}(p)(r)$ is contractible is a mere proposition, and thus is a fortiori an $m$ -type as $m\geq-1$ . Thus, by truncation induction, we may assume the inhabitant of $\left\|{\sum_{y:Y}Q\left(x_{0}\right)(y)}\right\|_{m}$ is of the form $\left|\left(y_{0},q_{00}\right)\right|_{m}$ for some $y_{0}:Y$ and $q_{00}:Q\left(x_{0}\right)\left(y_{0}\right)$ . ∎

In summary, our goal is to prove Lemma 6.4 while assuming some fixed $y_{0}:Y$ and $q_{00}:Q\left(x_{0}\right)\left(y_{0}\right)$ . This is broken into the definition of $\mathtt{code}$ and its contractibility in the following two sections.

6.3 Definition of $\mathtt{code}$

We begin with the wedge connectivity lemma. For two pointed types $A$ and $B$ , there is a canonical map $i:A\vee B\rightarrow A\times B$ given by $\lambda a.\left(a,b_{0}\right)$ and $\lambda b.\left(a_{0},b\right)$ . By Proposition 4.31 and the induction principle of wedge sums, the next lemma is an equivalent way to state that if $A$ is $n$ -connected and $B$ is $m$ -connected, then $i$ is $(m+n)$ -connected.

Lemma 6.6 (Wedge connectivity lemma, [Uni13, Lemma 8.6.2]).

Suppose $\left(A,a_{0}\right)$ and $\left(B,b_{0}\right)$ are $n$ - and $m$ -connected pointed types, respectively, with $n,m\geq-1$ ¹¹1This lemma works for $m=-1$ or $n=-1$ although [Uni13, Lemma 3.1.3] requires $n,m\geq 0$ [Hou17].. Then given

•

$P:A\rightarrow B\rightarrow(n+m)\mathtt{\text{-}Type}$ ,
•

$f:(a:A)\rightarrow P(a)\left(b_{0}\right)$ ,
•

$g:(b:B)\rightarrow P\left(a_{0}\right)(b)$ and
•

$p:f\left(a_{0}\right)=_{P\left(a_{0}\right)\left(b_{0}\right)}g\left(b_{0}\right)$ ,

there is $h:(a:A)\rightarrow(b:B)\rightarrow P(a)(b)$ with homotopies $q:h(-)\left(b_{0}\right)\sim f$ and $r:h\left(a_{0}\right)\sim g$ such that $p={q\left(a_{0}\right)}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}r\left(b_{0}\right)$ .

In shorter terms, to define the dependent function $h:(a:A)\rightarrow(b:B)\rightarrow P(a)(b)$ , it suffices to consider respectively the cases when $b$ coincides with $b_{0}$ by giving $f$ and when $a$ coincides with $a_{0}$ by giving $g$ , and then ensure the consistency when both cases happen simultaneously by giving $p$ .

The following is an application of the wedge connectivity lemma, used in defining $\mathtt{code}$ .

Corollary 6.7.

Let $X,Y:\mathtt{Type}$ and $Q:X\rightarrow Y\rightarrow\mathtt{Type}$ . Let $x_{1}:X$ , $y_{0}:Y$ and $q_{10}:Q\left(x_{1}\right)\left(y_{0}\right)$ . Define type $A:\equiv\sum_{x_{0}:X}Q\left(x_{0}\right)\left(y_{0}\right)$ and type $B:\equiv\sum_{y_{1}:Y}Q\left(x_{1}\right)\left(y_{1}\right)$ . Assume that $A$ is $n$ -connected and $B$ is $m$ -connected. Define $C:A\rightarrow B\rightarrow(m+n)\mathtt{\text{-}Type}$ by

\lambda\left(x_{0},q_{00}\right).\lambda\left(y_{1},q_{11}\right).\left\|{{\mathtt{fib}_{\mathtt{glue}}}_{x_{0},y_{1}}\left(\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{glue}(q_{11})\right)}\right\|_{m+n},

where $q_{00}:Q\left(x_{0}\right)\left(y_{0}\right)$ and $q_{11}:Q\left(x_{1}\right)\left(y_{1}\right)$ . Then there is a dependent function $h:\left(\left(x_{0},q_{00}\right):A\right)\rightarrow\left(\left(y_{1},q_{11}\right):B\right)\rightarrow C\left(x_{0},q_{00}\right)\left(y_{1},q_{11}\right)$ .

Proof.

$A$ is pointed by $\left(x_{1},q_{10}\right)$ and $B$ is pointed by $\left(y_{0},q_{10}\right)$ . By Lemma 6.6, to obtain $h$ , it suffices to consider the following cases:

(1) When $\left(y_{1},q_{11}\right)$ coincides with $\left(y_{0},q_{10}\right)$ , for each $\left(x_{0},q_{00}\right):A$ , we need a term of the type

C\left(x_{0},q_{00}\right)\left(y_{0},q_{10}\right)\equiv\left\|{\mathtt{fib}_{\mathtt{glue}}\left(\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{glue}(q_{10})\right)}\right\|_{m+n},

which is inhabited by $\left|\left(q_{00},\square\right)\right|_{m+n}$ . (Here, $\square$ is the coherence path $\mathtt{glue}(q_{00})=\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{glue}(q_{10})$ .) (2) When $\left(x_{0},q_{00}\right)$ coincides with $\left(x_{1},q_{10}\right)$ , for each $\left(y_{1},q_{11}\right):B$ , we need to a term of the type

C\left(x_{1},q_{10}\right)\left(y_{1},q_{11}\right)\equiv\left\|{\mathtt{fib}_{\mathtt{glue}}\left(\mathtt{glue}(q_{10})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{glue}(q_{11})\right)}\right\|_{m+n},

which is inhabited by $\left|\left(q_{11},\square\right)\right|_{m+n}$ .

(3) Choices from (1) and (2) are consistent by coherence operations. ∎

We are now ready to define $\mathtt{code}$ .

Figure 6.2: Diagram for the definition of

\mathtt{code}

Proposition 6.8.

$\mathtt{code}:\left(p:X\sqcup^{Q}Y\right)\rightarrow\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)\rightarrow\mathtt{Type}.$

Proof.

See Figure 6.2. When $p$ is $\mathtt{inr}(y_{1})$ , for $r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}\mathtt{inr}(y_{1})\right)$ , as mentioned earlier,

\mathtt{code}(\mathtt{inr}(y_{1}))(r):\equiv\left\|{\sum_{q_{01}:Q\left(x_{0}\right)\left(y_{1}\right)}\mathtt{glue}(q_{01})=r}\right\|_{m+n}.

When $p$ is $\mathtt{inl}(x_{1})$ , for $s:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}\mathtt{inl}(x_{1})\right)$ ,

\mathtt{code}(\mathtt{inl}(x_{1}))(s):\equiv\left\|{\sum_{q_{10}:Q\left(x_{1}\right)\left(y_{0}\right)}\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}=s}\right\|_{m+n}.

$\blacksquare$ When $p$ ‘varies along’ $\mathtt{glue}(q_{11}):\mathtt{inl}(x_{1})=\mathtt{inr}(y_{1})$ , we need to find a path

\mathtt{apd}_{\mathtt{code}}\left(\mathtt{glue}(q_{11})\right):\mathtt{code}(\mathtt{inl}(x_{1}))=_{\mathtt{glue}(q_{11})}^{\mathtt{code}}\mathtt{code}(\mathtt{inr}(y_{1}))

By function extensionality, it suffices to find for every $r:\left(\mathtt{inl}(x_{0})=\mathtt{inr}(y_{1})\right)$ , a path

{\mathtt{glue}(q_{11})}_{\ast}\left(\mathtt{code}(\mathtt{inl}(x_{1}))\right)(r)=\mathtt{code}(\mathtt{inr}(y_{1}))(r).

Starting from the left-hand side, we have a path

	$\displaystyle{\mathtt{glue}(q_{11})}_{\ast}\left(\mathtt{code}(\mathtt{inl}(x_{1}))\right)(r)$	$\displaystyle={\mathtt{glue}(q_{11})}_{\ast}\left(\mathtt{code}(\mathtt{inl}(x_{1}))({\mathtt{glue}(q_{11})}_{\ast}^{-1}(r))\right)$
		$\displaystyle=\mathtt{code}(\mathtt{inl}(x_{1}))(r\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{11})}^{-1}),$

where the first equality is by Lemma 3.14 and the second by Lemma 3.11 and Lemma 3.15. Therefore, by univalence, it remains to find a function

\Phi:\mathtt{code}(\mathtt{inl}(x_{1}))(r\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{11})}^{-1})\rightarrow\mathtt{code}(\mathtt{inr}(y_{1}))(r),

(6.1)

and show that $\Phi$ is an equivalence. By the definition of $\mathtt{code}(\mathtt{inl}(x_{1}))$ and $\mathtt{code}(\mathtt{inr}(y_{1}))$ and truncation induction, it suffices to find for each $q_{01}:Q\left(x_{1}\right)\left(y_{0}\right)$ with $\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}=r\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,{\mathtt{glue}(q_{11})}^{-1}$ , a term of $\left\|{\sum_{q_{01}:Q\left(x_{0}\right)\left(y_{1}\right)}\mathtt{glue}(q_{01})=r}\right\|_{m+n}$ . By coherence operations, we then eliminate $r$ , and it suffices to find for each $q_{10}:Q\left(x_{1}\right)\left(y_{0}\right)$ , a term of

\left\|{\sum_{q_{01}:Q\left(x_{0}\right)\left(y_{1}\right)}\mathtt{glue}(q_{01})=\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\mathtt{glue}(q_{11})}\right\|_{m+n}.

We now invoke Corollary 6.7 (by using which the order of the variables is effectively rearranged), and the desired term is given by $h\left(x_{0},q_{00}\right)\left(y_{1},q_{11}\right)$ . We can similarly define the function on the other direction and show that the two functions are quasi-inverses, but the details are omitted. ∎

6.4 Contractibility of $\mathtt{code}$

We now show that for any $p:X\sqcup^{Q}Y$ and any $r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)$ , $\mathtt{code}(p)(r)$ is contractible, by first giving its centre of contraction.

Proposition 6.9.

$\mathtt{centre}:\left(p:X\sqcup^{Q}Y\right)\rightarrow\left(r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)\right)\rightarrow\mathtt{code}(p)(r).$

Proof.

By path induction, it suffices to assume $p\equiv\mathtt{inl}(x_{0})$ and $r\equiv\mathtt{refl}_{\mathtt{inl}(x_{0})}$ . Then

\mathtt{code}(\mathtt{inl}(x_{0}))(\mathtt{refl}_{\mathtt{inl}(x_{0})})\equiv\left\|{\sum_{q:Q\left(x_{0}\right)\left(y_{0}\right)}\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q)}^{-1}=\mathtt{refl}_{\mathtt{inl}(x_{0})}}\right\|_{m+n},

which is inhabited by $\left|\left(q_{00},\square\right)\right|_{m+n}$ . Then $\mathtt{centre}$ is explicitly given by

\mathtt{centre}(p)(r):\equiv\mathtt{transport}^{\hat{\mathtt{code}}}\left(\mathtt{pair}^{=}(r,t)\right)\left(\left|\left(q_{00},\square\right)\right|_{m+n}\right)

by Corollary 4.6, where $t:r_{\ast}\left(\mathtt{refl}_{x_{0}}\right)=r$ by Lemma 3.15 and

\hat{\mathtt{code}}:\sum_{p:X\sqcup^{Q}Y}\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)\rightarrow\mathtt{Type}

is the uncurried form of $\mathtt{code}$ . ∎

Before we proceed, the next lemma allows us to extract some valuable information from the hard-earned definition of $\mathtt{code}$ for calculating a certain $\mathtt{transport}$ function.

Lemma 6.10 ([Uni13, Lemma 8.6.10]).

Let $A:\mathtt{Type}$ , $B:A\rightarrow\mathtt{Type}$ , $C:(a:A)\rightarrow B(a)\rightarrow\mathtt{Type}$ , $a_{1},a_{2}:A$ , $m:a_{1}=_{A}a_{2}$ , and $b_{2}:B\left(a_{2}\right)$ . Denote $b_{1}:\equiv m_{\ast}^{-1}\left(b_{2}\right):B\left(a_{1}\right)$ . Also denote the uncurried form of $C$ as $\hat{C}:\sum_{a:A}B(a)\rightarrow\mathtt{Type}$ . Then the function

\mathtt{transport}^{\hat{C}}\left(\mathtt{pair}^{=}(m,t)\right):C\left(a_{1}\right)\left(b_{1}\right)\rightarrow C\left(a_{2}\right)\left(b_{2}\right),

where $t:m_{\ast}\left(b_{1}\right)=_{B\left(a_{2}\right)}b_{2}$ is an obvious path, is equal to the equivalence obtained by applying $\mathtt{idtoeqv}$ to the composite of paths

C\left(a_{1}\right)\left(b_{1}\right)=m_{\ast}\left(C\left(a_{1}\right)\left(m_{\ast}^{-1}\left(b_{2}\right)\right)\right)=m_{\ast}\left(C\left(a_{1}\right)\right)\left(b_{2}\right)=C\left(a_{2}\right)\left(b_{2}\right),

where the first equality is by definition and by Lemma 3.11, the second is by Lemma 3.14, and the third is by $\mathtt{happly}(\mathtt{apd}_{C}(m))\left(b_{2}\right)$ .

Figure 6.3: Diagram for Lemma 6.10.

The blue

m_{\ast}

represents

\mathtt{transport}^{B}(m)

, the green

m_{\ast}

represents

\mathtt{transport}^{\lambda a.\mathtt{Type}}(m)

and the red

m_{\ast}

represents

\mathtt{transport}^{C}(m)

. The arrow between

m_{\ast}\left(C\left(a_{1}\right)\right)

and

C\left(a_{2}\right)

represents the dependent path

\mathtt{apd}_{C}(m)

$\blacksquare$ We apply Lemma 6.10 with $A\equiv X\sqcup^{Q}Y$ , $B\equiv(p:A)\rightarrow\left(\mathtt{inl}(x_{0})=p\right)$ , $C\equiv\mathtt{code}$ , $a_{1}\equiv\mathtt{inl}(x_{1})$ , $a_{2}\equiv\mathtt{inr}(y_{1})$ , $m\equiv\mathtt{glue}(q_{11})$ , and $b_{2}\equiv\mathtt{glue}(q_{01})$ . Then $b_{1}\equiv m_{\ast}^{-1}\left(b_{2}\right)=\mathtt{glue}(q_{01})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{11})}^{-1}$ by Lemma 3.15. Then Lemma 6.10 says that to calculate the function

T:\equiv\mathtt{transport}^{\hat{\mathtt{code}}}\left(\mathtt{pair}^{=}(\mathtt{glue}(q_{11}),t)\right):\mathtt{code}(\mathtt{inl}(x_{1}))(b_{1})\rightarrow\mathtt{code}(\mathtt{inr}(y_{1}))(b_{2}),

the major work lies in recovering our definition of $\mathtt{apd}_{\mathtt{code}}\left(\mathtt{glue}(q_{11})\right)$ , which is a part of the definition of $\mathtt{code}$ . Noticing that the type of $T$ matches the type of $\Phi$ (Function 6.1), to calculate $T$ , we can utilise our definition of $\Phi$ , with $r$ replaced with $b_{2}\equiv\mathtt{glue}(q_{01})$ . The definition of $\Phi$ relies on Corollary 6.7 and we know explicitly the output under the two special cases in the proof of Corollary 6.7. In particular, suppose that $\left(x_{0},q_{00}\right)$ coincides with $\left(x_{1},q_{10}\right)$ and further $q_{01}$ coincides with $q_{11}$ . By case (2) in the proof of Corollary 6.7 and the full statement of Lemma 6.6, $h$ gives the term $\left|\left(q_{01},\square\right)\right|_{m+n}$ (up to a path), and thus so does $\Phi$ and $T$ .

Finally, we show that each $\mathtt{code}(p)(r)$ is contractible, written as:

Proposition 6.11.

$\left(p:X\sqcup^{Q}Y\right)\rightarrow\left(r:\left(\mathtt{inl}(x_{0})=_{X\sqcup^{Q}Y}p\right)\right)\rightarrow\left(c:\mathtt{code}(p)(r)\right)\rightarrow\left(\mathtt{centre}(p)(r)=c\right).$

Remark 6.12.

$\blacksquare$ It is tempting to assume by path induction that $p$ is $\mathtt{inl}(x_{0})$ and $r$ is $\mathtt{refl}_{\mathtt{inl}(x_{0})}$ . The desired path is in $\mathtt{code}(p)(r)$ and thus is an $(m+n-1)$ -type (and a fortiori an $(m+n)$ -type), so by truncation induction, we assume that $c$ is $\left|\left(q,\ell_{1}\right)\right|_{m+n}$ , where $q:Q\left(x_{0}\right)\left(y_{0}\right)$ and $\ell_{1}:\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q)}^{-1}=\mathtt{refl}_{\mathtt{inl}(x_{0})}$ . By the definition of $\mathtt{centre}$ , we need a path $\left|\left(q_{00},\square\right)\right|_{m+n}=\left|\left(q,\ell_{1}\right)\right|_{m+n}.$ Now we can only attempt to prove $q_{00}=q$ , which is unattainable, as $\ell_{1}$ only indicates $\mathtt{glue}(q_{00})=\mathtt{glue}(q)$ .

Proof.

By Corollary 4.4, it suffices to assume that $p$ is $\mathtt{inr}(y_{0})$ and $r$ is $\mathtt{glue}(q_{00})$ . We need a re-generalisation for the $y$ ‘s and claim the following:

\left(y_{1}:Y\right)\rightarrow\left(r:\mathtt{inl}(x_{0})=\mathtt{inr}(y_{1})\right)\rightarrow\left(c:\mathtt{code}(\mathtt{inr}(y_{1}))(r)\right)\rightarrow\left(\mathtt{centre}(\mathtt{inr}(y_{1}))(r)=c\right).

By truncation induction, it suffices to assume $c$ is $\left|\left(q_{01},\ell_{2}\right)\right|_{m+n}$ for some $q_{01}:Q\left(x_{0}\right)\left(y_{1}\right)$ and $\ell_{2}:\mathtt{glue}(q_{01})=r$ . Noticing that since $r$ is a free point in the type $\mathtt{inl}(x_{0})=\mathtt{inr}(y_{1})$ , we use path induction on $\ell_{2}$ to assume that $r$ is $\mathtt{glue}(q_{01})$ and $\ell_{2}$ is $\mathtt{refl}_{\mathtt{glue}(q_{01})}$ . It then remains to show that, for any $y_{1}:Y$ and any $q_{01}:Q\left(x_{0}\right)\left(y_{1}\right)$ , we have

\mathtt{centre}(\mathtt{inr}(y_{1}))(\mathtt{glue}(q_{01}))=\left|\left(q_{01},\mathtt{refl}_{\mathtt{glue}(q_{01})}\right)\right|_{m+n}.

By the definition of $\mathtt{centre}$ , the left-hand side is

\mathtt{transport}^{\hat{\mathtt{code}}}\left(\mathtt{pair}^{=}(\mathtt{glue}(q_{01}),t)\right)\left(\left|\left(q_{00},\square\right)\right|_{m+n}\right).

This is precisely the case we discussed after Lemma 6.10 (since now $x_{0}$ , $q_{00}$ and $q_{01}$ coincide with $x_{1}$ , $q_{10}$ , $q_{11}$ ), thus it is indeed equal to $\left|\left(q_{01},\square\right)\right|_{m+n}$ . ∎

We have hence completed the proof of Lemma 6.4 and thus Theorem 6.3.

6.5 Consequences

The contractibility of $\mathtt{code}$ in fact implies more than Theorem 6.3, which only uses the case $\mathtt{code}(\mathtt{inr}(y_{1}))$ . Our definition for the other case, $\mathtt{code}(\mathtt{inl}(x_{1}))$ , immediately implies the following:

Proposition 6.13.

$\blacksquare$ Assume the same as in Theorem 6.3. Also let $x_{0}:X$ , $y_{0}:Y$ and $q_{00}:Q\left(x_{0}\right)\left(y_{0}\right)$ . Then for any $x_{1}:X$ , the map

\tau:\lambda q_{10}.\mathtt{glue}(q_{00})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{glue}(q_{10})}^{-1}:Q\left(x_{1}\right)\left(y_{0}\right)\rightarrow\left(\mathtt{inl}(x_{0})=\mathtt{inl}(x_{1})\right)

is $(m+n)$ -connected.

Proof.

For any $s:\left(\mathtt{inl}(x_{0})=\mathtt{inl}(x_{1})\right)$ , $\left\|{\mathtt{fib}_{\tau}(s)}\right\|_{m+n}$ matches our definition for $\mathtt{code}(\mathtt{inl}(x_{1}))(s)$ , which is contractible by 6.11. ∎

Curiously, it is this dual case, instead of Theorem 6.3, that directly gives rise to the Freudenthal suspension theorem as below.

Theorem 6.14 (Freudenthal suspension theorem).

Let $C$ be an $n$ -connected type with $n\geq-1$ and $x_{0}:C$ . Then the map $\sigma:C\rightarrow\Omega(\Sigma C)$ , given by $\sigma:\equiv\lambda x.\mathtt{merid}(x)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{merid}(x_{0})}^{-1}$ , is $2n$ -connected.

Proof.

$\blacksquare$ We rewrite the suspension using the generalised pushout. The suspension $\Sigma C$ is the pushout of the diagram $\mathbb{1}_{N}\overset{f}{\leftarrow}C\overset{g}{\rightarrow}\mathbb{1}_{S}$ , so $X\equiv\mathbb{1}_{N}$ , $Y\equiv\mathbb{1}_{S}$ and $f,g$ are two trivial maps. Then in this case, $Q\left(\star_{N}\right)\left(\star_{S}\right)$ is equivalent to $C$ and under this equivalence the map $\mathtt{glue}_{NS}$ is identified with $\mathtt{merid}$ . The connectivity assumptions are easily seen to be satisfied, so by Proposition 6.13, the map $\sigma^{\prime}:\equiv\lambda x.\mathtt{merid}(x_{0})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{merid}(x)}^{-1}:C\rightarrow\Omega(\Sigma C)$ is $2n$ -connected. Hence, for any $s:\Omega(\Sigma C)$ , the fibre $\mathtt{fib}_{\sigma^{\prime}}(s)$ is $2n$ -connected. Notice that by the definition of fibres and coherence operations,

	$\displaystyle\mathtt{fib}_{\sigma^{\prime}}\left(s^{-1}\right)$	$\displaystyle\equiv\sum_{x:C}\left(\mathtt{merid}(x_{0})\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{merid}(x)}^{-1}=s^{-1}\right)$
		$\displaystyle\simeq\sum_{x:C}\left(\mathtt{merid}(x)\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}{\mathtt{merid}(x_{0})}^{-1}=s\right)\equiv\mathtt{fib}_{\sigma}(s),$

so $\sigma$ is also $2n$ -connected. ∎

Corollary 6.15 (Freudenthal equivalence, [Uni13, Corollary 8.6.14]).

Suppose that type $X$ is $n$ -connected and pointed with $n\geq-1$ . Then $\left\|X\right\|_{2n}\simeq\left\|{\Omega(\Sigma X)}\right\|_{2n}$ .

Proof.

By Theorem 6.14 and Lemma 4.32.∎

We can now complete Table 1.1 as promised.

Corollary 6.16 (Stability for spheres).

If $k\leq 2n-2$ , then $\pi_{k+1}\left({\mathbb{S}}^{n+1}\right)\simeq\pi_{k}\left({\mathbb{S}}^{n}\right)$ .

Proof.

By Example 4.26, ${\mathbb{S}}^{n}$ is $(n-1)$ -connected. By Corollary 6.15 and by the definition of $n$ -spheres, $\left\|{\mathbb{S}}^{n}\right\|_{2(n-1)}\simeq\left\|{\Omega(\Sigma{\mathbb{S}}^{n})}\right\|_{2(n-1)}\equiv\left\|{\Omega({\mathbb{S}}^{n+1})}\right\|_{2(n-1)}$ . Since $k\leq 2(n-1)$ , by Lemma 4.14, $\left\|{\mathbb{S}}^{n}\right\|_{k}\simeq\left\|{\Omega({\mathbb{S}}^{n+1})}\right\|_{k}$ . We then calculate

	$\displaystyle\pi_{k+1}\left({\mathbb{S}}^{n+1}\right)$	$\displaystyle\equiv\left\\|{\Omega^{k+1}\left({\mathbb{S}}^{n+1}\right)}\right\\|_{0}\equiv\left\\|{\Omega^{k}\left(\Omega\left({\mathbb{S}}^{n+1}\right)\right)}\right\\|_{0}$
		$\displaystyle\simeq\Omega^{k}\left\\|{\Omega\left({\mathbb{S}}^{n+1}\right)}\right\\|_{k}\simeq\Omega^{k}\left\\|{\mathbb{S}}^{n}\right\\|_{k}\simeq\left\\|{\Omega^{k}\left({\mathbb{S}}^{n}\right)}\right\\|_{0}\equiv\pi_{k}\left({\mathbb{S}}^{n}\right),$

where we use Proposition 4.13 twice to exchange truncation and $\Omega$ . ∎

Corollary 6.17.

$\pi_{n}\left({\mathbb{S}}^{n}\right)\simeq{\mathbb{Z}}$ for $n\geq 1$ .

Proof.

$\pi_{1}\left({\mathbb{S}}^{1}\right)\simeq{\mathbb{Z}}$ by Corollary 4.21 and $\pi_{2}\left({\mathbb{S}}^{2}\right)\simeq{\mathbb{Z}}$ by Corollary 5.22. When $n\geq 2$ , $n\leq 2(n-1)$ , so by induction on $n$ and Corollary 6.16, the result follows. ∎

Corollary 6.18.

$\pi_{3}\left({\mathbb{S}}^{2}\right)\simeq{\mathbb{Z}}$ .

Proof.

$\pi_{3}\left({\mathbb{S}}^{2}\right)\simeq\pi_{3}\left({\mathbb{S}}^{3}\right)\simeq{\mathbb{Z}}$ by Corollary 5.22 and Corollary 6.17. ∎

Chapter 7 Conclusion and Further Discussions

Table 7.1: Main results and their locations in this dissertation.

Result	Location(s)
Homotopy groups of $n$ -spheres in Table 1.1	Corollaries 4.21, 4.29, 5.22, 6.17, 6.18
Fibre sequence of a pointed map	Corollary 5.8
Long exact sequence of a pointed map	Theorem 5.12
Hopf construction	Theorem 5.18
Hopf fibration	Theorem 5.21
Blakers–Massey theorem	Theorem 6.3
Freudenthal suspension theorem	Theorem 6.14
Stability for spheres	Corollary 6.16

The main results in this dissertation are listed in Table 7.1. We briefly evaluate our synthetic approach to homotopy theory:

•

Formal abstraction allows us to focus on the homotopy properties without considering the underlying set-theoretic structures. For example, unlike in classical topology, the universal cover of ${\mathbb{S}}^{1}$ (see Lemma 4.20) does not involve real numbers $\mathbb{R}$ (a helix projecting onto the circle [Hat02, Theorem 1.7]) but only a fibration with fibre $\mathbb{Z}$ , a primitive notion expressed by a dependent type.
•

All notions are homotopy-invariant. For example, our formation of the Blakers–Massey theorem Theorem 6.3 replaces set intersection and union in classical text [Hat02, Theorem 4.23] with pushouts, leading to a more general statement.
•

Proof relevance has enabled us to effectively reuse pieces of previous results, such as the application of Lemma 6.10 in Section 6.4.
•

The rule-based nature of type theory has rendered many proofs routine applications of induction principles and existing lemmas. Whilst this allows for mechanised reasoning, occasionally proofs lack geometric intuition and appear overly technical. Nevertheless, there are methods, such as [Uni13, Section 8.1.5] for $\pi_{k}({\mathbb{S}}^{1}$ ) and [HFLL16, Section 4] for the Blakers–Massey theorem, that offer easier homotopy-theoretic interpretations.

These further topics are related to our work:

•

The proof of the Blakers–Massey theorem [HFLL16] has inspired a ‘reverse engineered’ classical proof [Rez15] and a generalised version valid in an arbitrary higher topos and with respect to an arbitrary modality [ABFJ20].

•

The Blakers–Massey theorem is used in proving that there is a natural number $n$ such that $\pi_{4}\left({\mathbb{S}}^{3}\right)\simeq{\mathbb{Z}}/n{\mathbb{Z}}$ [Bru16, Proposition 3.4.4].

•

There is a fibration over ${\mathbb{S}}^{4}$ with fibre ${\mathbb{S}}^{3}$ and total space ${\mathbb{S}}^{7}$ by giving an H-space structure on ${\mathbb{S}}^{3}$ [BR16, Theorem 4.10]. In particular, $\pi_{7}\left({\mathbb{S}}^{4}\right)$ has an element of infinite order [BR16, Corollary 4.11].

We also state some speculations based on current work:

•

Chapter 6 has left the proofs on coherence operations unattended. As remarked by [Bru16, p. 23], in general, these proofs can be done via recursive application of path inductions. Here we speculate that there may be automated methods or some ‘meta-theorem’ that reduce the need to spell out these proofs.
•

Our diagrams, combined with the topological interpretations of HoTT, suggest the potential for a systematic approach to HoTT-based graphical reasoning. For example, proof assistants may be equipped with a graphical interface, where the application of rules is visualised through shape transformations. This could potentially offer more intuition and simplify mechanised reasoning.

Appendix A The ‘hub and spoke’ construction of $n$ -truncations

Definition A.1.

Let $A:\mathtt{Type}$ and $n\geq-1$ . Define $\left\|A\right\|_{n}$ , the $n$ -truncation of $A$ , as the higher inductive type generated by two point constructors $|-|_{n}:A\rightarrow\left\|A\right\|_{n}$ and $h:\left({\mathbb{S}}^{n+1}\rightarrow\left\|A\right\|_{n}\right)\rightarrow\left\|A\right\|_{n}$ and a path constructor

s:\left(r:\left({\mathbb{S}}^{n+1}\rightarrow\left\|A\right\|_{n}\right)\right)\rightarrow\left(x:{\mathbb{S}}^{n+1}\right)\rightarrow\left(r(x)=_{\left\|A\right\|_{n}}h(r)\right).

The rationale behind $h$ and $s$ (‘hub and spoke’) in this definition will be clear in the following proof, visualised by Figure A.1.

Figure A.1: Diagram for Lemma A.2.

n

is thought of as

0

and the dashed circle represents the ‘image’ of

{\mathbb{S}}^{n+1}

under

r

Lemma A.2 ([Uni13, Lemma 7.3.1]).

For any $A:\mathtt{Type}$ and $n\geq-1$ , $\left\|A\right\|_{n}$ is an $n$ -type.

Proof.

Take any $b:\left\|A\right\|_{n}$ and let $\left\|A\right\|_{n}$ be pointed by $b$ . By Theorem 4.8, it suffices to prove that $\Omega^{n+1}\left\|A\right\|_{n}$ is contractible. By a corollary of the ‘suspension-loop adjunction’ [Uni13, Lemma 6.5.4], it then suffices to prove that the type $M:\equiv\mathtt{Map}_{\star}\left({\mathbb{S}}^{n+1},\left\|A\right\|_{n}\right)$ is contractible. The constant function $c_{b}:\equiv\lambda x.b$ is a pointed map of type $M$ witnessed by $\mathtt{refl}_{b}$ , and we claim it to be the centre of contraction, i.e., for any pointed function $r:{\mathbb{S}}^{n+1}\rightarrow_{\star}\left\|A\right\|_{n}$ witnessed by $r_{\star}:r\left(\mathtt{base}\right)=b$ , we have $\left(c_{b},\mathtt{refl}_{b}\right)=_{M}\left(r,r_{\star}\right)$ . By Proposition 3.21, it suffices to find a path $q:c_{b}=r$ such that $q_{\ast}\left(\mathtt{refl}_{b}\right)=r_{\star}$ . Using function extensionality, to find $q$ , it suffices to find a path $q_{x}:b=r(x)$ for any $x:{\mathbb{S}}^{n+1}$ . Now using the constructors we have a point $h(r):\left\|A\right\|_{n}$ along with two paths $s(r)(x):r(x)=h(r)$ and $s(r)\left(\mathtt{base}\right):r\left(\mathtt{base}\right)=h(r)$ . Thus the desired path $q_{x}$ is given by

r_{\star}^{-1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,s(r)\left(\mathtt{base}\right)\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,s{(r)(x)}^{-1}.

Finally, by Lemma 3.13, $q_{\ast}\left(\mathtt{refl}_{b}\right)=r_{\star}$ is equivalent to $q_{\mathtt{base}}^{-1}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,\mathtt{refl}_{b}\,\mathchoice{\mathbin{\raisebox{2.15277pt}{$\displaystyle\centerdot$}}}{\mathbin{\raisebox{2.15277pt}{$\centerdot$}}}{\mathbin{\raisebox{1.07639pt}{$\scriptstyle\,\centerdot\,$}}}{\mathbin{\raisebox{0.43057pt}{$\scriptscriptstyle\,\centerdot\,$}}}\,\mathtt{refl}_{b}=r_{\star}$ , which holds by the definition of $q_{\mathtt{base}}$ and coherence operations. ∎

References

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Synthetic Homotopy Theory

Abstract

Acknowledgements

Note

Chapter 1 Introduction

Chapter 2 Dependent type theory

2.1 Function types

2.2 Inductive types

2.2.1 Σ\Sigma-types

2.2.2 Other inductive types

2.3 Identity types

Chapter 3 Homotopy type theory

3.1 The ∞\infty-groupoid structure of types

Lemma 3.1 ([Uni13, Lemma 2.1.1]).

Remark 3.2.

Proof.

Lemma 3.3 ([Uni13, Lemma 2.1.2]).

Proof.

Lemma 3.4 ([Uni13, Lemma 2.1.4]).

Proof.

Definition 3.5.

Example 3.6.

Definition 3.7.

3.2 Dependent paths

Lemma 3.8 ([Uni13, Lemma 2.2.1]).

Lemma 3.9 ([Uni13, Lemma 2.3.1]).

Lemma 3.10 ([Uni13, Lemma 2.3.4]).

Lemma 3.11 ([Uni13, Lemma 2.3.5, Lemma 2.3.8]).

Proposition 3.12 ([Uni13, Lemma 2.3.9]).

Lemma 3.13.

Lemma 3.14 ([Uni13, (2.9.4)]).

Lemma 3.15 ([Uni13, Lemma 2.11.2]).

3.3 Homotopies and equivalences

Definition 3.16.

Definition 3.17.

Proposition 3.18 ([Uni13, Section 2.4]).

Lemma 3.19 ([Uni13, Example 2.4.9]).

Proof.

Definition 3.20.

Proposition 3.21 ([Bru16, Proposition 1.6.8]).

3.4 Univalence axiom and function extensionality

Proposition 3.22 ([Uni13, Lemma 2.10.1]).

Proof.

Axiom 3.23 (Univalence).

Remark 3.24.

Proposition 3.25.

Proof.

Theorem 3.26 (Function extensionality).

3.5 Higher inductive types

3.5.1 Circle

Remark 3.27 ([Uni13, Remark 6.2.4]).

3.5.2 Pushouts

Chapter 4 Homotopy nn-types and nn-connected types

4.1 Contractible types

Definition 4.1.

Example 4.2.

Proposition 4.3.

Proof.

Corollary 4.4 (Path induction, strengthened).

Proof.

Remark 4.5.

Corollary 4.6.

Proof.

4.2 Homotopy nn-types

Definition 4.7.

Theorem 4.8 ([Uni13, Theorem 7.2.9]).

Proposition 4.9 ([Uni13, Theorem 7.1.7]).

4.3 nn-truncations

Proposition 4.10.

Proof.

Proposition 4.11 ([Uni13, Corollary 7.3.7]).

Proof.

Proposition 4.12 ([Uni13, Theorem 7.3.12]).

Proposition 4.13 ([Uni13, Corollary 7.3.14]).

Lemma 4.14 ([Uni13, Lemma 7.3.15]).

4.4 Interlude: homotopy groups of the circle

Definition 4.15.

Remark 4.16.

Definition 4.17.

Example 4.18.

2.2.1 $\Sigma$ -types

3.1 The $\infty$ -groupoid structure of types

Chapter 4 Homotopy $n$ -types and $n$ -connected types

4.2 Homotopy $n$ -types

4.3 $n$ -truncations

4.5 $n$ -connected types

Example 4.25 ( $\blacksquare$ Inspired by [Hou17, Figure 3.4]).

Proposition 4.30 (Induction principle for $n$ -connected types).

Proposition 4.31 (‘Induction principle’ for $n$ -connected maps, [Uni13, Lemma 7.5.7]).

Lemma 5.11 ([Uni13, Theorem 8.4.6] ¹¹1 $\blacksquare$ A few typos in [Uni13, Theorem 8.4.6] are fixed in this proof.).