Dagger $n$-categories

Categories with an (anti-)involution have been studied since the early days of category theory [ML61, Pup62]. A familiar example is the category of Hilbert spaces and bounded linear maps, with an involution $(-)^{\dagger}$ given by the usual notion of adjoint map. The notion of a complex $*$-category and notions of $\rm C^{*}$- and $\rm W^{*}$-category in the complex linear setting were introduced in [GLR85] and have been studied extensively in the $\rm C^{*}$ tensor setting [FK93, DR89, LR97, Wen98, M0̈3, Yam04].

Roughly, an involution $(-)^{\dagger}$ should reverse the direction of morphisms in a compatible way. In [Sel07], the following notion was introduced in the non-linear setting. A category $\mathcal{C}$ is a dagger category if it is equipped with a bijection $(-)^{\dagger}:\hom(x,y)\to\hom(y,x)$ for each pair $(x,y)$ of objects, such that $(f^{\dagger})^{\dagger}=f$ and $(f\circ g)^{\dagger}=g^{\dagger}\circ f^{\dagger}$ for all composable morphisms $f,g$. In other words, $(-)^{\dagger}$ is a functor $\mathcal{C}\to\mathcal{C}^{\mathrm{op}}$ which is involutive and the identity on (the set of) objects.

This definition is evil in the following sense [Hen15]: since it explicitly references the set of objects in $\mathcal{C}$, dagger structures cannot be readily transported along general equivalences of categories. In detail: given a dagger category $(\mathcal{C},(-)^{\dagger})$ and an equivalence $F:\mathcal{C}\simeq\mathcal{D}$ with (weak) inverse $F^{-1}$, the transported “dagger” functor^‡‡‡A different formula for the transported structure supplies a dagger structure on $\mathcal{D}$ such that $F^{-1}$ is a dagger functor [Kar18, Lemma 2.1.16], but then $F$ will typically not only fail to be a dagger equivalence, it will fail to be compatible with the dagger structures at all. $F\circ(-)^{\dagger}\circ F^{-1}:\mathcal{D}\to\mathcal{D}^{\mathrm{op}}$ is naturally part of a weak involution in the sense that its square is coherently-isomorphic to $\mathrm{id}_{\mathcal{D}}$, but it will almost never be an identity on objects. We violated the general rule of thumb: from a category, one can coherently extract its groupoid of objects up to isomorphism, but not its set of objects. Non-evil notions refer only to coherently-extractable data.

Nevertheless, there is a well-developed “dagger category theory” that parallels the usual theory of categories, with dagger versions of functor, natural transformation, and the like [GLR85, LR97, Yam04, HP17, JP17, Kar18, HK16, HK19, Sri21]. From the perspective of dagger category theory, the coherently-extractable space of objects is the groupoid of objects and unitary isomorphism.

With this in mind, [SS23] developed the following ideas. Consider the $(2,1)$-category $\mathbf{Cat}$ of $1$-categories, functors and natural isomorphisms. This $(2,1)$-category carries a $\mathbb{Z}/2\mathbb{Z}$-action which sends $\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$. An anti-involutive category is a homotopy fixed point for this action: explicitly, it consists of a category $\mathcal{C}$, a categorical equivalence $\dagger:\mathcal{C}\overset{\sim}{\to}\mathcal{C}^{\mathrm{op}}$, and a natural isomorphism $\eta:\dagger^{\mathrm{op}}\circ\dagger\overset{\sim}{\Rightarrow}\mathrm{id}_{\mathcal{C}}$, such that for each object $x\in\mathcal{C}$, $\eta_{x^{\dagger}}^{-1}=(\eta_{x})^{\dagger}$.

If $\mathcal{G}$ is a groupoid, then $\mathcal{G}$ and $\mathcal{G}^{\mathrm{op}}$ are canonically equivalent via the functor that is the identity on objects and sends morphisms to their inverses. In other words, the $\mathbb{Z}/2\mathbb{Z}$-action $\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ on $\mathbf{Cat}$ trivializes when restricted to the full sub-$(2,1)$-category $\mathbf{Gpd}\subset\mathbf{Cat}$ of groupoids. In particular, given an anti-involutive category $\mathcal{C}$, the groupoid $\iota_{0}\mathcal{C}$ of objects of $\mathcal{C}$ inherits a coherent action by $\mathbb{Z}/2\mathbb{Z}$. Let $(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ denote the groupoid of homotopy fixed points for this action.

As a motivating example, consider $\mathcal{C}=\mathbf{Vec}^{\mathrm{fd}}_{\mathbb{C}}$ the category of finite-dimensional vector spaces, and $\dagger:V\mapsto\overline{V}^{\vee}$ the functor that assigns the complex-conjugate dual of a vector space. Then $(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ is the groupoid of finite-dimensional Hermitian vector spaces, i.e. vector spaces equipped with a nondegenerate conjugate-symmetric sesquilinear form which might not be positive-definite, and “unitary” isomorphisms thereof. In order to encode the theory of Hilbert spaces, we must specify extra data: we must mark some of the Hermitian spaces as preferred, and leave out the others. This marking restricts the set of objects of $(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$, but keeps all unitary isomorphisms between them; in other words, the marked subspace $\mathcal{C}_{0}\subset(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ should be a full subgroupoid. If there were a vector space that did not admit a Hilbert structure, then we should have already excised it from $\mathcal{C}$; in other words, the composition $\mathcal{C}_{0}\subset(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}$ should be essentially surjective.

Coherent dagger structures are non-evil: if $\mathcal{C}$ is a dagger category and $\mathcal{C}\cong\mathcal{D}$ an equivalence with an ordinary category, then $\mathcal{D}$ acquires a natural coherent dagger structure. The main theorem of [SS23] says that coherent dagger categories are equivalent in the appropriate sense to dagger categories as traditionally defined. Namely, coherent dagger categories can be strictified to dagger categories as traditionally defined:

Definition 1.1 immediately generalizes to the $(\infty,1)$-world, where it was independently proposed by [Hen22]. An $(\infty,1)$-category is a homotopically-coherent version of “category enriched in spaces.” As is standard in higher category theory, we will interchangeably use the word $\infty$-groupoid and space for “nice topological space considered up to homotopy.” Then an $(\infty,1)$-category has not a set of morphisms between any two objects, but rather a space of morphisms; composition is associative up to parameterized homotopy; this homotopy itself satisfies higher and higher homotopies relating different parenthesizations. There are many ways of axiomatizing the notion of “$(\infty,1)$-category,” and we refer the reader to [Ber10, AC16] for nice surveys.

Ignoring size issues, there is an $(\infty,1)$-category $\mathbf{Cat}_{(\infty,1)}$ whose objects are $(\infty,1)$-categories and whose morphisms are $(\infty,1)$-functors. There is a functor $\iota_{0}\colon\mathbf{Cat}_{(\infty,1)}\to\mathbf{Space}$, corepresented by the terminal $(\infty,1)$-category $\{\mathrm{pt}\}$, which assigns to each $(\infty,1)$-category its space of objects. Moreover, $\mathbf{Cat}_{(\infty,1)}$ carries a canonical $\mathbb{Z}/2\mathbb{Z}$-action sending $\mathcal{C}\mapsto\mathcal{C}^{\mathrm{op}}$ [Toë05]. An anti-involutive $(\infty,1)$-category is a (homotopy) fixed point for this action.

As in the 1-categorical case, the $\mathbb{Z}/2\mathbb{Z}$-action on $\mathbf{Cat}_{(\infty,1)}$ trivializes on the full subcategory $\mathbf{Space}\subseteq\mathbf{Cat}_{(\infty,1)}$. Hence, the space of objects $\iota_{0}\mathcal{C}$ of any anti-involutive $(\infty,1)$-category $\mathcal{C}$ inherits a coherent $\mathbb{Z}/2\mathbb{Z}$ action, whose homotopy fixed points we continue to denote by $(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}.$

Anticipating our $n$-dimensional generalization, let us note that part of the data of a dagger $(\infty,1)$-category is the space $\mathcal{C}_{0}$, the $(\infty,1)$-category $\mathcal{C}$, and the essential surjection $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$. This structure is called a flagged $(\infty,1)$-category. Any specific $1$-category $\mathcal{C}$ in the traditional sense supplies a flagged $(\infty,1)$-category: one takes $\mathcal{C}_{0}$ to be the actual set of objects, whereas $\iota_{0}\mathcal{C}$ is the groupoid of objects and isomorphisms. In other words, flaggings are a way of remembering “sets” of objects in a way that nevertheless transports coherently. The requirement that $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ be essentially surjective simply means that the data of the homomorphisms between elements of $\mathcal{C}_{0}$ suffices to recover all of $\mathcal{C}$ up to categorical equivalence. A flagging is called univalent^§§§This is reminiscent of the completeness condition of Segal spaces. if it does not in fact remember any further data: if the map $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ is not just essentially surjective but also fully faithful.

By analogy, in a dagger $(\infty,1)$-category, the requirement that $\mathcal{C}_{0}\to(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$ be fully faithful will be called the univalence axiom, even though it does not force any map to be an equivalence. And by analogy, dropping the univalence axiom leads to a useful weakening of Definition 1.3.

Our motivating example of a flagged dagger $(\infty,1)$-category is the higher category of bordisms, which we discuss in Example 6.7.

Whether univalent or not, the flagging should be thought of as recording the “identity on objects” condition in the traditional definition of dagger category. In a flagged dagger category which is not univalent, the groupoid $\mathcal{C}_{0}$ selects a notion of equivalence between objects that is finer than unitary equivalence.

Write $\mathbf{Fl{\dagger}Cat}_{(\infty,1)}$ for the $(\infty,1)$-category of flagged dagger $(\infty,1)$-categories: the homomorphisms are the obvious ones which preserve the equivariance and the flaggings. Write ${\dagger}\mathbf{Cat}_{(\infty,1)}$ for the full subcategory of $\mathbf{Fl}{\dagger}\mathbf{Cat}_{(\infty,1)}$ on the univalent ones.

One benefit of Statement 1.5 is that it can be easier to present flagged dagger categories than their univalentizations. Indeed, most $(\infty,1)$-categories $\mathcal{C}$ of interest (for example the bordism categories discussed in Section 6) come with distinguished flaggings $\mathcal{C}_{0}\to\mathcal{C}$, and sometimes this flagged category is naturally anti-involutive in the sense that $\mathcal{C}$ is anti-involutive and $\mathcal{C}_{0}\to\iota_{0}\mathcal{C}$ is $\mathbb{Z}/2\mathbb{Z}$-equivariant. To promote this to a flagged dagger structure then merely requires a trivialization of the $\mathbb{Z}/2\mathbb{Z}$-action on $\mathcal{C}_{0}$. Speaking very approximately, it can be easier to do this when $\mathcal{C}_{0}$ has very few morphisms requiring trivialization data.

Instead of dropping the univalence axiom, we could instead drop the essential surjectivity condition:

Comparing to Definition 1.1, a coflagged dagger $(\infty,1)$-category can be thought of as an anti-involutive $(\infty,1)$-category equipped with the data of Hermitian pairings on some but not necessarily all objects.

Note that every anti-involutive $(\infty,1)$-category is naturally a coflagged dagger $(\infty,1)$-category with $\mathcal{C}_{0}=(\iota_{0}\mathcal{C})^{\mathbb{Z}/2\mathbb{Z}}$. Through this observation, Statement 1.7 specializes to a higher version of the “Hermitian completion” functor of [SS23].

The notion of $(\infty,n)$-category is designed to formalize categories with an $n$-fold hierarchy of directions of composition. As with the $(\infty,1)$-case, there are many models, some of which are surveyed in [BR13, BR20, BSP21]. Informally, an $(\infty,n)$-category has a space of objects; for each pair of objects, a space of $1$-morphisms between them and a coherently-associative and coherently-unital composition law; for each pair of parallel $1$-morphisms, a space of $2$-morphisms between them, and two (coherently-associative and coherently-unital) composition laws, one of which lifts the composition law of $1$-morphisms and the other of which is in the new second dimension; and so on up to dimension $n$.

In this informal description of $(\infty,n)$-categories in the previous paragraph, we did not specify which maps between them should be considered equivalences. There is a most natural guess if one explained the idea without supplying the name “category”: a homomorphism could be declared an equivalence when it induces equivalences on all spaces of $k$-morphisms, including on the space of objects. Comparing with the traditional notion of strict 1-category, this choice would select the strict isomorphisms and not the categorical equivalences. In contrast, categorical equivalence does not remember any specific sets or spaces of $k$-morphisms, but merely the higher groupoids thereof. As in the $(\infty,1)$-case, the extra data of spaces of morphisms in a presentation of an $(\infty,n)$-category is called a flagging of that $(\infty,n)$-category [AF18]. Given a correct theory of $(\infty,n)$-categories, the notion of flagged $(\infty,n)$-category can be defined as follows:

The definition of dagger $(\infty,1)$-category used as an essential ingredient the fact that every $(\infty,1)$-category $\mathcal{C}$ has an opposite $\mathcal{C}^{\mathrm{op}}$ that reverses the direction of composition. Similarly, given an $(\infty,n)$-category $\mathcal{C}$, one can produce new $(\infty,n)$-categories by reversing any of the $n$ directions of composition. This supplies an action of $(\mathbb{Z}/2\mathbb{Z})^{n}$ on $\mathbf{Cat}_{(\infty,n)}$, the $(\infty,1)$-category of $(\infty,n)$-categories. In fact, this action is completely canonical: one of the theorems of [BSP21], generalizing [Toë05], says that this map

is an isomorphism of (higher) groups. Given $1\leq k\leq n$, we will occasionally write $(\mathbb{Z}/2\mathbb{Z})_{k}\subset(\mathbb{Z}/2\mathbb{Z})^{n}$ for the $k$th coordinate $\mathbb{Z}/2\mathbb{Z}$, which acts on $(\infty,n)$-categories by reversing the composition of $k$-morphisms; we will write that action as $\mathcal{C}\mapsto\mathcal{C}^{k\mathrm{op}}$.

As in the 1-category case, volutive structures do not capture the theory of daggers: they do not include an analogue of the requirement that “dagger is the identity on objects.” To encode the latter, we use flaggings. To set up the definition, we note the following. Every $(\infty,k)$-category is in particular an $(\infty,k+1)$-category, and the inclusion $\mathbf{Cat}_{(\infty,k)}\hookrightarrow\mathbf{Cat}_{(\infty,k+1)}$ is stable under the ambient $(\mathbb{Z}/2\mathbb{Z})^{k+1}$-action. Indeed, the first $k$ involutions $(\mathbb{Z}/2\mathbb{Z})^{k}\subset(\mathbb{Z}/2\mathbb{Z})^{k+1}$ act on $\mathbf{Cat}_{(\infty,k)}$ via the canonical action, and the last involution $(\mathbb{Z}/2\mathbb{Z})_{k+1}$ has a canonical trivialization. Recall that a fixed point for the trivial action of a group $G$ on a category $\mathcal{X}$ is precisely an action of $G$ on some object $X\in\mathcal{X}$. Thus if $\mathcal{C}$ is an $(\infty,k)$-category thought of as an $(\infty,k+1)$-category, then a fully-volutive structure in the $(\infty,k+1)$-sense is precisely a fully-volutive structure in the $(\infty,k)$-sense together with a coherently-compatible $\mathbb{Z}/2\mathbb{Z}$-action. In particular, by picking the trivial $\mathbb{Z}/2\mathbb{Z}$-action, any fully-volutive $(\infty,k)$-category can be thought of as a fully-volutive $(\infty,k+1)$-category.

We remind the reader that, in the higher-categorical world, requesting that “this is a map of these things” is requesting for extra structure on the map. In the case at hand, this structure can be unpacked as follows. A map $\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ is the same as a map $\mathcal{C}_{k}\to\iota_{k}\mathcal{C}_{k+1}$. The fully-volutive structure on $\mathcal{C}_{k+1}$ induces a fully-volutive structure on $\iota_{k}\mathcal{C}_{k+1}$ in the $(\infty,k)$-sense together with a typically-nontrivial action of $\mathbb{Z}/2\mathbb{Z}$. The requested structure unpacks to a map of fully-volutive $(\infty,k)$-categories $\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathbb{Z}/2\mathbb{Z}}$. Comparing with Definitions 1.1 and 1.3, we exactly recover the “flagged dagger categories”; in particular, the “essential surjectivity” axiom is enforced by asking that $\mathcal{C}_{0}\to\mathcal{C}_{1}\to\dots\to\mathcal{C}_{n}$ be a flagged $(\infty,n)$-category independent of the volutive, but we do not have any univalence axiom. We add that univalence axiom now.

Beware that a fully-dagger $(\infty,n)$-category is not the same as a flagged fully-dagger $(\infty,n)$-category of which the underlying flagging is univalent.

Definitions 2.3 and 2.4 refer only to the successive inclusions $\mathcal{C}_{k}\to(\iota_{k}\mathcal{C}_{k+1})^{\mathbb{Z}/2\mathbb{Z}}\to\mathcal{C}_{k+1}$. But they imply more general conditions, and these more general conditions will be needed when studying groups $G$ other than $(\mathbb{Z}/2\mathbb{Z})^{n}$.

We will write $G{\dagger}\mathbf{Cat}_{(\infty,n)}$ for the $(\infty,1)$-category of $G$-dagger $(\infty,n)$-categories. The most interesting case for examples is when $G=(\mathbb{Z}/2\mathbb{Z})_{n}$, in which case we will also refer to $(\mathbb{Z}/2\mathbb{Z})_{n}$-dagger $(\infty,n)$-categories as top-dagger. More generally, the cases that arise in examples are when $G\subset(\mathbb{Z}/2\mathbb{Z})^{n}$ is a product of some of the coordinate $(\mathbb{Z}/2\mathbb{Z})_{k}$s. For such a $G$, Definition 2.6 simplifies: since in this case the maps $\prod_{j={k+1}}^{l}G(\{j\})\to G(\{k+1,\dots,l\})$ are all isomorphisms, one can restrict to just the successive inclusions $\mathcal{C}_{k}\to\mathcal{C}_{k+1}$ in the definition and invoke a $G$-version of Statement 2.5. We have suggested a more general definition in anticipation of our discussion of unitary duality in Section 5.

Let $\mathcal{V}$ be a symmetric monoidal $(\infty,1)$-category, with monoidal structure $\otimes_{\mathcal{V}}\colon\mathcal{V}\times\mathcal{V}\to\mathcal{V}$ and monoidal unit $1_{\mathcal{V}}\in\mathcal{V}$. There is a notion of $\mathcal{V}$-enriched $(\infty,1)$-category, which we will abbreviate as $\mathcal{V}$-category. A full definition is provided by [GH15]; we will recall the main ingredients. A flagged $\mathcal{V}$-category $\mathcal{C}$ consists of a space $\mathcal{C}_{0}$ of objects and, for each $x,y\in\mathcal{C}_{0}$, an object $\hom(x,y)\in\mathcal{V}$, together with a coherently-associative and coherently-unital composition law $\hom(y,z)\otimes_{\mathcal{V}}\hom(x,y)\to\hom(x,z)$. The global sections functor $\Gamma:=\hom(1_{\mathcal{V}},-)\colon\mathcal{V}\to\mathbf{Space}$ is lax-symmetric-monoidal, and so $\mathcal{C}$ induces a flagged plain $(\infty,1)$-category $\Gamma\mathcal{C}$ with the same space $\mathcal{C}_{0}$ of objects, and with spaces of morphisms given by $\Gamma\hom(-,-)$. The flagged $\mathcal{V}$-category $\mathcal{C}$ is called univalent if $\Gamma\mathcal{C}$ is univalent. By definition, a $\mathcal{V}$-category is a univalent flagged $\mathcal{V}$-category. We will write $\mathbf{Cat}[\mathcal{V}]$ for the $(\infty,1)$-category of $\mathcal{V}$-categories. The primordial example: $\mathbf{Cat}[\mathbf{Cat}_{(\infty,n-1)}]\cong\mathbf{Cat}_{(\infty,n)}$.

Using the symmetry on $\mathcal{V}$, for each $\mathcal{V}$-category $\mathcal{C}$ it should be possible to define an opposite $\mathcal{V}$-category $\mathcal{C}^{\mathrm{op}}$ with the same objects and morphisms but the opposite order of composition. This supplies an involution $\mathbb{Z}/2\mathbb{Z}\to\operatorname{Aut}(\mathbf{Cat}[\mathcal{V}])$. We can immediately generalize Definition 1.3:

We will write ${\dagger}\mathbf{Cat}[\mathcal{V}]$ for the $(\infty,1)$-category of dagger $\mathcal{V}$-categories. We expect that the equivalence $\mathbf{Cat}[\mathbf{Cat}_{(\infty,n-1)}]\cong\mathbf{Cat}_{(\infty,n)}$ generalizes to dagger categories:

In particular, by iterating Statement 3.2, one arrives at an alternative model of fully-dagger $(\infty,n)$-category than the one given in Definition 2.4. Mixing $\mathbf{Cat}[-]$ and ${\dagger}\mathbf{Cat}[-]$ provides alternative models of the versions with dagger structures on only some levels:

For example, iterating the second of these equivalences supplies an alternative model for top-dagger $(\infty,n)$-categories.

In this section, we will elaborate on how Definitions 2.4 and 2.6 play out in the case of bicategories. Our goal is to outline a strictification for coherent dagger bicategories analogous to Statement 1.2 for $1$-categories. Even though our definition of a fully-dagger bicategory unpacks to something complicated, it can be strictified to a “more traditionally defined” fully-dagger bicategory involving less data (but more “evil”):

Unpacking Definition 4.1, it is an enriched-type definition in the sense of Section 3, so that Statement 3.2 for $n=2$ gives a justification for the hope that bi-involutive bicategories are a model for fully dagger bicategories. In other words, similarly to how a bicategory is a category weakly enriched in categories, a bi-involutive bicategory is a $\dagger$-category ($\dagger_{1}$) weakly enriched in $\dagger$-categories ($\dagger_{2}$).

We decided on the name “bi-involutive bicategory” because it generalizes the notion of a bi-involutive tensor category of [HP17] to a bicategory with more than one object.

In the rest of this section, we provide some ideas for a proof of the following strictification result, which is a bicategorical analogue of the main theorem of [SS23].

Firstly, a fully-volutive structure on a bicategory $\mathcal{B}$ consists of a pair of equivalences $\psi_{1}\colon\mathcal{B}\to\mathcal{B}^{1\mathrm{op}}$ and $\psi_{2}\colon\mathcal{B}\to\mathcal{B}^{2\mathrm{op}}$, together with natural isomorphisms $\Omega_{1}\colon\psi_{1}^{1\mathrm{op}}\circ\psi_{1}\to\mathrm{id}_{\mathcal{B}}$, $\Omega_{2}\colon\psi_{2}^{2\mathrm{op}}\circ\psi_{2}\to\mathrm{id}_{\mathcal{B}}$, and $\Omega_{12}\colon\psi_{2}^{1\mathrm{op}}\circ\psi_{1}\to\psi_{1}^{2\mathrm{op}}\circ\psi_{2}$, and various modifications which satisfy various coherence conditions. To provide a fully-dagger structure on this volutive bicategory, we need to understand the relevant fixed point categories and specify our flaggings.

The bigroupoid $(\iota_{0}\mathcal{B})^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$ has objects consisting of an element of $\mathcal{B}$ equipped with fixed point data for both $\psi_{1}$ and $\psi_{2}$ together with compatibility data between them, satisfying various coherence conditions. Similarly to the $1$-categorical case, this in particular consists of isomorphisms $h_{b}^{1}\colon\psi_{1}(b)\to b$ and $h_{b}^{2}\colon\psi_{2}(b)\to b$. The space $\mathcal{B}_{0}$ in the definition of a dagger bicategory can be identified with a subspace of $(\iota_{0}\mathcal{B})^{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}}$ such that the map to $\mathcal{B}$ is essentially surjective. This picks out at least one preferred $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$-fixed point structure for every object $b\in\mathcal{B}$.

The bicategory $\iota_{1}\mathcal{B}^{\mathbb{Z}/2\mathbb{Z}}$ has objects given by an element of $\mathcal{B}$ equipped with fixed point data for $\psi_{2}$. The 1-morphisms consist of pairs of a 1-morphism $f\colon b\to b^{\prime}$ in $\mathcal{B}$ and 2-isomorphisms

satisfying a natural coherence condition. One can think of this $2$-morphism as the data specifying how $f$ is a unitary $1$-morphism. The bicategory $\mathcal{B}_{1}$ is equivalent to a subcategory of $\iota_{1}\mathcal{B}^{\mathbb{Z}/2\mathbb{Z}}$, whose objects can be identified with those of $\mathcal{B}_{0}$, since $\mathcal{B}_{0}\to\mathcal{B}_{1}$ is essentially surjective. The 1-morphisms pick out at least one preferred $\mathbb{Z}/2\mathbb{Z}$-fixed point structure on every 1-morphism. The 2-morphisms are fixed by the condition that the map to $\iota_{1}\mathcal{B}^{\mathbb{Z}/2\mathbb{Z}}$ is fully faithful on 2-morphisms. Note how this additionally fixes the 1-morphisms of $\mathcal{B}_{0}$: they are exactly those whose fixed point structure restricts to one of the chosen fixed point structures in $\mathcal{B}_{1}$. In summary, the structure of a dagger bicategory is equivalent to a fully-volutive bicategory $\mathcal{B}$ together with a choice of at least one $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$-fixed point on every object $b\in\mathcal{B}$ and at least one compatible $\mathbb{Z}/2\mathbb{Z}$-fixed point on every 1-morphism.

From this data we can construct a bi-involutive bicategory (analogous to the construction in [SS23]) as follows. Consider the bicategory $\mathcal{B}^{\prime}$ of which the objects are the objects of $\mathcal{B}_{0}$ and 1-morphisms the 1-morphisms of $\mathcal{B}_{1}$. The 2-morphisms are those of $\mathcal{B}$, requiring no compatibility with the fixed point data. By construction, there is a forgetful functor $\mathcal{B}^{\prime}\to\mathcal{B}$, which is an equivalence of bicategories. The anti-involutions $\psi_{1}$ and $\psi_{2}$ induce compatible anti-involutions on $\mathcal{B}^{\prime}$.^‡‡‡‡‡‡We expect the equivalence $\mathcal{B}^{\prime}\to\mathcal{B}$ to come equipped with a canonical datum saying it preserves the fully-volutive structures. Namely, $\dagger_{1}$ is defined by the same formula as in the $1$-categorical case:

We do not spell out the fixed point structure (2) on the $1$-morphism $f^{\dagger_{1}}$ here. There is a natural isomorphism $\phi:\dagger_{1}^{2}\to\mathrm{id}_{\mathcal{B}^{\prime}}$ which is the identity on objects and uses the fixed point data on $b$ and $b^{\prime}$ to get a $2$-isomorphism $f^{\dagger_{1}\dagger_{1}}\cong f$ for a $1$-morphism $f:b\to b^{\prime}$. The top-dagger $\dagger_{2}\colon\mathcal{B}^{\prime}\longrightarrow\mathcal{B}^{\prime 2\mathrm{op}}$ is the identity on objects and 1-morphisms. It sends a 2-morphism $\vartheta\colon f\Rightarrow g$ to a version of $\psi_{2}(\vartheta)$, where we identify its domain and target with $g$ and $f$ respectively using their fixed point data. The functor $\dagger_{2}$ strictly squares to the identity. The two daggers commute strictly, and $\phi$ is $\dagger_{2}$-unitary. We have thus constructed a canonical bi-involutive bicategory $\mathcal{B}^{\prime}$ from the coherent full $\dagger$ bicategory $\mathcal{B}$.

An $(\infty,n)$-category $\mathcal{C}$ is said to have adjoints, if for each $1\leq k\leq n-1$ and each $k$-morphism $f:x\to y$ (between parallel $(k-1)$-morphisms $x,y$), there exist $k$-morphisms $f^{R},f^{L}:y\to x$ and $(k+1)$-morphisms $\eta^{R}\colon\mathrm{id}_{x}\to f^{R}\circ f$, $\epsilon^{R}\colon f\circ f^{R}\to\mathrm{id}_{y}$, $\eta^{L}\colon\mathrm{id}_{y}\to f\circ f^{L}$, $\epsilon^{L}\colon f^{L}\circ f\to\mathrm{id}_{x}$ such that the zig-zag compositions, which after suppressing coherence (e.g. unitor and associator) information become

are equivalent to identities. Let $\mathbf{AdjCat}_{(\infty,n)}\subset\mathbf{Cat}_{(\infty,n)}$ denote the full sub-$(\infty,1)$-category on the $(\infty,n)$-categories with adjoints.

The theory of $(\infty,n)$-categories with adjoints has been well-studied, but there are many questions remaining. The most famous work pertains to the symmetric monoidal case, in which case dualizability is also imposed on objects. We will call a symmetric monoidal category rigid if it has adjoints and also admits duals for objects. The Cobordism Hypothesis of [BD95, Lur09b] asserts that the free rigid symmetric monoidal $(\infty,n)$-category generated by a single object is the $(\infty,n)$-category $\mathbf{Bord}_{n}^{\mathrm{fr}}$ of framed $n$-dimensional bordisms — a framing $\tau$ on a smooth $n$-manifold $M$ is a trivialization of its tangent bundle $\tau:\mathrm{T}_{M}\cong\mathbb{R}^{n}$. In other words, if $\mathcal{C}$ is a rigid symmetric-monoidal $(\infty,n)$-category, with space of objects $\iota_{0}\mathcal{C}$, then there is a canonical equivalence

The group $\mathrm{O}(n)$ obviously acts on the framings on a given $n$-manifold, by rotating the trivialization, and so acts on the $(\infty,n)$-category $\mathbf{Bord}_{n}^{\mathrm{fr}}$. This in turn supplies a famous action of $\mathrm{O}(n)$ on $\iota_{0}\mathcal{C}$ for any rigid symmetric monoidal $(\infty,n)$-category $\mathcal{C}$.

In fact, there is a larger group that acts, as explained in [Lur09b, Remark 2.4.30]. It makes sense to talk about a “tangent bundle” of a piecewise linear (a.k.a. PL) manifold, but it is not a vector bundle: whereas the tangent bundle of a smooth $n$-manifold $M$ is classified by a map $\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{O}(n)\simeq\mathrm{B}\mathrm{Diff}(n)$, a tangent bundle of a PL manifold is classified by a map $\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$. In particular, it makes sense to talk about framed PL manifolds: they are PL manifolds equipped with a trivialization of $\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$. By construction, any smoothing of a PL manifold lifts $\mathrm{T}_{M}:M\to\mathrm{B}\mathrm{PL}(n)$ through $\mathrm{B}\mathrm{Diff}(n)\simeq\mathrm{B}\mathrm{O}(n)$. We now quote a nontrivial fact of differential topology, called the Main Theorem of Smoothing Theory:

An immediate corollary is that a framed PL manifold has a unique (up to a contractible space) smoothing that is compatible with the framing. In particular, the framed bordism categories built from smooth or from PL manifolds are equivalent. But $\mathrm{PL}(n)$ acts by rotating the framings of $n$-dimensional PL-manifolds, and so acts on the PL version of $\mathbf{Bord}^{\mathrm{fr}}_{n}$, and so acts on the space $\iota_{0}\mathcal{C}$. This is the largest group that acts universally on the objects of rigid $(\infty,n)$-categories:

The proof of Statement 5.2 has circulated among experts, but is not available in print.

The story in the absence of symmetric monoidal structures is less well-studied. The Tangle Hypothesis and Cobordism Hypothesis with Singularities amount to the existence of a graphical calculus for $(\infty,n)$-categories with adjoints, generalizing the graphical calculi described for example in [Sel11]. The precise details of this graphical calculus have not been worked out in the literature. Roughly, the main ingredients are the following:

1. (a)

   The diagrams in the graphical calculus are drawn on networks of submanifolds of the standard $\mathbb{R}^{n}$.

2. (b)

   Strata of codimension $k$ are labeled by $k$-morphisms.

3. (c)

   Strata are normally framed: if $X\subset\mathbb{R}^{n}$ is a codimension-$k$ submanifold, then it comes with a trivialization $\nu:\mathrm{N}_{X}\cong\mathbb{R}^{k}$ of its normal bundle. More generally, substrata of strata are relatively normally framed. This normal framing encodes source and target information (and so must be consistent with the labelings).

4. (d)

   Strata are tangentially framed: if $X\subset\mathbb{R}^{n}$ is a codimension-$k$ submanifold, then it comes with a trivialization $\tau:\mathrm{T}_{X}\cong\mathbb{R}^{n-k}$ of its tangent bundle. This tangential framing encodes the direction of composition internal to the morphism.

5. (e)

   The various framing data are compatible. For example, if $X\subset\mathbb{R}^{n}$ is a non-sub stratum, it must come equipped with a nullhomotopy of the composite isomorphism

   $$\mathbb{R}^{n}=\mathrm{T}_{\mathbb{R}^{n}}=\mathrm{N}_{X}\oplus\mathrm{T}_{X}\xrightarrow{\nu\oplus\tau}\mathbb{R}^{k}\oplus\mathbb{R}^{n-k}=\mathbb{R}^{n},$$

   where the left-hand equality simply uses that $X$ is a submanifold of the standard $\mathbb{R}^{n}$. For substrata $Y\subset\dots\subset X\subset\mathbb{R}^{n}$, there are similar but more complicated compatibility conditions.

When $n=2$, one finds the well known calculus of “string diagrams.” The framing-compatibility is probably the least familiar component of this calculus. Along a 1-dimensional stratum, it is a nullhomotopy of an element of $\mathrm{O}(2)$. That element is nullhomotopic only when it lives in $\mathrm{SO}(2)$. This forces the normal and tangential framings to determine each other. But there is more to a nullhomotopy than just its existence: an element of $\mathrm{SO}(2)$ has a $\mathbb{Z}$-torsor of nullhomotopies. Explicitly, the framing compatibility consists of a “winding number” carried by each wire. This axiom prevents closed circles; it is what allows the string diagram calculus to apply to non-pivotal 2-categories.

To say that $(\infty,n)$-categories with adjoints admit some graphical calculus is to say that $(\infty,n)$-categories with adjoints are precisely “interpreters” for such a graphical calculus. Suppose we are given such an interpreter. Here is another interpreter: precompose your interpreter with some element of $\mathrm{O}(n)$ acting on all input diagrams. Thus, assuming that there is indeed such a graphical calculus, one finds an action of $\mathrm{O}(n)$ on $\mathbf{AdjCat}_{(\infty,n)}$. We could act by any element of $\mathrm{Diff}(n)$; since $\mathrm{O}(n)\hookrightarrow\mathrm{Diff}(n)$ is a homotopy equivalence, this supplies “the same” action. But we were not precise about what regularity of manifolds are allowed. Statement 5.1 and its corollary about unique smoothings for framed PL manifolds says that one might as well work with a PL diagrams, and these are in any case more natural to draw. As such, we find an action of $\mathrm{PL}(n)$ on $\mathbf{AdjCat}_{(\infty,n)}$. Given Statement 5.2, we expect:

For the remainder of this article, we will assume part (a) of Conjecture 5.3, and we will be motivated by our belief in part (b). We emphasize that, not only is the current literature far from a proof of part (b), it does not even supply a rigorous construction of this map in part (a) except when $n$ is very low: