A Categorical Approach to Möbius Inversion via Derived Functors

For a category $\mathcal{C}$, we denote by $\mathrm{Hom}_{\mathcal{C}}(A,B)$ the set of morphisms from $A$ to $B$. Recall that an adjunction between two categories $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors $F:\mathcal{C}\to\mathcal{D}$ and $G:\mathcal{D}\to\mathcal{C}$, together with natural isomorphisms of sets $\Phi:\mathrm{Hom}_{\mathcal{D}}(F(A),B)\cong\mathrm{Hom}_{\mathcal{C}}(A,G(B))$. In this situation, we say $F$ is left adjoint to $G$ and $G$ is right adjoint to $F$, denoted $F\dashv G$. An important property of left adjoints is that they are cocontinuous, while right adjoints are continuous, meaning left adjoints preserve colimits and right adjoints preserve limits.

A category $\mathcal{C}$ is preadditive if for every set $\mathrm{Hom}_{\mathcal{C}}(A,B)$ there is an abelian group structure, and composition is bilinear, i.e.,

A functor $F:\mathcal{C}\to\mathcal{D}$ between preadditive categories is additive if for all objects $A,B$, the induced map $\mathrm{Hom}_{\mathcal{C}}(A,B)\to\mathrm{Hom}_{\mathcal{D}}(F(A),F(B))$ is a group homomorphism.

A biproduct of objects $A_{1},\ldots,A_{n}$ in a preadditive category $\mathcal{C}$ is an object $A_{1}\oplus\cdots\oplus A_{n}$ with morphisms $\pi_{k}:A_{1}\oplus\cdots\oplus A_{n}\to A_{k}$ and $\iota_{k}:A_{k}\to A_{1}\oplus\cdots\oplus A_{n}$ satisfying:

In this case, $\big{(}A_{1}\oplus\cdots\oplus A_{n},\pi_{k}\big{)}$ is a product, and $\big{(}A_{1}\oplus\cdots\oplus A_{n},\iota_{k}\big{)}$ is a coproduct. A preadditive category is additive if it admits all finite biproducts.

A preadditive category $\mathcal{C}$ is abelian if it has a zero object, all binary biproducts, kernels and cokernels, and every monomorphism and epimorphism is normal. Functors between abelian categories are typically additive, and if $F\dashv G$, both $F$ and $G$ are automatically additive.

Exactness is central in abelian categories. An additive functor $F:\mathcal{C}\to\mathcal{D}$ between abelian categories is:

• •

  left-exact if it preserves exactness of sequences of the form

  $$0\to A\to B\to C,$$

• •

  right-exact if it preserves exactness of sequences of the form

  $$A\to B\to C\to 0.$$

A functor is exact if it is both left-exact and right-exact. If $F\dashv G$, then $F$ is automatically right-exact and $G$ is left-exact.

A monoidal category consists of a category $\mathcal{C}$, a bifunctor $\otimes:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ (called the tensor product), an object ${\mathbf{1}}$ (the unit object), and natural isomorphisms:

These isomorphisms satisfy certain coherence conditions (see [10, VII]). A monoidal category is symmetric if there is a natural isomorphism $s_{A,B}:A\otimes B\cong B\otimes A$ that satisfies its own coherence conditions. A (symmetric) monoidal category $(\mathcal{C},\otimes,{\mathbf{1}})$ is closed if for each $A\in\mathcal{C}$, the functor $-\otimes A:\mathcal{C}\to\mathcal{C}$ has a right adjoint. This right adjoint is denoted ${\mathcal{H}\kern-0.5pt\mathrm{om}}_{\mathcal{C}}(A,-):\mathcal{C}\to\mathcal{C}$ and is called the internal hom.

We will focus on categories that are both abelian and closed symmetric monoidal. A key compatibility condition between the abelian and monoidal structures is that the tensor product must be additive in both variables. When the monoidal structure is closed, this compatibility condition is automatically satisfied because biproducts serve as both limits and colimits, and the tensor product is left adjoint to the internal hom functor.

Henceforth, $\mathcal{C}$ is a small, complete, abelian, and closed symmetric monoidal category.

Fix a functor $S:\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\mathcal{D}$. A wedge of $S$, denoted $e:W\to S$, is an object $W\in\mathcal{D}$ together with maps $e_{A}:W\to S(A,A)$ such that for every $f:A\to B$, the diagram

commutes. The end of $S$, denoted $\int_{A}S(A,A)$, is the universal wedge, meaning any wedge $e:W\to S$ factors uniquely through it. That is, if $e:W\to S$ is any wedge of $S$, then there is a unique map $\phi:W\to\int_{A}S(A,A)$ satisfying $\pi_{A}\circ\phi=e_{A}$, for all objects $A\in\mathcal{C}$.

The end of an ordinary functor $T:\mathcal{C}\to\mathcal{D}$, denoted $\int_{A}T(A)$, is defined by taking the end of the composition $S=\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\stackrel{{\scriptstyle p}}{{\to}}\mathcal{C}\stackrel{{\scriptstyle T}}{{\to}}\mathcal{D}$, where $p$ denotes the projection functor onto the second factor. In this case, the end reduces to the limit of $T$.

Given functors $F,G:\mathcal{C}\to\mathcal{D}$, the set of natural transformations $\mathrm{Nat}(F,G)$ from $F$ to $G$ is given (up to isomorphism) by the formula

This formula and generalizations of it are central to our constructions. If $\mathcal{D}$ is a closed category, then we can replace $\mathrm{Hom}_{\mathcal{D}}(F(A),G(A))$ in the formula above by ${\mathcal{H}\kern-0.5pt\mathrm{om}}_{\mathcal{D}}(F(A),G(A))$ to make $\mathrm{Nat}(F,G)$ an object of $\mathcal{D}$ (see Definition 3.5 below).

An object $I$ in $\mathcal{C}$ is called injective if, for every monomorphism $f:A\hookrightarrow B$ and any morphism $g:A\to I$, there exists a morphism $h:B\to I$ making the following diagram commute:

Arbitrary products of injective objects are injective. Moreover, in an abelian category, finite direct sums (i.e., biproducts) of injective objects remain injective since finite sums coincide with products. However, arbitrary direct sums do not necessarily preserve injectivity.

An abelian category $\mathcal{C}$ is said to have enough injectives if every object $A$ admits a monomorphism $A{\hookrightarrow}I$ into an injective object $I$.

An injective resolution of an object $X$ in $\mathcal{C}$ is an exact sequence

where each $I^{d}$ is an injective object in $\mathcal{C}$.

If $\mathcal{C}$ has enough injectives, every object $X$ admits an injective resolution via the standard construction. Begin by embedding $X$ into an injective object $I_{0}$, and successively embed each cokernel into a new injective object $I_{i+1}$ as follows:

The Grothendieck ring of a category $\mathcal{C}$, denoted $\mathcal{R}(\mathcal{C})$, is the free abelian group generated by the isomorphism classes $[A]$ of objects $A$ in $\mathcal{C}$, subject to the relation $[B]=[A]+[C]$ for every short exact sequence $0\to A\to B\to C\to 0$ in $\mathcal{C}$. The ring structure on $\mathcal{R}(\mathcal{C})$ is induced by the monoidal structure of $\mathcal{C}$, where the product is given by $[A]\cdot[B]=[A\otimes B]$, with $\otimes$ denoting the tensor product in $\mathcal{C}$. The resulting ring $\mathcal{R}(\mathcal{C})$ is commutative and unital, with the class of the unit object $[\mathbf{1}]$ acting as the multiplicative identity: $[A]\cdot[\mathbf{1}]=[A]$. The multiplication is both commutative and associative, reflecting the symmetric monoidal structure of $\mathcal{C}$.

We begin by defining posets and their interpretation as categories, along with the notion of $P$-modules. Principal cofree modules and cofree modules are introduced as building blocks for working with $P$-modules.

Henceforth, we assume all posets are finite.

Having introduced principal cofree modules, we now shift our focus to enriched hom-objects and tensor product constructions for $P$-modules.

This subsection defines enriched hom-objects and describes how the symmetric monoidal structure extends from the base category to the category of $P$-modules. We introduce tensor products of $P$-modules and establish an adjunction between the enriched hom functor and the tensor product functor.

If $\mathcal{C}$ is an abelian category, then the category of $P$-modules, $\mathcal{C}^{P}$, is also abelian. Moreover, if $(\mathcal{C},\otimes,{\mathbf{1}})$ is a symmetric monoidal category, then $(\mathcal{C}^{P},\otimes,{\mathbf{1}})$ inherits a symmetric monoidal structure as follows: for two $P$-modules $M$ and $N$, their tensor product $M\otimes N$ is given by the composition

The unit $P$-module, denoted ${\mathbf{1}}:P\to\mathcal{C}$, assigns the object ${\mathbf{1}}$ to each element of $P$, with identity morphisms acting between them. The coherence conditions for this structure follow directly from those in $\mathcal{C}$.

While we do not require $\mathcal{C}^{P}$ to be closed, we do require it to be enriched over $\mathcal{C}$. Specifically, for each $P$-module $M$, there must be a functor

that is right adjoint to a functor of the form $-\otimes A:\mathcal{C}\to\mathcal{C}^{P}$, for each object $A\in\mathcal{C}$.

Before proving the enriched hom adjunction, we introduce constant $P$-modules. For an object $A\in\mathcal{C}$, denote by $A^{P}:P\to\mathcal{C}$ as the $A$-constant $P$-module. That is $A^{P}(a)=A$, for all $a\in P$, all the morphisms are set to the identity morphism on $A$.

We define principal injective modules and elementary injective modules, which are necessary for constructing injective resolutions for $P$-modules. This section parallels the theory of injective resolutions in the underlying abelian category.

Clearly, principal injective modules are principal cofree modules, and elementary injective modules are cofree. With the concept of elementary injective modules in hand, we now establish that every elementary injective module is injective in the category of $P$-modules.

We show that if the base category has enough injectives, then the category of $P$-modules inherits this property, enabling us to construct injective resolutions for all $P$-modules.

By Proposition 3.12, every $P$-module admits a monomorphism into an elementary injective module, leading to the following corollary.

This subsection introduces Ext-functors, which arise as derived functors of the enriched hom.

Recall Proposition 3.7: for a fixed $P$-module $M$, the tensor product functor

is left adjoint to the enriched hom functor

This adjunction makes $\underline{\mathrm{Hom}}(M,-)$ a left-exact functor between the two abelian categories.

Now, let $N$ be a second $P$-module. Consider an injective resolution

and apply the functor $\underline{\mathrm{Hom}}(M,-)$ to obtain the following cochain complex in $\mathcal{C}$:

The cohomology objects of this complex, denoted by $R^{d}\underline{\mathrm{Hom}}(M,N)$, are the higher derived functors of the enriched hom and are commonly referred to as Ext-functors. These are denoted by

We are particularly interested in the Ext-functor for a specific class of indicator $P$-modules $M$, which we now describe.

As a direct consequence of Propositions 2.1 and 3.8, we have the following corollary.

Henceforth, all Ext-functors will be of the form $\underline{\mathrm{Ext}}^{\ast}({\mathbf{1}}_{Z},N)$, where $Z\subset P$ is a spread.

This subsection focuses on calculating Ext-functors using cofree resolutions. While injective resolutions are theoretically useful for defining the Ext-functor, cofree resolutions offer computational advantages, particularly for explicit calculations.

A cofree resolution of a $P$-module $N$ is an exact sequence in $\mathcal{C}^{P}$:

where each $F^{d}$ is a cofree $P$-module. A specific cofree resolution, called the standard cofree resolution, is particularly useful for computations.

We use $\sigma\lhd_{i}\tau$ to indicate that $\dim\tau-\dim\sigma=i$. For a simplex $\tau=a_{0}<\cdots<a_{d}$, we denote $\min\tau$ as $a_{0}$ and $\max\tau$ as $a_{d}$. If $\sigma\unlhd\tau$, then $\min\tau\leq\min\sigma$ and $\max\sigma\leq\max\tau$.

Now consider an elementary cofree $P$-module $X_{a}^{\downarrow a}$. Since $\mathcal{C}$ has enough injectives, $X_{a}$ admits an injective resolution:

which induces an injective resolution of the elementary $P$-module:

Applying the functor $\underline{\mathrm{Hom}}({\mathbf{1}}_{Z},-)$, and using Propositions 3.8 and 2.1, we obtain, assuming $a\in Z$, the following acyclic cochain complex:

If $a\notin Z$, the resulting chain complex is zero. Thus, $\underline{\mathrm{Ext}}^{d}({\mathbf{1}}_{Z},X_{a}^{\downarrow a})=0$ for all $d>0$. In other words, elementary cofree $P$-modules, and hence cofree $P$-modules in general, are $\underline{\mathrm{Hom}}({\mathbf{1}}_{Z},-)$-acyclic, leading to the following proposition.

Having established how Ext-functors are computed using cofree resolutions, we now define the Euler characteristic as an element of the Grothendieck ring.

The following proposition gives an explicit formula for computing $\chi({\mathbf{1}}_{Z},N)$.

We begin by specializing the notion of indicator modules ${\mathbf{1}}_{Z}$ to singleton sets. For any element $a\in P$, the set $\{a\}\subset P$ is a spread, allowing us to define its indicator module in the same way as for more general indicator $P$-modules. For simplicity, we denote the $P$-module ${\mathbf{1}}_{\{a\}}$ by ${\mathbf{1}}_{a}$.

By Equation 4 and Corollary 4.2, the Möbius cohomology corresponds to the cohomology of the following cochain complex: