ON THE COHOMOLOGY RINGS OF SMALL CATEGORIES

FEI XU

Department of Mathematical Sciences University of Aberdeen Aberdeen AB24 3UE Scotland, U.K.

Abstract. Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.

Keywords. Cohomology ring, EI-categories, finite generation, restriction, extension, generalized LHS spectral sequence.

1. Introduction

Let C be a small category and Ab the category of abelian groups. We denote by Cmod the abelian category of all covariant functors from C to Ab. The nth cohomology group of C with coefficients in a functor F ∈ C-mod, H^{n}(C; F), can be defined as the nth higher inverse limit \underleftarrow{lim}_{C}^{n} F. If A is an abelian group and \underline{A} is the corresponding constant functor which sends every object to A and every morphism to the identity, then H^{n}(C; \underline{A}) ≅ H^{n}(|C|, A), where |C| is the topological realization of NC–the nerve of C. We are particularly interested in the case where A is a commutative ring with identity, because then H^{∗}(C; \underline{A}) ≅ H^{∗}(|C|, A) will become a graded commutative ring. To this end, we will study the cohomology of C with coefficients in a functor F : C → R-mod for a commutative ring with identity R (not just the ring of integers ℤ). We call H^{∗}(C; \underline{R}) ≅ H^{∗}(|C|, R) the cohomology ring of C (with coefficients in R).

For each small category C and a fixed ring R, one can define an associative Ralgebra, called the category algebra RC of C (see Xu [38]), generalizing the notion of the group algebra of a group and the notion of the incidence algebra of a poset. If Ob C is finite, Mitchell [26] showed that RC-mod ≃ (R-mod)^{C}, which implies that every functor is an RC-module, and vise versa. This equivalence allows us to use representation theory of finite-dimensional algebras to investigate homological properties of functor categories. Since all categories here in this paper are assumed to have finite object sets, we will just denote by RC-mod the category of all covariant functors from C to R-mod. Upon this equivalence of categories, H^{n}(C; F) can be identified with Ext_{RC}^{n}(\underline{R}, F) and H^{∗}(C; \underline{R}) ≅ Ext_{RC}^{∗}(\underline{R}, \underline{R}). For the reader’s convenience, in the appendix we will give more information about cohomology of small categories and, more generally, Ext and Tor over category algebras.

Partially supported by a British EPSRC grant GR/S94667/01 through a research fellowship at Aberdeen.

Current address: Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 Rue de la Houssinière, 44322 Nantes, France. E-mail : xu@math.univ-nantes.fr.

When a finite group G is regarded as a category with a single object (written as Ĝ), Evens [10] and Venkov [34] proved that H^{∗}(Ĝ; \underline{R}) ≅ H^{∗}(G, R) is finitely generated when R is Noetherian. This is one of the fundamental theorems in group cohomology theory, and is the starting point of the theory of varieties for modules [4]. Other categories whose cohomology rings are finitely generated include finite posets and centric linking systems (when they exist) in the theory of p-local finite groups established by Broto, Levi and Oliver [7]. Motivated by these known results, one wants to know whether the finite generation property is true in greater generality. Since many categories considered in group representation and cohomology theory are the so-called EI-categories and are finite, it is natural to focus our investigation on the special class of finite EI-categories. By definition, a category is finite if Mor C is a finite set and is EI if every endomorphism is an isomorphism. Given an EI-category, it’s easy to see that the isomorphism classes of objects in it possesses a natural poset structure. Hence to some extent a finite EI-category may be regarded as an amalgam of a finite poset and several finite groups. In fact, Słomińska [32] showed that the classifying space of C, |C|, is homotopy equivalent to the homotopy colimit of some functor from a finite poset to the category of finite groupoids. Since cohomology rings of finite groups and finite posets are finitely generated, one wishes to generalize the finite generation property of cohomology rings to finite EI-categories. The finite generation property of such a cohomology ring would be useful in computing higher limits over a category. For example, in the theory of p-local finite groups [7], in order to know the existence and uniqueness of a centric linking system for a given fusion system F over a p-group S, one needs to calculate the higher limits of the central functor Z over a full subcategory F^{c} ⊂ F, which sends a subgroup of S to its center. A characterization of H^{∗}(F^{c}; \underline{ℤ}), where ℤ is the ring of integers, would be useful in computing \underleftarrow{lim}_{F^{c}}^{∗} Z because the cohomology ring acts on it. The finite generation property of cohomology rings with coefficients in a field would also lead us to a theory of varieties for modules over category algebras. We note that since the category algebra RC for a finite EI-category is not a Hopf algebra in general, the finite generation property doesn’t follow from the main theorem of Friedlander-Suslin [12].

It turns out that the cohomology ring of a finite EI-category is not finitely generated in general, even after modulo nilpotents. The reader can find an example in Section 2 of the present paper. However, it would be of great interest if the finite generation could be proved for a certain subclass of finite EI-categories that includes the important cases in group representations and cohomology (for example, one may want to consider the concrete categories [27] whose morphisms are monomorphic).

In Section 2, we compute the cohomology rings of some finite EI-categories. The examples exhibit the failure of the finite generation of the cohomology ring in general, even after modulo nilpotents. Then we continue to study the cohomology of finite EI-categories with the goal of extending some classical results in group cohomology. In Section 3, given a finite EI-category C, we show that the cohomology ring of C modulo some nilpotents is isomorphic to a subring of the cohomology ring of a disjoint union of finite groups. Furthermore we prove that, for a fixed prime p, there exist certain subcategories such that the cohomology ring of C modulo nilpotents can be embedded into the cohomology ring of such a subcategory modulo nilpotents.

Theorem A Let C be a finite EI-category and D a subcategory. Suppose p is a prime. If Ob D = {x ∈ Ob C : p | | Aut_{C}(x)|} and Aut_{D}(x) contains a Sylow p-subgroup of Aut_{C}(x), then the map induced by the restriction H^{∗}(C; \underline{𝔽_{p}})/N_{C} → H^{∗}(D; \underline{𝔽_{p}})/N_{D} is injective, where N_{C} and N_{D} are the ideals of nilpotents in H^{∗}(C; \underline{𝔽_{p}}) and H^{∗}(D; \underline{𝔽_{p}}), respectively.

We note that for any finite EI-category C, a subcategory D satisfying the conditions in Theorem A always exists. At last in Section 4, we consider the extensions of small categories and deduce a generalized LHS spectral sequence. We introduce the concepts of opposite extensions and subextensions.

Proposition B Given a functor F : C → R-mod, there are two spectral sequences associated with an extension K → E → C as follows:

(1) a homology spectral sequence

E_{ij}^{2} = H_{i}(C; H_{j}(K; F)) ⇒ H_{i+j}(E; F);

and

(2) a cohomology spectral sequence

E_{2}^{ij} = H^{i}(C^{op}; H^{j}(K^{op}; F)) ⇒ H^{i+j}(E^{op}, F).

Based on the second spectral sequence, we can find connections between the cohomology ring of a category and those of its extensions.

Theorem C Suppose there is an extension K → E → C. If ι_{D} : D ↪ C is the inclusion such that the undercategory ι_{D} ↓^{y} is contractible for every y ∈ Ob C, then H^{∗}(E; F) ≅ H^{∗}(E_{D}; F) for any contravariant functor F : E → R-mod. Here E_{D} ⊂ E is the subextension corresponding to D.

Suppose there is an opposite extension K → E → C. If ι_{D} : D ↪ C is the inclusion such that overcategory ι_{D} ↓_{y} is contractible for every y ∈ Ob C, then H^{∗}(E; F) ≅ H^{∗}(E_{D}; F) for any covariant functor F : E → R-mod. Here E_{D} ⊂ E is the opposite subextension corresponding to D.

Conventions We use curly letters, such as C, to denote small categories in this paper. Symbols like D and E are used to denote subcategories and extensions of C, respectively. When G is a group, we use Ĝ to denote the corresponding category with a single object. For instance, let x be an object of C. We frequently refer to its automorphism group Aut_{C}(x) via the corresponding subcategory { \atop Aut_{C}(x)} ⊂ C. For each x ∈ Ob C, we denote by [x] ⊂ Ob C the set of all objects isomorphic to x in C. We also use ˆ[x] ⊂ C to denote the groupoid which consists of all objects in [x] and all isomorphisms among them. The symbol A is reserved for the subcategory of C which is the disjoint union of all these groupoids ˆ[x], [x] ⊂ Ob C.

For each category C, we use the corresponding blackboard bold letter ℂ to denote the naturally constructed chain complex from its nerve. For simplicity, when R is understood, sometimes we omit R or \underline{R} in the cohomology ring of a category C and will write it as H^{∗}(C) instead of H^{∗}(|C|, R) or H^{∗}(C; \underline{R}). A category C is said to be contractible if |C| is.

We mainly work with finite EI-categories in this paper. Given an EI-category C, there is a natural poset structure defined on the set of isomorphism classes of objects in C. For any [x], [y] ⊂ Ob C, we define [x] ≤ [y] if Hom_{C}(x, y) ≠ ∅. The resulting poset is written as [C], and there is a canonical functor C → [C]. Based on the existence of the partial order on the set of isomorphism classes of objects, for any x ∈ Ob C, we can define a full subcategory C_{≤x} ⊂ C such that Ob C_{≤x} = {y ∈ Ob C| Hom_{C}(y, x) ≠ ∅}. We can similarly define full subcategories such as C_{<x}, C_{≥x} and C_{>x} et cetera.

Acknowledgements The work started when I was doing my PhD in the University of Minnesota. I wish to thank my advisor Peter Webb, as well as Aurélien Djament, Vincent Franjou, Hans-Werner Henn, Ran Levi, Markus Linckelmann, Bob Oliver and Laurent Piriou for many useful conversations. Especially Bob informed me about some of Słomińska’s results directly related to the present paper. Based on that information, Aurélien, Laurent and I were able to find an example against the finite generation conjecture on cohomology rings of finite EI-categories. Hans-Werner was able to produce another example soon after a conversation with me. I would like to express my gratitude to Aurélien and Laurent for allowing me to include the example here. Finally I thank the referee for helpful suggestions.

2. The cohomology ring modulo nilpotents

In this section, we first recall the main construction and theorem in Słomińska’s paper [32]. Using her decomposition formula for classifying spaces of categories, we are able to determine the homotopy types of the classifying spaces of certain categories. Then we investigate the structures of the cohomology rings of these categories. Our main examples says that the cohomology ring of a finite EI-category modulo nilpotent is not necessarily finite generated.

The materials in this section are pretty much independent of the general cohomology theory of small categories. For those who wants to know more about the general theory, please read the appendix.

2.1. Decompositions of classifying spaces. Let C be a finite EI-category. Let π : C → [C] be the natural functor from C to its underlying poset. According to Słomińska’s main result in [32], one has a homotopy equivalence

|C| ≃ hocolim_{sd[C]}|π̃|,

where sd means subdivision and π̃ : sd[C] → sCats, the category of small categories, is a functor whose values are groupoids. For the reader’s convenience, we recall the definition of π̃. Assume v. = [c_{0}→^{v_{1}}c_{1}→^{v_{2}} ⋯ →^{v_{n}}c_{n}] is an object of sd[C]. Then π̃(v.) is a small category whose objects are of the form x_{0}→^{α_{1}}x_{1}→^{α_{2}} ⋯ →^{α_{n}}x_{n} such that π(x_{i}) = c_{i} and π(α_{i}) = v_{i} and the morphisms are the (n + 1)-tuples of isomorphisms in C which make the following diagram commutative:

\xymatrix{x_{0} \ar[r]^{α_{1}} \ar[d]^{g_{0}} & x_{1} \ar[r]^{α_{2}} \ar[d]^{g_{2}} & ⋯ \ar[r] & x_{n−1} \ar[d]^{g_{n−1}} \ar[r]^{α_{n}} & x_{n} \ar[d]^{g_{n}} \\ x′_{0} \ar[r]^{α′_{1}} & x′_{1} \ar[r]^{α′_{2}} & ⋯ \ar[r] & x′_{n−1} \ar[r]^{α_{n}} & x′_{n} & .}

2.2. Categories with two objects. We want to compute the cohomology rings of a special class of finite EI-categories of the following form

[figure: x; G; G/H; y; K] ,

in which the automorphism groups are two finite groups G and H. The morphisms from x to y are given by the set of left cosets G/H for some subgroup H ⊂ G. We let G act on Hom_{C}(x, y) in the natural way and K act on it trivially. Then Słomińska’s formula gives an explicit form of |C| up to homotopy equivalence. In fact, the homotopy colimit becomes the homotopy pushout of the following diagram

\xymatrix{BH × BK \ar[r] \ar[d] & BK \\ BG & ,}

where the two maps are projections followed by inclusions. A short calculation will show that the homotopy pushout is the join of BG and BK, wedge sum the mapping cone of the inclusion BH ↪ BG. Hence |C| ≃ (BG ∗ BK) ∨ (BG/BH), where ∗ denotes the join of two spaces (see [15], [11]). Since the cup product in H^{∗}(BG ∗ BK) is trivial, up to nilpotents the structure of H^{∗}(C) is completely determined by that of H^{∗}(BG/BH).

As an example, when G = H in the above category, the homotopy pushout is simply the join of BG and BK. Hence the cup product of any two elements of positive degree in H^{∗}(|C|) is trivial and H^{∗}(|C|) modulo nilpotents is the base ring. More generally, one may describe the chomology ring structure of the following category C_{n}, n > 1,

[figure: x_{1}; G_{1}; α_{1}; x_{2}; G_{2}; α_{2}; ⋯; α_{n−1}; x_{n}; G_{n}] ,

where {α_{i}}·G_{i} = {α_{i}} = G_{i+1}·{α_{i}} as sets for all i. If the structures of the cohomology rings of G_{i} are provided, then one can write down explicitly the structure of H^{∗}(C). For those who are not familiar with the join construction, a direct computation of the cohomology ring of C_{n} (when G_{i} ≅ ℤ_{2}, the cyclic group of order 2) with coefficients in a field of characteristic 2 can be found in the appendix.

Next we consider an explicitly constructed category C as follows.

[figure: x; 1_{x}; g; k; gk; α; β; y; {1_{y}}] ,

where g^{2} = k^{2} = 1_{x}, gk = kg, αk = α, αg = β, βk = β and βg = α. If we name G = Aut_{C}(x) and H = {1_{x}, k}, up to homotopy equivalence |C| is the homotopy pushout of the following diagram

\xymatrix{BH × B\{1_{y}\} \ar[r] \ar[d] & B\{1_{y}\} \\ BG & .}

Hence the classifying space |C| ≃ BG/BH. One can compute the cohomology of BG/BH via the relative cohomology ring H^{∗}(BG, BH). In fact, H^{∗}(BG, BH) can be identified with the (maximal) ideal of H^{∗}(BG/BH) consisting of all positive degree elements. Let k be a field of characteristic 2. Then one can use the long exact sequence for relative cohomology, along with the well-known structures of the mod-2 cohomology rings of elementary 2-groups, to establish a ring isomorphism

H^{∗}(BG, BH) ≅ H^{∗}(ℤ_{2}) ⊗_{k} H̃^{∗}(ℤ_{2}),

in which H̃^{∗}(ℤ_{2}) is the reduced cohomology ring. The ring on the right has no nontrivial nilpotents and it is not finitely generated since H̃^{∗}(ℤ_{2}) doesn’t have a unit. Finally we comment that the non finite generation of the relative group cohomology ring was first shown by Blower [1]

3. The cohomology ring modulo nilpotents and the restriction

Although the cohomology ring of a finite EI-category modulo nilpotents is not finitely generated in general, it would be very useful if one could prove the finite generation for certain special classes of finite EI-categories in group representation and cohomology theory. This is still an ongoing research project.

In Sections 3 and 4, we try to generalize some classic results in group cohomology.

3.1. Comparing the cohomology of a category with those of its subcategories. Let C be a small category and D a subcategory. Then the inclusion ι : D → C naturally induces the restriction H^{∗}(C) → H^{∗}(D). We want to compare the cohomology rings of C and of its various subcategories. In [7], [18], [14] and [38] the authors studied the case where D ⊂ C is a full subcategory which has fewer objects, and showed under certain assumptions one can have H^{∗}(C) ≅ H^{∗}(D). Here we investigate subcategories D ⊂ C with the same set of objects but with fewer morphisms. Our approach is quite elementary.

Let’s consider the subcategory A ⊂ C consisting of all objects and exactly all isomorphisms in C. Then the chain complexes associated with the nerves of these two categories give a short exact sequence of complexes

0 → 𝔸→^{i}ℂ→^{π}ℂ/𝔸 → 0,

where i is the inclusion and π is the natural surjection. Note that for any abelian group A, H^{∗}(C; \underline{A}) = H^{∗}(ℂ, A) and H^{∗}(A; \underline{A}) = H^{∗}(𝔸, A) ≅ ⊕_{[x]⊂Ob C} H^{∗}(Aut_{C}(x), A), where [x] is the isomorphism class of an object x ∈ Ob C. Suppose D ⊂ C is a subcategory such that Ob D = Ob C and 𝔻 is the chain complex associated with the nerves of D. Let D ∩ A ⊂ A be the obvious subcategory consisting of objects and morphisms in both D and A. Then we have the following commutative diagram

\xymatrix{0 \ar[r] & 𝔸 \ar[r]^{i} & ℂ \ar[r]^{π} & ℂ/𝔸 \ar[r] & 0 \\ 0 \ar[r] & 𝔸_{𝔻} \ar[r]_{i} \ar[u]_{ι} & 𝔻 \ar[r]_{π} \ar[u]_{ι} & 𝔻/𝔸_{𝔻} \ar[r] \ar[u]_{κ} & 0,}

where 𝔸_{𝔻} is 𝔸 ∩ 𝔻, the chain complex of D ∩ A. Let A be an abelian group. This diagram gives rise to an infinite commutative diagram with a typical segment as follows

\xymatrix{H^{n−1}(𝔸, A) \ar[r] \ar[d]_{ι^{∗}} & H^{n}(ℂ/𝔸, A) \ar[r]^{π^{∗}} \ar[d]_{κ^{∗}} & H^{n}(ℂ, A) \ar[r]^{i^{∗}} \ar[d]_{ι^{∗}} & H^{n}(𝔸, A) \ar[r] \ar[d]_{ι^{∗}} & H^{n+1}(ℂ/𝔸, A) \ar[d]_{κ^{∗}} \\ H^{n−1}(𝔸_{𝔻}, A) \ar[r] & H^{n}(𝔻/𝔸_{𝔻}, A) \ar[r]_{π^{∗}} & H^{n}(𝔻, A) \ar[r]_{i^{∗}} & H^{n}(𝔸_{𝔻}, A) \ar[r] & H^{n+1}(𝔻/𝔸_{𝔻}, A),}

which induces a commutative diagram for each n

\xymatrix{H^{n}(C; \underline{A})/I_{C}^{n} \ar[r]^{i′} \ar[d]_{ι′} & H^{n}(A; \underline{A}) \ar[d]^{ι^{∗}} \\ H^{n}(D; \underline{A})/I_{D}^{n} \ar[r]_{i′} & H^{n}(A_{D}; \underline{A}),}

where ι′ is induced by ι^{∗} : H^{n}(C; \underline{A}) → H^{n}(D; \underline{A}), I_{C}^{n} = π^{∗} H^{n}(ℂ/𝔸, A) and I_{D}^{n} = π^{∗} H^{n}(𝔻/𝔸_{𝔻}, A) satisfying ι^{∗}(I_{C}^{n}) = π^{∗}κ^{∗}(H^{n}(ℂ/𝔸, A)) ⊂ π^{∗}(H^{n}(𝔻/𝔸_{𝔻}, A)) = I_{D}^{n}. By the exactness of the long exact sequences, both i′ are injective.

Remark 3.1.1. When C is a finite group, I_{C}^{n} and I_{D}^{n} vanish and the map ι′ coincides with the usual restriction map.

We show, when A = R is a ring, I_{C} = ⊕I_{C}^{n} and I_{D} = ⊕I_{D}^{n} are nilpotent ideals in H^{∗}(C) and H^{∗}(D), respectively. Thus we have a commutative diagram of rings, not just of groups, with injective horizontal homomorphisms.

Proposition 3.1.2. Let C be a finite category. The image of H^{∗}(ℂ/𝔸), denoted by I_{C} = ⊕_{n≥1}I_{C}^{n}, is an ideal of H^{∗}(C) consisting of nilpotents, and i^{∗}(H^{∗}(C)) = i′(H^{∗}(C)/I_{C}) ≅ H^{∗}(C)/I_{C} is a subalgebra of H^{∗}(A).

Proof. The first observation comes from the long exact sequence induced by

0 → 𝔸 →^{i} ℂ →^{π} ℂ/𝔸 → 0,

⋯ → H^{n−1}(𝔸) → H^{n}(ℂ/𝔸) →^{π^{∗}} H^{n}(ℂ) →^{i^{∗}} H^{n}(𝔸) → H^{n+1}(ℂ/𝔸) → · · · .

It’s a canonical result that i^{∗} induces a ring homomorphism, still denoted by i^{∗}. The image I_{C} = π^{∗}(H^{∗}(ℂ/𝔸)) ⊂ ⊕_{n≥1} H^{n}(ℂ) because H^{0}(ℂ/𝔸) = 0. It implies i^{∗}(H^{∗}(C)) contains the identity in H^{∗}(A). Since I_{C} is the kernel of the ring homomorphism i^{∗}, it’s an ideal in H^{∗}(C). Furthermore elements of I_{C} are nilpotent because the category C is finite. If k is the maximum length of chains of non-isomorphisms in C, then for any f ∈ I_{C}, we must have f^{k+1} = 0 by direct calculations using the definition of cup product. □

Note that a subalgebra of a finitely generated algebra doesn’t have to be finitely generated.

In the commutative diagram we considered, we can artificially add an upwards map, which is the transfer, pictured as follows

\xymatrix{H^{∗}(C)/I_{C} \ar[d]^{ι′} \ar[r]^{incl.} & H^{∗}(A) \ar[d]^{ι^{∗}} \\ H^{∗}(D)/I_{D} \ar[r]_{incl.} \ar@{.>}[u]^{?} & H^{∗}(A_{D}). \ar[u]^{tr}}

Although it’s unclear whether or not the transfer restricts to a well-defined map H^{∗}(D)/I_{D} → H^{∗}(C)/I_{C}, the composite tr ∘ι^{∗} : H^{∗}(A) → H^{∗}(A) is given by

α = (α_{x})_{[x]⊂Ob C} ↦ tr ∘ι^{∗}(α) = (n_{x}α_{x})_{[x]⊂Ob C},

by a standard result in group cohomology theory, where α = (α_{x})_{[x]⊂Ob C} is an element of H^{∗}(A) = ⊕_{[x]⊂Ob C} H^{∗}(Aut_{C}(x)) such that for each [x] ⊂ Ob C α_{x} ∈ H^{∗}(Aut_{C}(x)) and n_{x} := | Aut_{C}(x) : Aut_{D}(x)|.

Corollary 3.1.3. Let n_{C} be the least common multiple of the integers {| Aut_{C}(x)|}, where x runs over the set of isomorphism classes of objects whose automorphism groups have order not invertible in R. Then, for any i > 0 and α ∈ H^{i}(C)/I_{C}, n_{C}α = 0.

Remark 3.1.4. If A is an abelian group and \underline{A} is the corresponding constant functor, usually H^{∗}(C; \underline{A}) doesn’t have a ring structure. Hence the preceding proposition makes no sense in this case. However, the restriction H^{∗}(C; \underline{A}) → H^{∗}(A; \underline{A}) is interesting in its own right.

Let G be a finite group, k an algebraic closed field of characteristic p and F a fusion system of a block of the group algebra kG, see Linckelmann [24]. Let \underline{k^{×}} be the constant functor from F to Ab, sending every object in F to the multiplicative group k^{×} of k. People are interested in the cohomology groups of the full subcategory F^{c} ⊂ F, consisting of F-centric subgroups, and its quotient category (orbit category of F^{c}) F̄^{c}. In particular, one wants to understand the restrictions H^{2}(F^{c}; Z) → H^{2}(A_{F^{c}}; Z) and H^{2}(F̄^{c}; \underline{k^{×}}) → H^{2}(AF̄^{c}; \underline{k^{×}}), because they may provide important information for studying Alperin’s Conjecture in modular representation theory of finite groups, see Linckelmann [21] Section 4.1. There is a commutative diagram

\xymatrix{H^{2}(F̄^{c}; \underline{k^{×}}) \ar[r]^{ι^{∗}} \ar[d]_{π^{∗}} & H^{2}(AF̄^{c}; \underline{k^{×}}) \ar[d]^{π^{∗}} \\ H^{2}(F^{c}; \underline{k^{×}}) \ar[r]^{ι^{∗}} & H^{2}(A_{F^{c}}; \underline{k^{×}})}

where the maps are induced by the functors π : F^{c} → F̄^{c}, ι : A_{F^{c}} → F^{c} and ι : AF̄^{c} → F̄^{c}.

3.2. Restriction to subcategories with fewer isomorphisms. We assume that Ob D = Ob C and D contains all non-isomorphisms in Mor(C). We show there exist subcategories D of C such that D shares the same objects and non-isomorphisms with C, while for every x ∈ Ob C = Ob D the automorphism group Aut_{D}(x) is a designated subgroup (up to isomorphism) of Aut_{C}(x).

Lemma 3.2.1. Let C be a finite EI-category. For each [y] ⊂ Ob C we fix a representative x ∈ [y] and choose a subgroup H_{x} ⊂ Aut_{C}(x). Then there exists a subcategory D such that D contains all objects and non-isomorphisms in C and such that Aut_{D}(z) is isomorphic to H_{x} if z ≅ x.

Suppose for different choices of sets of representatives x′ and corresponding subgroups H_{x′} ⊂ Aut_{C}(x′), x′ ∈ [y] ⊂ Ob C, there is a D′ with the same properties as D. Then D′ ≅ D, if H_{x} = gH_{x′}g^{−1} for some g ∈ Is_{C}(x′, x) whenever [x] = [x′].

Proof. Since we want to keep all objects and all non-isomorphisms, the key to finding such a subcategory is to find a set of isomorphisms in C such that they can form Mor(D) along with the non-isomorphisms. In fact, we need to construct for each isomorphism class [x] ⊂ Ob C a subgroupoid of [x] whose skeleton is isomorphic to the chosen Ĥ_{x}.

We proceed in two steps. First of all, we want to find a unique isomorphism α_{ij} ∈ Hom_{C}(x_{i}, x_{j}) for every pair of objects in [x] = {x_{1}, ⋯, x_{i}, ⋯, x_{j}, ⋯ } such that this set of isomorphisms is closed under composition, and such that α_{ii} = 1_{x_{i}} and α_{ij} = α_{ji}^{−1}. Let’s do this by induction on the number of objects in [x]. When [x] only has one or two objects, the construction is trivial. When there are three objects in [x], say x_{1}, x_{2}, x_{3}, we can choose arbitrary α_{12} ∈ Hom_{C}(x_{1}, x_{2}), α_{13} ∈ Hom_{C}(x_{1}, x_{3}) and then define α_{23} = α_{13}α_{12}^{−1}. Suppose we have constructed such sets for all isomorphism classes with less than n objects. For an isomorphism class of objects in C with n objects, we fix a set of compatible isomorphisms for the subgroupoid consisting of any n − 1 objects, say x_{1}, ⋯, x_{n−1}. Then any choice of an isomorphism α_{1,n} ∈ Hom_{C}(x_{1}, x_{n}) can be used to get a compatible set of isomorphisms for [x], by composing 1_{x_{n}}, α_{1,n} and its inverse with existing isomorphisms among x_{1}, ⋯, x_{n−1}.

Second of all, without loss of generality, for a fixed object x_{1} = x ∈ [x], we let H_{x_{1}} = H_{x}. Then for each x_{i} ∈ [x], we define H_{x_{i}} to be α_{1i}H_{x_{1}}α_{1i}^{−1}. These groups of automorphisms, along with the compatible set of isomorphisms {α_{ij}} define a subgroupoid in [x] whose skeleton is isomorphic to Ĥ_{x}.

Let x run over the set of isomorphism classes of objects in C. Then we get a collection of isomorphisms from subgroupoids constructed above. These isomorphisms, together with all non-isomorphisms in C, form the morphism set of a subcategory D satisfying our conditions.

Now let’s prove the second half of the lemma. We want to show with the given property there exists a g_{y} ∈ Aut_{C}(y) for each y ∈ Ob C such that such that ^{g_{y}} Aut_{D′}(y) = Aut_{D}(y). If this is true, then we can define a functor τ : D′ → D by setting τ(y) = y for each object y, and τ(α) = g_{z}^{−1}αg_{y} if α ∈ Hom_{D′}(y, z). It is easy to verify that τ is an isomorphism of categories.

Let {x′} be a different set of representatives of objects and {H_{x′}} the corresponding chosen subgroups. We just proved there exists a D′ satisfying conditions in the first part of our statement. Without loss of generality we can assume {x′} = {x} and then use new symbols {H′_{x}}, instead of {H_{x′}}, for the set of subgroups different from {H_{x}}. Suppose H_{x} = g_{x}H′_{x}g_{x}^{−1} for some g_{x} ∈ Aut_{C}(x), any chosen x. We need to find such an element for every object in C, not just the given representatives {x}. Let’s take a compatible set of isomorphisms {α_{ij}^{x}} for each [x] ⊂ D and another compatible set {β_{ij}^{x}} in D′. Then for any x_{i}, x_{j} ∈ [x], we must have Aut_{D}(x_{j}) = α_{ij}^{x} Aut_{D}(x_{i})α_{ji}^{x} and Aut_{D′}(x_{j}) = β_{ij}^{x} Aut_{D′}(x_{i})β_{ji}^{x}. Assume x = x_{1}. Since Aut_{D}(x) = H_{x} = g_{x}H′_{x}g_{x}^{−1} = g_{x} Aut_{D′}(x)g_{x}^{−1}, for each x_{i} ∈ [x] we can find g_{x_{i}} = α_{1i}^{x}g_{x}β_{i1}^{x} ∈ Aut_{C}(x_{i}) satisfying g_{x_{i}} Aut_{D′}(x_{i})g_{x_{i}}^{−1} = Aut_{D}(x_{i}). Hence we’re done. □

In particular, for a fixed prime p we can choose Aut_{D}(x) to be a Sylow p-subgroup of Aut_{C}(x), where x runs over the set of isomorphism classes in Ob C. We’ll use C_{p} to denote a representative of such subcategories, because they are isomorphic to each other.

Corollary 3.2.2. Suppose p is a prime and C is a finite EI-category. Then there exists a unique subcategory C_{p} up to isomorphism such that Ob C_{p} = Ob C, C_{p} contains all non-isomorphisms in C and for each object x Aut_{C_{p}}(x) is a Sylow p-subgroup of Aut_{C}(x).

One may call each C_{p} a Sylow p-subcategory of C since if D is a subcategory such that it shares all objects and all non-isomorphisms with C and such that Aut_{D}(x) is a p-subgroup of Aut_{C}(x), for each x ∈ Ob C, then D is contained in a C_{p}. To some extent, C_{p} plays the role of a Sylow p-subgroup of a finite group. If we consider the chain complexes of categories C, A, C_{p} and A_{p} = C_{p} ∩ A, we’ll have

\xymatrix{0 \ar[r] & 𝔸 \ar[r]^{i} & ℂ \ar[r]^{π} & ℂ/𝔸 \ar[r] & 0 \\ 0 \ar[r] & 𝔸_{p} \ar[r]_{i} \ar[u]_{ι} & ℂ_{p} \ar[r]_{π} \ar[u]_{ι} & ℂ_{p}/𝔸_{p} \ar[r] \ar[u]_{κ} & 0,}

and hence

\xymatrix{H^{∗}(C)/I_{C} \ar[r]^{i′} \ar[d]_{ι′} & H^{∗}(A) \ar[d]^{ι^{∗}} \\ H^{∗}(C_{p})/I_{C_{p}} \ar[r]_{i′} & H^{∗}(A_{p}).}

The next result follows from the standard fact that the restriction ι^{∗} : H^{∗}(A; \underline{𝔽_{p}}) → H^{∗}(A_{p}; \underline{𝔽_{p}}) is injective.

Proposition 3.2.3. Let C be a finite EI-category and p a prime. Then the map

ι′ : H^{∗}(C; \underline{𝔽_{p}})/I_{C} → H^{∗}(C_{p}; \underline{𝔽_{p}})/I_{C_{p}}

is an injection, where I_{C} = π^{∗} H^{∗}(ℂ/𝔸) and I_{C_{p}} = π^{∗} H^{∗}(ℂ_{p}/𝔸_{p}) are two nilpotent ideals.

At present, it’s not clear that whether or not ι′ is an isomorphism if the other restriction ι^{∗} : H^{∗}(A, 𝔽_{p}) → H^{∗}(A_{p}, 𝔽_{p}) is an isomorphism.

3.3. Restriction to subcategories with fewer non-isomorphisms. We assume that Ob D = Ob C and D and C have the same set of isomorphisms. Then the commutative diagram of short exact sequences reads as follows

\xymatrix{0 \ar[r] & 𝔸 \ar[r]^{i} & ℂ \ar[r]^{π} & ℂ/𝔸 \ar[r] & 0 \\ 0 \ar[r] & 𝔸 \ar[r]_{i} \ar[u]_{Id} & 𝔻 \ar[r]_{π} \ar[u]_{ι} & 𝔻/𝔸 \ar[r] \ar[u]_{κ} & 0.}

It induces a commutative diagram

\xymatrix{H^{∗}(C)/I_{C}^{∗} \ar[r]^{i′} \ar[d]_{ι′} & H^{∗}(A) \ar[d]^{Id} \\ H^{∗}(D)/I_{D}^{∗} \ar[r]_{i′} & H^{∗}(A),}

where i′ is induced by i^{∗}, ι′ is induced by ι^{∗} : H^{∗}(C) → H^{∗}(D), I_{C}^{∗} = π^{∗} H^{∗}(ℂ/𝔸) and I_{D}^{∗} = π^{∗} H^{∗}(𝔻/𝔸) are two nilpotent ideals satisfying ι^{∗}(I_{C}^{∗}) = π^{∗}κ^{∗}(H^{∗}(ℂ/𝔸)) ⊂ π^{∗}(H^{∗}(𝔻/𝔸)).

Proposition 3.3.1. Let C be a finite EI-category. Assume there exists a subcategory D ⊂ C such that Ob D = Ob C and Aut_{D}(x) = Aut_{C}(x) for each x ∈ Ob C. Then ι′ : H^{∗}(C)/I_{C} → H^{∗}(D)/I_{D} is injective for any base ring R.

Proof. It follows directly from Proposition 3.1.2 and the commutative diagram above it. □

As a simple example, one can verify that if the orders of all automorphism groups in C are invertible in the base ring R, then H^{∗}(C)/I_{C} = R. If for some x ∈ Ob C, | Aut_{C}(x)|^{−1} ∈ R, then the isomorphism class [x] can be “dropped” due to the following result. Let x ∈ Ob C. We define C_{x} to be the full subcategory of C consisting of all objects but those isomorphic to x. Corollary 3.3.2. Let C be a finite EI-category. Suppose x ∈ Ob C such that | Aut_{C}(x)| is invertible in R. Then H^{∗}(C)/N_{C} → H^{∗}(C_{x})/N_{C_{x}} is injective.

Proof. The subcategory D = C_{x} ⊔ ˆ[x] (a disjoint union) obviously satisfies the condition in our previous proposition. Thus we have an injective map ι′ : H^{∗}(C)/I_{C} → H^{∗}(D)/I_{D}. Since H^{∗}(D) = H^{∗}(C_{x}) ⊕ H^{∗}(ˆ[x]) and H^{∗}(ˆ[x]) vanishes at any positive degree, pr ∘ ι′ : H^{∗}(C)/I_{C} → H^{∗}(C_{x})/I_{D} is an injective homomorphism, where pr is the natural projection H^{∗}(D)/I_{D} → H^{∗}(C_{x})/I_{D}. Hence the statement follows. □

Thus to some extent, when considering mod p cohomology one can focus on categories whose automorphism groups of objects are p-groups. Combining Propositions 3.2.3 and 3.3.1 and Corollary 3.3.2, we have the following main result. Theorem 3.3.3. Let C be a finite EI-category and D a subcategory. Suppose p is a prime. If Ob D = {x ∈ Ob C : p | | Aut_{C}(x)|} and Aut_{D}(x) contains a Sylow p-subgroup of Aut_{C}(x), then the map induced by the restriction H^{∗}(C; \underline{𝔽_{p}})/N_{C} → H^{∗}(D; \underline{𝔽_{p}})/N_{D} is injective, where N_{C} and N_{D} are the ideals of nilpotents in H^{∗}(C; \underline{𝔽_{p}}) and H^{∗}(D; \underline{𝔽_{p}}), respectively.

A nontrivial example can be found in the theory of p-local finite groups, see [6], [7] and [20] for a complete description of results stated in the next example. Example 3.3.4. Let (S, F, L) be a p-local finite group [7] with finite covering data (see [20]) (θ, Γ, H), where Γ is a p-group or a p′-group, θ : S → Γ is a group homomorphism and H ⊂ Γ is a subgroup. Define S_{H} = θ^{−1}(H). Then there exists a fusion system F_{H} over S_{H} ([6] Proposition 3.8) and a p-local finite group (S_{H}, F_{H}, L_{H}) ([6] Theorem 3.9) such that |L_{H}| is a covering space of |L|. Let Γ = H = 1. Then S_{1} = S and L_{1} ⊂ L satisfies the condition that, for any P ∈ Ob L = Ob L_{1}, Aut_{L_{1}}(P) contains a Sylow p-subgroup of Aut_{L}(P) ([6] Proposition 3.8 c). By the preceding theorem, we get an injection H^{∗}(L)/N_{L} → H^{∗}(L_{1})/N_{L_{1}}.

On the other hand, Levi-Ragnarsson [20] showed that in this case one can construct a transfer map for p-local finite groups which composes with the restriction is the identity: H^{∗}(L) → H^{∗}(L_{1}) → H^{∗}(L). Especially, it implies Res : H^{∗}(L) → H^{∗}(L_{1}) is injective hence another proof of the injectivity of the induced map H^{∗}(L)/N_{L} → H^{∗}(L_{1})/N_{L_{1}}.

We conclude this section with a couple of final remarks. Let C be a small category. One can define the subdivision S(C) of C, which is homotopy equivalent to C (see for example [22]). The subdivision is a category with subobjects [28], so especially all morphisms are monomorphic. Thus when studying cohomology rings of small categories one can just focus on categories whose morphisms are monomorphic. We comment that in the theory of p-local finite groups [7], as well as in local representation theory [33], many categories have the property that either all morphisms are monomorphic or all of them are epimorphic.

4. Extensions of categories and the generalized LHS spectral sequences

In last section we considered certain subcategories of a category C and the relationship between the cohomology rings of those subcategories and of C. In this section, we do the “converse” and consider the cohomology rings of extensions of C, which have the same sets of objects but have “larger” morphism sets. At the beginning we go over some basic knowledge about extensions of categories. The reader is referred to Hoff [17] or Webb [35] for more information about extensions of categories.

Given a functor μ : E → C, Gabriel and Zisman [13] showed there exists a spectral sequence converging to the homology of E. When the category E is an extension of C, we deduce a spectral sequence, based on Gabriel and Zisman’s, as a generalization of the LHS spectral sequence for group extensions. It will be used to compare cohomology rings of E and C.

4.1. Extensions of categories. An extension E of a category C via a category K, in the sense of Hoff [17], is a sequence of functors

K→^{ι}E→^{π}C,

which has the following properties:

(1) Ob K = Ob E = Ob C, ι is injective and π is surjective on morphisms;

(2) if π(α) = π(β), for two morphisms α, β ∈ Mor(E), if and only if there is a unique g ∈ Mor(K) such that β = ι(g)α;

(3) if αι(h) exists for α ∈ Mor(E) and h ∈ Mor(K), then there exists a unique h′ ∈ Mor(K) such that ι(h′)α = αι(h);

(4) for any α ∈ Hom_{C}(x, y), K(y) acts regularly on π^{−1}(α). Note that (3) and (4) can be deduced from (1) and (2) which are normally used to define an extension of a category. It’s known by Hoff [17] that K is a disjoint union of the groups π^{−1}(1_{x}) for all 1_{x} ∈ Mor(C) (regarded as categories), and can be identified with a functor K : E → Groups. Usually from the context, one can easily see when we take K to be a category and when it is regarded as a functor.

An extension is split if it admits a functor s : C → E such that π ∘ s = 1_{C}. In this case, E is a Grothendieck construction [35].

For future reference, we define an opposite extension E of C via K to be a sequence of functors K → E → C such that the following sequence is an extension of C^{op}

K^{op} → E^{op} → C^{op}.

When it won’t cause any confusion, we’ll just say K → E → C is an opposite extension of C.

Example 4.1.1. (1) Let G be a finite group and p a prime dividing the order of G. A collection C of p-subgroups of G is a set of p-subgroups which is closed under conjugations in G. The transporter category Tr_{C}(G) (see [9]) is an extension of the orbit category O_{C}(G)

K_{s} → Tr_{C}(G) → O_{C}(G),

where K_{s}(H) = H ⊂ Aut_{Tr_{C}(G)}(H) = N_{G}(H);

(2) Let (S, F, L) be a p-local finite group [7]. Then Z→^{δ}L→^{π}F^{c} is an opposite extension of F^{c}, where F^{c} ⊂ F is a full subcategory consisting of all F-centric subgroups and Z(P) = Z(P) is the so-called central functor.

4.2. Generalized Lyndon-Hochschild-Serre spectral sequence. Gabriel and Zismann ([13] Appendix II Theorem 3.6) gave a homology spectral sequence for a functor between two categories π : E → C

E_{ij}^{2} = H_{i}(C; H_{j}(π ↓_{?}; F)) ⇒ H_{i+j}(E; F),

where F ∈ RE-mod is a functor from E to R-mod.

Since we are more interested in cohomology, we also write out the cohomology version of their spectral sequence

E_{2}^{ij} = H^{i}(C^{op}; H^{j}(π ↓_{?}; F)) ⇒ H^{i+j}(E^{op}; F),

where F ∈ RE^{op}-mod, or a contravariant functor E → R-mod. When K → E → C is an opposite extension, we prove the cohomology groups of the overcategory with coefficients in F, H^{∗}(π ↓_{y}; F), can be identified with H^{∗}(K(y), F (y)) by a formula of Jackowski-Słomińska ([18] Proposition 5.4), which says if a functor μ : C_{1} → C_{2} satisfies the condition that every undercategory μ ↓^{y}, y ∈ Ob C_{2}, is contractible, then H^{∗}(C_{2}; F) ≅ H^{∗}(C_{1}; F ∘ μ) for any contravariant functor F : C_{2} → R-mod. Similarly one can get a Jackowski-Słomińska formula for the cases when F is a covariant functor and for homology.

Note that if there is a covariant functor π : E → C then any functor from C to R-mod induces a functor from E to R-mod (restriction along π).

Lemma 4.2.1. Let K → E → C be an extension. Then there exists a natural functor ι : { \atop K(y)} → π ↓_{y} such that every undercategory associated with it is contractible. Hence

(1) H_{∗}(π ↓_{?}; F) ≅ H_{∗}(K(?), F (?)) as functors in RC-mod for any F ∈ RC-mod; and

(2) H^{∗}(π ↓_{?}; F) ≅ H^{∗}(K(?), F (?)) as functors in RC^{op}-mod for any F ∈ RC^{op}-mod.

Proof. The category π ↓_{y} has objects of the form (x, α), where x ∈ Ob E = Ob C and α ∈ Hom_{C}(x, y). From the definition of π ↓_{y}, it’s easy to see the maximal objects are (y, g), g ∈ Aut_{C}(y), which are isomorphic to each other and have automorphism groups isomorphic to K(y).

Next we take the full subcategory of π ↓_{y}, consisting of all maximal objects. This full subcategory is denoted by [(y, 1_{y})] and its skeleton is isomorphic to the group { \atop K(y)}. Using Quillen’s Theorem A, we show the undercategories associated with ι : [(y, 1_{y})] ↪ π ↓_{y} are contractible, and thus we can apply Jackowski-Słomińska’s result we just mentioned to get the isomorphism of cohomology groups.

Fix an object (x, α) ∈ π ↓_{y}. The undercategory (π ↓_{y}) ↓^{(x,α)} has objects of the form (β, (y, g)), where β : (x, α) → (y, g) is an morphism in π ↓_{y} satisfying gπ(β) = α. Since π(β) = g^{−1}α, by the definition of a category extension, β = g^{−1}αk for a unique k ∈ K(x). From here we can deduce that (β, (y, g)) ≅ (β′, (y, g′)) for any (y, g′) and β′ : (x, α) → (y, g′), and that (β, (y, g)) ∈ (π ↓_{y}) ↓^{(x,α)} has a trivial automorphism group. These imply (π ↓_{y}) ↓^{(x,α)} is equivalent to a point, and hence is contractible.

The isomorphism of homology and cohomology groups follows from the result of Jackowski and Słomińska ([18] Proposition 5.4) we quoted above. It’s not hard to see the isomorphisms give rise to the desired isomorphisms of functors. □

Combining the above lemma and the spectral sequences of Gabriel-Zisman, one can write out the following spectral sequences for category extensions. For simplicity we use H^{∗}(K; F) etc, instead of H^{∗}(K(?); F (?)) etc, for the functors in Lemma 4.2.1.

Proposition 4.2.2. Given a functor F : C → R-mod, there are two spectral sequences associated with an extension K → E → C as follows:

(1) a homology spectral sequence

E_{ij}^{2} = H_{i}(C; H_{j}(K; F)) ⇒ H_{i+j}(E; F);

and

(2) a cohomology spectral sequence

E_{2}^{ij} = H^{i}(C^{op}; H^{j}(K^{op}; F)) ⇒ H^{i+j}(E^{op}, F).

Note that H^{j}(K^{op}; F) ≅ H^{j}(K; F).

Remark 4.2.3. From these two spectral sequence, one can obtain two five term exact sequences

H_{2}(E; F) → H_{2}(C; F) → H_{0}(C; H_{1}(K; F)) → H_{1}(E; F) → H_{1}(C; F) → 0,

and

0 → H^{1}(C^{op}; F) → H^{1}(E^{op}; F) → H^{0}(C^{op}; H^{1}(K^{op}; F)) → H^{2}(C^{op}; F) → H^{2}(E^{op}; F),

where F : C → R-mod is a functor. When K → E → C is a group extension then these two exact sequences are the usual five term sequence in group homology and cohomology.

Webb [36] has produced the same five term exact sequences using other techniques.

In general the finite generation of cohomology rings of both K and C doesn’t guarantee the cohomology ring of E has the same property. One of the examples, C_{2} in Section 2 and Appendix, we used to demonstrate that the cohomology rings of EI-categories are not finitely generated, is an extension of a contractible category:

[figure: y; {1_{y}}; y; ℤ_{2}; y; ℤ_{2}; ι; π; .; x; ℤ_{2}; x; ℤ_{2}; α; x; α; {1_{x}}]

However, when K is cohomologically trivial, the cohomology rings of E and C are isomorphic.

Corollary 4.2.4. Suppose K → E → C is an extension, and |K(x)| is invertible in R for every object x. Then for any F ∈ RC-mod

H_{∗}(E; F) ≅ H_{∗}(C; H_{0}(K; F)) ≅ H_{∗}(C; \underrightarrow{lim}_{K}F).
H^{∗}(E^{op}; F) ≅ H^{∗}(C^{op}; H^{0}(K^{op}; F)) ≅ H^{∗}(C^{op}; \underleftarrow{lim}_{K^{op}}F).

Since \underleftarrow{lim}_{K^{op}} \underline{R} ≅ \underline{R} in RC^{op}-mod, we have H^{∗}(C; \underline{R}) ≅ H^{∗}(C^{op}; \underline{R}) ≅ H^{∗}(E^{op}; \underline{R}) ≅ H^{∗}(E; \underline{R}) as algebras.

Proof. Under the assumption, the E_{2} (resp. E^{2}) page of the cohomology (resp. homology) spectral sequence collapses to the horizontal axis. □

4.3. Subextensions and reduction. Let K → E → C be an extension. The generalized LHS spectral sequences establish connections between the cohomology and homology of E and D. Since there is a natural correspondence between the subcategories of C and those of E, one would like to exploit further connections between the homological properties of C and E. Let D be a subcategory of C and E_{D} its “preimage” in E. We show the undercategories (or overcategories) associated with the inclusions are equivalent, when K → E → C is an extension (or an opposite extension) of C.

Definition 4.3.1. Let K → E → C be an extension and D ⊂ C a subcategory. The subextension of D in E via K|_{D}, named E_{D}, is a subcategory of E whose object set is the same as D and whose morphism set consists of morphisms in E which are preimages of morphisms in D.

If D is a full subcategory of C then E_{D} is a full subcategory of E. Given an extension K → E → C, { \atop Aut_{K}(x)} → { \atop Aut_{E}(x)} → { \atop Aut_{C}(x)} is a subextension for any x ∈ Ob C.

Proposition 4.3.2. Let K → E → C a sequence of functors and D a full subcategory of C with the inclusion ι_{D} : D → C. Then

(1) if E is an extension of C, K|_{D} → E_{D} → D is the subextension and ι_{E_{D}} : E_{D} → E is the inclusion, then for any y ∈ Ob C = Ob E, the undercategory ι_{D} ↓^{y} is isomorphic to a subcategory of the undercategory ι_{E_{D}} ↓^{y}, which is equivalent to ι_{E_{D}} ↓^{y};

(2) if E is an opposite extension of C, K|_{D} → E_{D} → D is the (opposite) subextension and ι_{E_{D}} : E_{D} → E is the inclusion, then for any y ∈ Ob C = Ob E, the overcategory ι_{D} ↓_{y} is isomorphic to a subcategory of the overcategory ι_{E_{D}} ↓_{y}, which is equivalent to ι_{E_{D}} ↓_{y}.

Proof. We’ll prove (2). In ι_{E_{D}} ↓_{y}, any two objects (x, α) and (x, β) are isomorphic if and only if π(α) = π(β). Let \overline{ι_{E_{D}} ↓_{y}} ⊂ ι_{E_{D}} ↓_{y} be the full subcategory consisting of one object from each isomorphism class of objects described above. Then \overline{ι_{E_{D}} ↓_{y}} and ι_{E_{D}} ↓_{y} are equivalent. We prove the former is isomorphic to ι_{D} ↓_{y}.

There is a natural bijection between objects sets of these two categories (x, α) → (x, π(α)) (π is surjective on morphisms). We show there is a bijection between the morphism sets and the bijections extend to a functor which gives an isomorphism between two categories. Any (x, α)→^{γ}(z, β) in Mor(\overline{ι_{E_{D}} ↓_{y}}) gives rise to a morphism

(x, π(α)) →^{π(γ)}(z, π(β)) in ι_{D} ↓_{y}. On the other hand, a morphism (x, π(α)) →^{π(γ)}(z, π(β)) in ι_{D} ↓_{y} implies π(β)π(γ) = π(α), which means there exists a unique g ∈ K(x) such

that βγ = αg. Thus we have a uniquely defined morphism (x, α)→^{g^{−1}}(x, αg)→^{γ}(z, β) =

(x, α) →^{γg^{−1}}(z, β) in Mor(\overline{ι_{E_{D}} ↓_{y}}). Note that a different γ′ such that π(γ′) = π(γ) gives

the same morphism (x, α) →^{γg^{−1}}(z, β), so the map from Mor(ι_{D} ↓_{y}) to Mor(\overline{ι_{E_{D}} ↓_{y}}) is well-defined. It’s straightforward to check these two assignments on morphisms are mutually inverse to each other.

In order to show the bijections on objects and morphisms defining an isomorphism between categories, we need to verify they preserve composition and identity. We’ll just prove the former and leave the proof of preserving identity to the reader. Suppose (x, α) → (z, β) → (w, γ) is a composite of two morphisms in \overline{ι_{E_{D}} ↓_{y}}. Then our map naturally sends it to a composite of morphisms (x, π(α)) → (z, π(β)) →

(w, π(γ)). Conversely, if (x, π(α)) →^{π(u)}(z, π(β)) →^{π(v)}(w, π(γ)) = (x, π(α)) →^{π(vu)}(w, π(γ)) is the composite of two morphisms in ι_{D} ↓_{y}, then we need to show the two morphisms

(x, α) →^{ug^{−1}}(z, β) →^{vh^{−1}}(w, γ) = (x, α) →^{vh^{−1}ug^{−1}}(w, γ) and (x, α) →^{vut^{−1}}(w, γ) are equal, where

g, h, t are isomorphisms, described in the preceding paragraph. Since π(vh^{−1}ug^{−1}) = π(vut^{−1}), there is a unique isomorphism s satisfying vh^{−1}ug^{−1} = vut^{−1}s. But then we have α = γvut^{−1} = γvh^{−1}ug^{−1} = γvut^{−1}s, and this forces s = 1 because K^{op} acts freely on morphisms in Mor(E^{op}). Hence we get vh^{−1}ug^{−1} = vut^{−1}. □

The following corollary is a natural outcome of the proposition. A space X is said to be R-acyclic, if the reduced homology groups H̃^{∗}(X, R) vanish.

Corollary 4.3.3. Let K → E → C be a sequence of functors and D ⊂ C a full subcategory with the inclusion ι_{D} : D → C. Then

(1) if E is an extension of C, then ι_{D} ↓^{y} is contractible (or R-acyclic or connected) if and only if ι_{E_{D}} ↓^{y} is;

(2) if E is an opposite extension of C, then ι_{D} ↓_{y} is contractible (or R-acyclic or connected) if and only if ι_{E_{D}} ↓_{y} is.

Example 4.3.4. Let K → E → C be an extension with a unique maximal object x such that Aut_{C}(x) acts freely and transitively on Hom_{C}(y, x) for any y ∈ Ob C. Then it’s easy to check that ι : { \atop Aut_{C}(x)} ↪ C induces a homotopy equivalence since all undercategories associated to it are contractible. Hence we know { \atop Aut_{E}(x)} ↪ E is a homotopy equivalence as well.

Since any category can be regarded as a trivial extension of itself, the following result is a generalization of Jackowski-Słomińska’s formula we mentioned in the previous subsection.

Corollary 4.3.5. Suppose there is an extension K → E → C. If ι_{D} : D ↪ C is an inclusion such that ι_{D} ↓^{y} is contractible for every y ∈ Ob C, then H^{∗}(E; F) ≅ H^{∗}(E_{D}; F) for any contravariant functor F : E → R-mod, and H_{∗}(E; F) ≅ H_{∗}(E_{D}; F) for any covariant functor F : E → R-mod. Here E_{D} is the subextension corresponding to D.

Suppose there is an opposite extension K → E → C. If ι_{D} : D ↪ C is an inclusion such that ι_{D} ↓_{y} is contractible for every y ∈ Ob C, then H^{∗}(E; F) ≅ H^{∗}(E_{D}; F) for any covariant functor F : E → R-mod. Here E_{D} is the opposite subextension corresponding to D.

Proof. We prove the statements for cohomology. Since ι_{D} ↓^{y} is contractible for every y ∈ Ob C, ι_{E_{D}} ↓^{y} is contractible for every y ∈ Ob E as well by Proposition 4.3.2. That means if we take the dual version P′_{∗} of the projective resolution P_{∗}, described in 2.1, of the contravariant functor \underline{R} : E_{D} → R-mod, the left Kan extension of P′_{∗} → \underline{R} → 0 is still a projective resolution K(P′_{∗}) → K(\underline{R}) ≅ \underline{R} → 0 of the contravariant functor \underline{R} : E → R-mod. Thus our results follows from the isomorphism of complexes of R-modules Hom_{RE_{D}}(P′_{∗}, F ∘ ι_{E_{D}}) ≅ Hom_{RE}(K(P′_{∗}), F).

When we have an opposite extension, using the same proposition we get ι_{E_{D}} ↓_{y} is contractible for every y ∈ Ob E. If P_{∗} → \underline{R} → 0 is the projective resolutions of \underline{R} as a covariant functor, then the left Kan extension of it, K(P_{∗}) → K(\underline{R}) ≅ \underline{R} → 0, is a projective resolution of \underline{R} as a covariant functor from E to R-mod. Hence we can obtain a similar isomorphism of complex using the adjunction of K and Res_{ι_{E_{D}}}, the restriction along ι_{E_{D}}. □

As an example when K → E → C is an extension (or an opposite extension) and C has a unique maximal (or minimal) object x and Aut_{C}(x) acts regularly on Hom_{C}(y, x) (or Hom_{C}(x, y)) for any y ∈ Ob C, we have H^{∗}(C; F) ≅ H^{∗}(Aut_{C}(x), F (x)) hence H^{∗}(E; F) ≅ H^{∗}(Aut_{E}(x), F (x)) for any contravariant (or covariant) functor F.

5. Appendix : Cohomology of small categories

This section intends to provide a self-contained introduction to the standard materials that the reader may want to know about the cohomology theory of small categories. Here we begin with a general setup, which is followed by a canonical resolution for computing the cohomology of small categories with coefficients in functors. Using this resolution, we calculate the cohomology of a family of categories named C_{n}, where n > 1 denotes the number of (non-isomorphic) objects.

Let R be a commutative ring with identity. In the introduction we mentioned that for any small category C one can define the category algebra RC to be a free R-module with basis the set of morphisms in C (see Xu [38]), in which multiplication is given by composition of basis elements. When Ob C is finite, Mitchell [26] showed the category of left RC-modules is equivalent to the category of covariant functors from C to Rmod, i.e. RC-mod ≃ (R-mod)^{C}. The equivalence is given as follows. If F ∈ (R-mod)^{C} is a functor, then we define an RC-module to be the R-module M_{F} = ⊕_{x∈Ob C}F (x) equipped with natural actions by morphisms in C. Conversely if M ∈ RC-mod, then we define a functor F_{M} ∈ (R-mod)^{C} such that F_{M}(x) = 1_{x}M for each x ∈ Ob C. It’s easy to verify that these two assignments are functors which are inverse to each other. Similarly, the category of right RC-modules (same as RC^{op}-mod) is equivalent to the category of contravariant functors from C to R-mod. For simplicity, we’ll stick with the terms of covariant and contravariant functors.

These equivalences allow us to consider Ext_{RC}^{∗}(F_{1}, F_{2}) = Ext_{(R-mod)^{C}}^{∗}(F_{1}, F_{2}) and Tor_{∗}^{RC}(M_{1}, M_{2}) = Tor_{∗}^{(R-mod)^{C}}(M_{1}, M_{2}), where F_{1}, F_{2} and M_{2} are covariant functors and M_{1} is a contravariant functor (for further information, see tom Dieck [8], Lück [25], Webb [35] or Xu [38]). For the reader’s convenience, we point out that the tensor product of a contravariant functor M and a covariant functor N as an R-module is the following M ⊗_{RC} N: ∑_{x∈Ob C} M(x) ⊗_{R} N(x)/ ∼, where the relation is given by m_{x}M(α)⊗_{R} n_{y} ∼ m_{x} ⊗_{R} N(α)n_{y} for any m_{x} ∈ M(x), n_{y} ∈ N(y) and α ∈ Hom_{C}(y, x).

The constant functor \underline{R} plays an important role in the cohomology theory of small categories. In fact it should be regarded as the generalization of the trivial module of a group algebra. We define H^{∗}(C; F) := Ext_{RC}^{∗}(\underline{R}, F) and H_{∗}(C; F) := Tor_{∗}^{RC}(F, \underline{R}). In order to compute them, we need to construct a projective resolution of \underline{R}. There is a canonical projective resolution of \underline{R} (see for example Grodal [14]) which can be defined using the so-called overcategories (see Mac Lane [27]) associated with the identity functor Id_{C} : C → C. Recall that a functor μ : D → C can be used to produce an overcategory μ ↓_{x} for each x ∈ Ob C. This overcategory consists of objects of the form (y, α), where y ∈ Ob D and α ∈ Hom_{C}(μ(y), x). A morphism between two objects (y, α) and (z, β) is given by some γ ∈ Hom_{D}(y, z) which makes the following diagram commutative:

\xymatrix{μ(y) \ar[dd]_{μ(γ)} \ar[dr]^{α} \\  & x \\ μ(z) \ar[ur]_{β} & .}

Dually, one can define an undercategory μ ↓^{x} for each object x ∈ Ob C. Now fix an object x ∈ Ob C. Then Id_{C} ↓_{x} is always contractible, by Quillen’s Theorem A, because it has a terminal object (x, 1_{x}). For each integer n ≥ 0 we can define a functor Id_{C} ↓_{?}^{n}: C → R-mod sending x to Id_{C} ↓_{x}^{n}—the free R-module spanned over the set of n-chains of morphisms in Id_{C} ↓_{x}. There is a natural way to assemble these functors into a sequence of functors, Id_{C} ↓_{?}^{∗}→ \underline{R} → 0, which evaluated at each x ∈ Ob C is a complex for computing the reduced homology of | Id_{C} ↓_{x} |. Since every | Id_{C} ↓_{x} | is contractible, the sequence is exact. In order to show it’s in fact a projective resolution of \underline{R}, we need to prove each Id_{C} ↓_{?}^{∗} is a projective object in RC-mod. This can be seen by rewriting the sequence in a slightly different form: P_{∗} → \underline{R} → 0, where P_{n} : C → R-mod is defined as follows (see Oliver [28]). The new form of this resolution is less conceptual and is easier to use in practice. For any x ∈ Ob C, P_{n}(x) is the free abelian group with a basis the set of all sequences [x_{0} → x_{1} → ⋯ → x_{n} → x] of morphisms in Mor(C) ending in x. For any morphism f ∈ Hom_{C}(x, y), P_{n}(f) is defined by its action on basis elements: [x_{0} → ⋯ → x_{n}→^{ϕ}x] ↦ [x_{0} → ⋯ → x_{n}→^{fϕ}y].

The boundary map σ = {σ_{x}} : P_{n} → P_{n−1} is given by setting

σ_{x}([x_{0} → x_{1} → ⋯ → x_{n} → x]) = ∑_{i=0}^{n}(−1)^{i}[x_{0} → ⋯ → x̂_{i} → ⋯ → x_{n} → x],

on base elements. Since

Hom_{RC}(P_{n}, F) ≅ ∏_{[x_{0}→x_{1}→···→x_{n}]∈N_{n}(C)} F (x_{n}),

where N(C) is the nerve of C, we know Hom_{RC}(P_{n},?) is exact and hence P_{n} is projective for any n ≥ 0. Thus H^{∗}(C; F) are the homology groups of the cochain complex 0 → Hom_{RC}(P_{∗}, F), or equivalently 0 → Hom_{RC}(Id_{C} ↓_{?}^{∗}, F). When F = \underline{A} is a constant functor, one can easily see H^{∗}(C; \underline{A}) ≅ H^{∗}(|C|, A), because 0 → Hom_{RC}(P_{∗}, \underline{A}) can be identified with a cochain complex used to compute H^{∗}(|C|, A).

5.1. Two torsion groups. Using Tor, one can define and compute the homology groups of C with coefficients in a contravariant functor F: H_{∗}(C; F) ≅ \underrightarrow{lim}_{C}^{∗} F ≅ Tor_{∗}^{RC}(F, \underline{R}). As in cohomology, similarly we have H_{∗}(C; \underline{A}) ≅ H_{∗}(|C|, A) for any constant functor \underline{A}. One can use the same projective resolution we described above to prove Tor_{∗}^{RC}(\underline{A}, \underline{R}) ≅ H_{∗}(|C|, A). Here we only intend to give a taste of the concrete calculations of (co-)homology groups, and do not plan to recall the general theory of Ext and Tor. With this purpose in mind, we first discuss two special torsion groups. They will be used in our further examples at the end of the appendix.

A functor is atomic if it takes non-zero values at only one isomorphism class of objects in C. For example, we can define an atomic bi-functor S_{x,R} : C → R-mod such that S_{x,R}(y) = R if y ≅ x and S_{x,R}(y) = 0 otherwise. Using the projective resolution P_{∗} ≅ Id_{C} ↓_{?}^{∗} of \underline{R}, described above, one can show that H_{∗}(C; S_{x,R}) ≅ Tor_{∗}^{RC}(S_{x,R}, \underline{R}) can be calculated as homology groups of the chain complex given by

{  ∐_{[x_{0}→x_{1}→···→x_{n}]∈N_{n}(C)} S_{x,R}(x_{n})} = {  ∐_{[x_{0}→x_{1}→···→x_{n}]∈N_{n}(C), x_{n}≅x} R} ,

where chain maps are induced by the face maps for N(C). We will encounter torsions like Tor_{∗}^{RC}(\underline{R}, S_{x,R}) as well in the next subsection. This time they are the homology groups of the chain complex given by

{  ∐_{[x_{0}→x_{1}→···→x_{n}]∈N_{n}(C)} S_{x,R}(x_{0})} = {  ∐_{[x_{0}→x_{1}→···→x_{n}]∈N_{n}(C), x_{0}≅x} R} ,

where chain maps are induced by the face maps for N(C).

5.2. further examples of cohomology rings. For each integer n > 1 we construct a category C_{n} which has exactly n objects and then examine the ring structure of H^{∗}(C_{n}) := H^{∗}(C_{n}; \underline{𝔽_{2}}) ≅ H^{∗}(|C_{n}|, 𝔽_{2}), where 𝔽_{2} is a field of characteristic 2. Let C_{n}, n > 1, be the following category

[figure: x_{1}; 1_{x_{1}}; g_{1}; α_{1}; x_{2}; 1_{x_{2}}; g_{2}; α_{2}; ⋯; α_{n−1}; x_{n}; 1_{xn}; g_{n}] ,

where α_{i} · g_{i} = α_{i} = g_{i+1} · α_{i} and g_{i}^{2} = 1_{x_{i}} for all i. We’ll first calculate the homology of C_{n}. Then the structure of H^{∗}(C_{n}; \underline{𝔽_{2}}) can be easily obtained via the Universal Coefficient Theorem.

Since H^{∗}(C_{n}; \underline{𝔽_{2}}) ≅ H^{∗}(|C_{n}|, 𝔽_{2}), one can just take the (normalized) chain complex associated to the nerve of C_{n} to compute its homology groups: ⋯ → C_{k} → ⋯ → C_{1} → C_{0} → 0, where C_{0} = ⊕_{i}𝔽_{2}x_{i} and C_{k}, k ≥ 1, is an 𝔽_{2}-vector space spanned by chains of k consecutive morphisms in C_{n}. For example in C_{2} of C_{2}, the basis is {(g_{1}, g_{1}), (g_{1}, α_{1}), (α_{1}, g_{2}), (g_{2}, g_{2})}.

In order to proceed, we need some auxiliary constructions. Suppose C is a finite EI-category. For each x ∈ Ob C, we’ll be interested in two full subcategories of C, C_{<x} ⊂ C_{≤x}, (see conventions in the introduction) and the associated complexes ℂ_{<x} ⊂ ℂ_{≤x} and the corresponding quotient complex, written as {(·, x⟩_{k}}_{k=0}^{∞}, which contains the linear combinations of chains ending at x. If x ≤ y ∈ Ob C, then we also define C_{[x,y]} to be a full subcategory of C consisting of objects z such that both Hom_{C}(x, z) and Hom_{C}(z, y) are not empty. There are naturally defined full subcategories of C_{[x,y]}: C_{(x,y]}, C_{[x,y)} and C_{(x,y)}. The quotient complexes of ℂ_{[x,y]} by ℂ_{(x,y]}, ℂ_{[x,y)} and the sum ℂ_{(x,y]} +ℂ_{[x,y)} are named {⟨x, y]_{k}}_{k=0}^{∞}, {[x, y⟩_{k}}_{k=0}^{∞} and {⟨x, y⟩_{k}}_{k=0}^{∞}, respectively.

For the sake of simplicity, when it won’t cause any confusion, we will use (·, x⟩, [x, y] and ⟨x, y⟩ et cetera to denote the quotient complexes defined above.

Proposition 5.2.1. Let C_{n}(n ≥ 2) be the category defined above. Then

H^{i}(C_{n}; \underline{𝔽_{2}}) = \{ \begin{matrix}𝔽_{2}, & if i = 0; \\ 0 , & if 1 ≤ i ≤ 2n − 2; \\ 𝔽_{2}^{m(i,n)}, & if i ≥ 2n − 1. \end{matrix}

Here m(i, n) ∈ ℤ is the number of chains of i consecutive morphisms in which every non-identity morphism in C_{n} appears at least once.

Proof. Suppose

⋯ → C_{n} → ⋯ → C_{2}→^{δ_{2}}C_{1}→^{δ_{1}}C_{0}→^{δ_{0}}0

is the normalized chain complex associated with the nerve of C_{n}. Let σ be any chain in any C_{i}. A significant property of δ_{i} in this particular case is that all the non-zero summands of δ_{i}(σ) contain the same isomorphisms as those in σ itself. For this reason, we can classify chains of morphisms in C_{∗} by the (ordered) lists of isomorphisms in them.

We say two chains in C_{∗} are of the same type if, after removing non-isomorphisms in them, the resulting ordered lists (possibly empty) of isomorphisms are the same.

We’ll denote the type of a chain by T = {t_{1}, t_{2}, ⋯, t_{n}}, where t_{i} is a non-negative integer and represents the number of g_{i} in the chain, for every i. The length of T is |T | = t_{1} + ⋯ + t_{n}. Due to our observation on δ_{∗}, chains of morphisms of the same type T form a subcomplex of C_{∗}, denoted by C_{∗}^{T}, which is always finite because there are finitely many non-isomorphisms. Furthermore C_{∗} = ⊕_{T} C_{∗}^{T}, where T runs over the (infinite) set of all possible types. In particular, the type {0, ⋯, 0} chains are chains of non-isomorphisms and if we demand C_{0} to be of type {0, ⋯, 0} then they form a natural subcomplex of C_{∗}, which is the order complex for the underlying poset [C_{n}] of C_{n}. In general, when a type T is given, all chains of type T have lengths greater than or equal to |T | (that is, C_{∗}^{T} begins at degree |T |).

Now we start to calculate H_{∗}(C_{n}). In order to compute the homology groups of C_{∗}, we just have to know how to do it for the subcomplex formed by chains of each type. Fix a type T, it is not hard to see that C_{∗}^{T} is isomorphic to the product of finitely many chain complexes of the forms [x_{1}, y⟩, ⟨x, y⟩ and ⟨x, x_{n}] coming from the underlying poset [C_{n}] of C_{n} (since isomorphisms have been excluded and there is only one morphism between any two non-isomorphic objects.). Hence one can use the Künneth Theorem to compute the homology of each C_{∗}^{T} if the homology groups of the three explicitly constructed quotient complexes are known.

Next we re-interpret the homology groups of the following chain complexes [x, y⟩, ⟨x, y⟩ and ⟨x, y] for objects x, y belonging to a finite poset. Note that when x = y, these complexes are trivial.

Lemma 5.2.2. Let C be a finite poset. For any two objects x < y, the homology groups of [x, y⟩, ⟨x, y⟩ and ⟨x, y] are isomorphic to Tor_{∗}^{RD}(S_{y,R}, \underline{R}), Tor_{∗}^{RD}(S_{y,R}, S_{x,R}) and Tor_{∗}^{RD}(\underline{R}, S_{x,R}), respectively, where D is the subposet C consisting of all objects z such that x ≤ z ≤ y.

The groups Tor_{i}^{RD}(\underline{R}, S_{y,R}) and Tor_{i}^{RD}(S_{x,R}, \underline{R}) vanish unless i = 0, when they are equal to R. If there exists x < z < y then Tor_{i}^{RD}(S_{y,R}, S_{x,R}) = 0. Otherwise there is only one non-zero torsion Tor_{1}^{RD}(S_{y,R}, S_{x,R}) = R.

Assuming the lemma is proved, we use it to calculate the ith homology of C_{∗} when i ≥ 1

H_{i}(C_{∗}) = ⊕_{T} H_{i}(C_{∗}^{T}).

By Künneth Theorem, we have

\begin{aligned}H_{i}(C_{∗}^{T}) & = & ⊕_{i_{1}+···+i_{k}=i}H_{i_{1}}([·, ·⟩) ⊗ H_{i_{2}}(⟨·, ·⟩) ⊗ ⋯ ⊗ H_{i_{k}}(⟨·, ·]) \\ & = & ⊕_{i_{2}+···+i_{k−1}=i}H_{i_{2}}(⟨·, ·⟩) ⊗ ⋯ ⊗ H_{i_{k−1}}(⟨·, ·⟩), \end{aligned}

where the dots represent appropriate objects in C_{n}.

The second equality is true because H_{i_{1}}([·, ·⟩) and H_{i_{k}}(⟨·, ·]) are non-zero if and only if i_{1} = i_{k} = 0, when they equal R by the previous lemma. The same lemma also implies that H_{i_{2}}(⟨·, ·⟩) ⊗ ⋯ ⊗ H_{i_{k−1}}(⟨·, ·⟩) is non-zero if and only if i_{2} = ⋯ = i_{k−1} = 1 and each ⟨·, ·⟩ is some ⟨x_{s}, x_{s+1}⟩, or equivalently every isomorphism in Mor(C) occurs at least once in T. When it’s non-zero, the tensor product must be R. Thus H_{i}(C_{∗}) = ⊕_{T} H_{i}(C_{∗}^{T}) = R^{m(i,n)}, where m(i, n) is the number of i-chains of all possible types in which every morphism in C_{n} appears at least once. □

Proof of the lemma. The homology groups of [x, y⟩ are exactly Tor_{∗}^{RD}(S_{y,R}, \underline{R}) by definition of [x, y⟩ and discussion on Tor_{∗}^{RD}(S_{y,R}, \underline{R}) in 2.2. All positive degree torsions are zero because \underline{R} = R Hom(x,?) is a representable functor. Similarly, we can prove H_{∗}(⟨x, y]) ≅ Tor_{∗}^{RD}(\underline{R}, S_{x,R}) and all positive degree torsions are zero because \underline{R} = R Hom(?, x) is a contravariant representable functor. As for ⟨x, y⟩, we note that Tor_{∗}^{RD}(S_{y,R}, S_{x,R}) can be computed through the long exact sequence

⋯ → Tor_{∗}^{RD}(S_{y,R}, \underline{R}_{>x}) → Tor_{∗}^{RD}(S_{y,R}, \underline{R}) → Tor_{∗}^{RD}(S_{y,R}, S_{x,R}) → ⋯

coming from the following short exact sequence of functors: 0 → \underline{R}_{>x} → \underline{R} → S_{x,R} → 0. Here \underline{R}_{>x} is the maximal subfunctor of \underline{R} (note that if there is no z such that x < z < y then \underline{R}_{>x} ≅ S_{y,R}). Since Tor_{∗}^{RD}(S_{y,R}, \underline{R}) ≅ H_{∗}([x, y⟩), Tor_{∗}^{RD}(S_{y,R}, \underline{R}_{>x}) ≅ H_{∗}((x, y⟩) and 0 → (x, y⟩ → [x, y⟩ → ⟨x, y⟩ → 0 also induces a long exact sequence, a Five Lemma argument shows that Tor_{∗}^{RD}(S_{y,R}, S_{x,R}) ≅ H_{∗}(⟨x, y⟩).

Finally from the long exact sequence for Tor_{∗}^{RD} and the values of Tor_{∗}^{RD}(S_{y,R}, \underline{R}_{>x}) and Tor_{∗}^{RD}(S_{y,R}, \underline{R}), we conclude that if there exists z such that x < z < y then Tor_{i}^{RD}(S_{y,R}, S_{x,R}) = 0. Otherwise there’s only one torsion Tor_{1}^{RD}(S_{y,R}, S_{x,R}) = R. □

Corollary 5.2.3. For each n > 1, the cohomology ring H^{∗}(C_{n}; \underline{𝔽_{2}}) is not finitely generated.

Proof. Over the field 𝔽_{2} we always have H^{∗}(C_{n}) ≅ Hom_{𝔽_{2}}(H_{∗}(C_{n}), 𝔽_{2}) in a natural way as vector spaces by the Universal Coefficient Theorem. It implies as a vector space H^{∗}(C_{n}) is infinite dimensional. It also means, for any k ≥ 0, H^{k}(C_{n}) is spanned by functions in Hom_{𝔽_{2}}(C_{k}, 𝔽_{2}) which are dual to the base elements for H_{k}(C_{n}). From the description of H_{∗}(C_{n}) we know what the generators of H^{∗}(C_{n}) are and thus we can compute the cup product of any two of these functions, which is always zero. Hence the statement is proved. □

Note that C_{2} is the smallest (in terms of number of objects and/or morphisms) non-contractible category that is neither a group nor a poset.

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