On inverse categories and transfer in cohomology

Markus Linckelmann

Abstract It follows from methods of B. Steinberg [22], extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrising forms. We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that for certain categories C, being a Mackey functor on C is equivalent to being extendible to a suitable inverse category containing C. We show further that extensions of inverse categories by abelian groups are again inverse categories.

1 Introduction

Given a small category C, a subcategory D and a commutative ring k, the restriction from C to D induces an algebra homomorphism on cohomology from H^{∗}(C; k) = Ext_{C}^{∗}(k, k) to H^{∗}(D; k), where we abusively denote by k the constant functor with value k on C and on D. It is not known whether there is a transfer map H^{∗}(D; k) → H^{∗}(C; k) with good formal properties, or transfer maps between the Hochschild cohomology algebras HH^{∗}(kC) and HH^{∗}(kD) of the category algebras kC, kD over k, unless the left and right Kan extensions of a functor D → C coincide (cf. [18]). This seems to be a rare phenomenon, however. Besides the categories of finite groups and their subgroups, one known example is the canonical functor from a transporter category of p-centric subgroups of a finite group to its centric linking systems in [3, 1.2, 1.3]. Finite inverse categories and their subcategories provide further examples for this phenomenon because these can be played back to groupoids. Following [14], a small category C is called an inverse category if for any morphism s : X → Y in C there is a unique morphism ŝ: Y → X such that s ∘ ŝ∘ s = s and ŝ ∘ s ∘ ŝ = ŝ. Just as in the case of inverse semigroups, the morphism set of C admits a partial order. We will review the relevant background in section 2. An algebra A over a commutative ring k is called symmetric if A is finitely generated projective as k-module and if A is isomorphic, as an A-A-bimodule, to its k-dual Hom_{k}(A, k). A symmetrising form of A is a linear map τ : A → k corresponding to 1_{A} under some bimodule isomorphism A ≅ Hom_{k}(A, k). A symmetric algebra over a field is in particular self-injective. We refer to [4] for a more detailed account on properties of symmetric algebras. Essentially by extending arguments due to B. Steinberg in the context of inverse semigroups in [22] we will show that category algebras of finite inverse categories are symmetric and admit canonical choices of symmetrising forms:

Theorem 1.1. Let k be a commutative ring and C a finite inverse category. Then kC is isomorphic to a direct product of matrix algebras over finite group algebras; in particular, kC is symmetric. Moreover, the linear map τ : kC → k sending a morphism s in C to the number of idempotent morphisms e in C satisfying e ≤ s is a symmetrising form for kC.

Thus if k is a field then the algebra kC is self-injective. There are examples, due to Fei Xu [24], of finite categories for which not even the quotient of the Hochschild cohomology by its nil-radical is finitely generated. In contrast, for finite inverse categories, the above theorem has the following consequence:

Corollary 1.2. Let C be a finite inverse category and k a commutative Noetherian ring. Then HH^{∗}(kC) and H^{∗}(C; k) are finitely generated graded commutative k-algebras.

One can describe the Hochschild and ordinary cohomology of finite inverse categories more precisely in terms of products of the Hochschild cohomology and ordinary cohomology of group algebras; this will be an easy consequence of the explicit description in 4.1 below of an isomorphism between kC and a direct product of matrix algebras over group algebras. For the same reason, standard results on Schur multipliers for finite groups carry over to finite inverse categories, such as:

Corollary 1.3. Let C be a finite inverse category and k an algebraically closed field. The abelian group H^{2}(C; k^{×}) is finite.

The classes in H^{2}(C; k^{×}) correspond to certain extensions of C, which are shown to be inverse categories as well in 2.7 below. The point of the next result is that it implies that for certain subcategory algebras of finite inverse categories we do indeed have transfer maps in cohomology.

Theorem 1.4. Let k be a commutative ring, C a finite inverse category and D an inverse subcategory of C having the property that for any object Y in D, the endomorphism monoid End_{D}(Y) contains all idempotents in End_{C}(Y). Set i = 1_{kD} = ∑_{Y ∈Ob(D)} Id_{Y}, the sum taken in kC. Then kCi is finitely generated projective as left kC-module and as right kD-module; similarly, ikC is is finitely generated projective as left kD-module and as right kC-module. Moreover, we have ikC ≅ Hom_{k}(kCi, k) as kC-kD-bimodules.

Proofs of the above results and some further consequences will be given in section 4. In terms of functor categories over C and D instead of module categories over kC and kD, respectively, theorem 1.4 implies that the left and right Kan extensions associated with the inclusion functor D → C coincide. Indeed, restriction of functors along this inclusion corresponds on modules to the functor ikC ⊗_{kC} −, and this functor has a left adjoint and a right adjoint which are both isomorphic to the functor kCi ⊗_{kD} −. One can show this also by directly calculating the relevant left and right Kan extensions. Any A-B-bimodule M for two symmetric algebras A, B which is finitely generated projective as left and right module gives rise to an array of transfer maps. This is based on the fact that the k-dual M^{∗} = Hom_{k}(M, k) is a B-A-bimodule which is again finitely generated projective as left and right module. This fact implies that the functors M ⊗_{B} − and M^{∗} ⊗_{A} − between the module categories of A and B are left and right adjoint to each other, and every choice of symmetrising forms on A and B determines adjunction isomorphisms. Thus, more precisely, the bimodule M together with a choice of symmetrising forms on A and B induces transfer maps between the Hochschild cohomology rings of the two algebras (cf. [16, 2.9]), between the graded centers of their stable or derived categories (cf. [18, 4.1]) as well as between Ext-groups of modules and their images under tensoring with the bimodule and its dual, satisfying reciprocity and compatibility properties described in [18, 4.3] [18, 4.8] and [17, 5.1]. Specialised to the situation of the above theorem - where we have a canonical choice of symmetrising forms by 1.1 - this means that induction and restriction between the module categories over kC and kD, truncated by the idempotent i, induce transfer maps htr_{D}^{C} = tr_{kCi} : HH^{∗}(kD) → HH^{∗}(kC), htr_{C}^{D} = tr_{ikC} : HH^{∗}(kC) → HH^{∗}(kD), as well as transfer maps tr_{D}^{C}(M, N) : Ext_{kC}^{∗}(Res_{kD}^{kC}(M), Res_{kD}^{kC}(N)) → Ext_{kC}^{∗}(M, N) for any two kC-modules M, N, and tr_{C}^{D}(U, V) : Ext_{kC}^{∗}(Ind_{kD}^{kC}(U), Ind_{kD}^{kC}(V)) → Ext_{kD}^{∗}(U, V) for any two kD-modules U, V. We describe, for future reference, the special case with constant coefficients. We write abusively tr_{D}^{C} instead of tr_{D}^{C}(k, k), we identify as usual H^{0}(C; k) = lim_{C}(k) with the set of families (λ_{X})_{X∈Ob(C)} of elements λ_{X} ∈ k satisfying λ_{X} = λ_{Y} for any two objects X, Y in C for which the morphism set Hom_{C}(X, Y) is non-empty, and we denote by 1 the unit element of H^{0}(D; k). We need furthermore some notation which will be reviewed in some more detail in the background section 2. We denote by μ the Möbius function of the partially ordered set Mor(C) of morphisms in C. Two idempotent endomorphisms e, f in C are called isomorphic if there is a morphism s in C such that s ∘ ŝ = e and ŝ∘ s = f. The endomorphisms s satisfying s ∘ ŝ = e = ŝ ∘ s form a group, with unit element e, which we denote by C_{e}. Using a similar notation for D, if f is an idempotent endomorphism in D then D_{f} is a subgroup of C_{f}. For any idempotent endomorphism e in C we set n(e) = ∑_{f}[C_{f} : D_{f}], where the sum is taken over a set of representatives of the D-isomorphism classes of idempotents in Mor(D) contained in the C-isomorphism class of e, with the usual convention n(e) = 0 if the sum is empty. We set π_{D}^{C} = ∑_{(e,f)} n(e)μ(f, e)f, where (e, f) runs over the set of ordered pairs of idempotents e, f in Mor(C) satisfying f ≤ e. We have π_{D}^{C} ∈ HH^{0}(kC) = Z(kC). We denote by τ_{D}^{C} the image of π_{D}^{C} in H^{0}(C; k) = lim_{C}(k) under the canonical algebra homomorphism HH^{∗}(kC) → H^{∗}(C; k).

Theorem 1.5. Let k be a commutative ring, C a finite inverse category and D an inverse subcategory of C. Suppose that for any object Y in D the endomorphism monoid End_{D}(Y) contains all idempotent endomorphisms in End_{C}(Y). Denote by α : HH^{∗}(kC) → H^{∗}(C; k) and β : HH^{∗}(kD) → H^{∗}(D; k) the canonical algebra homomorphisms. There are graded k-linear transfer map tr_{D}^{C} : H^{∗}(D; k) → H^{∗}(C; k) and htr_{D}^{C} : HH^{∗}(kD) → HH^{∗}(kC) with the following properties:

(i) tr_{D}^{C}(θ) · ζ = tr_{D}^{C}(θ · Res_{D}^{C}(ζ)) for any ζ ∈ H^{∗}(C; k) and any θ ∈ H^{∗}(D; k). (ii) α(htr_{D}^{C}(η)) = tr_{D}^{C}(β(η)) for any η ∈ HH^{∗}(kD). (iii) We have htr_{D}^{C}(1) = π_{D}^{C} and π_{D}^{C} is invertible in Z(kC) if and only if n(e) is invertible in k for all idempotent morphisms e in C. (iv) We have tr_{D}^{C}(1) = τ_{D}^{C} and τ_{D}^{C} is invertible in lim_{C}(k), if and only if n(e) is invertible in k for all minimal idempotent morphisms e in C.

(v) If τ_{D}^{C} is invertible then Res_{D}^{C} is injective and tr_{D}^{C} is surjective.

This will be proved at the end of section 5. We show further in 6.2 that any small category C in which all morphisms are monomorphisms and in which pullbacks exist can be embedded canonically into an inverse category, preserving cohomology with constant coefficients:

Theorem 1.6. Let C be a small category in which all morphisms are monomorphisms and in which pullbacks exist. There is an inverse category Ĉ containing C such that for any module A over a commutative ring k, viewed as constant contravariant functor, the restriction induces an isomorphism H^{∗}(Ĉ; A) ≅ H^{∗}(C; A).

In the situation of 1.6, being a Mackey functor on C in the sense of Dress [5, Part I] and Jackowski-McClure [11] is essentially equivalent to being extendible to the inverse category Ĉ; see 6.4.

2 Basic properties of inverse categories

We collect in this section basic ideas from the theory of semigroups and categories of partial maps translated to inverse categories; see [15, Chapter 1], [14], [8], [9]. Let C be an inverse category. Then in particular, for any object X in C, the monoid End_{C}(X) is an inverse monoid, and hence any two idempotents in End_{C}(X) commute (cf. [15, 1.1, Theorem 3]). For any morphism s : X → Y the morphisms ŝ ∘ s and s ∘ ŝ are idempotents in End_{C}(X) and End_{C}(Y), respectively. For any idempotent e in End_{C}(X) we have ê = e, and for any two composable morphisms s, t in C we have t̂ ∘ s = ŝ∘ t̂; in particular, an inverse category C is isomorphic to its opposite C^{op}. If e is an idempotent in End_{C}(X) and s : X → Y a morphism in C then f = s∘e∘ŝ is an idempotent in End_{C}(Y) satisfying s ∘ e = f ∘ s; indeed, using that the idempotents ŝ ∘ s and e commute we have f ∘ f = s ∘ e ∘ ŝ ∘ s ∘ e ∘ ŝ = s ∘ e ∘ e ∘ ŝ ∘ s ∘ ŝ = s ∘ e ∘ ŝ = f and f ∘ s = s ∘ e ∘ ŝ ∘ s = s ∘ ŝ ∘ s ∘ e = s ∘ e. Similarly, if f is an idempotent in End_{C}(Y) then e = ŝ ∘ f ∘ s is an idempotent in End_{C}(X) satisfying s ∘ e = f ∘ s. We use these elementary computational rules without further comment. As in the case of inverse semigroups one can define a partial order on the set of morphisms in C. For any two morphisms s, t : X → Y we write s ≤ t if s = t ∘ e for some idempotent e ∈ End_{C}(X). The proof of the - well-known - fact that this is actually a partial order and the proof, below, of an extension of the Preston-Wagner representation theorem to this context follow closely the proofs of corresponding statements for inverse semigroups in [15, Chapter 1]; we include details for the convenience of the reader.

Lemma 2.1. Let C be an inverse category and let s, t : X → Y be morphisms in C. The following statements are equivalent:

(i) s ≤ t. (ii) s = f ∘ t for some idempotent f ∈ End_{C}(Y). (iii) ŝ ≤ t̂. (iv) s = s ∘ ŝ ∘ t.

(v) s = t ∘ ŝ ∘ s.

Proof. The equivalence (i) ⇔ (ii) follows from the remarks preceding the lemma. The equality s = t ∘ e for some idempotent e ∈ End_{C}(X) is equivalent to ŝ = ê ∘ t̂ = e ∘ t̂, whence the equivalence (i) ⇔ (iii). If s = t ∘ e for some idempotent e ∈ End_{C}(X) then s = s ∘ ŝ ∘ s = t ∘ e ∘ ŝ ∘ s = t ∘ ŝ ∘ s ∘ e = t ∘ ŝ ∘ s, and since ŝ ∘ s is an idempotent we get the equivalence (i) ⇔ (v). A similar argument yields the equivalence (ii) ⇔ (iv).

Proposition 2.2. Let C be an inverse category, let X, Y, Z be objects in C and let s, t : X → Y and u, v : Y → Z be morphisms in C. The relation “≤” defines a partial order on the set of morphisms which is compatible with composition of morphisms in C; that is, if s ≤ t and u ≤ v then u ∘ s ≤ v ∘ t.

Proof. Since s = s ∘ ŝ∘ s we have s ≤ s. If s ≤ t and t ≤ s then, using that the idempotents ŝ∘ s and t̂∘ t commute, we get that s = t ∘ ŝ∘ s = s ∘ t̂∘ t ∘ ŝ∘ s = s ∘ t̂∘ t = t. If s ≤ t and t ≤ w for some further morphism w : X → Y then s = s ∘ ŝ ∘ t = s ∘ ŝ ∘ t ∘ t̂∘ w. Since the idempotents s ∘ ŝ and t ∘ t̂ commute, their product is an idempotent as well, and thus s ≤ w. This shows that “ ≤′′ is a partial order on the set of morphisms in C. If s ≤ t and u ≤ v there are idempotents f, g in End_{C}(Y) satisfying s = f ∘ t and u = v ∘ g. Thus u ∘ s = v ∘ g ∘ f ∘ s. The idempotents f, g commute, hence g ∘ f is an idempotent, and thus by the remarks at the beginning of this section there is an idempotent h in End_{C}(Z) such that v ∘ g ∘ f = h ∘ v. Together we get that u ∘ f = h ∘ v ∘ t, hence u ∘ s ≤ v ∘ t.

Lemma 2.3. Let C be an inverse category, let s : X → Y be a morphism in C and U an object in C. We have s ∘ Hom_{C}(U, X) = s ∘ ŝ ∘ Hom_{C}(U, Y).

Proof. If u ∈ Hom_{C}(U, X) then s ∘ u = s ∘ ŝ∘ s ∘ u ∈ s ∘ ŝ∘ Hom_{C}(U, Y), and if v ∈ Hom_{C}(U, Y) then s ∘ ŝ ∘ v ∈ s ∘ Hom_{C}(U, X), whence the result.

A category C is called idempotent complete if for any object X and any idempotent e ∈ End_{C}(X) there is an object Y as well as morphisms s : X → Y and t : Y → X such that e = t ∘ s and s ∘ t = Id_{Y}. Any category C can be embedded into its idempotent completion Ĉ, also called Karoubienne, constructed as follows: the objects of Ĉ are pairs (X, e) consisting of an object X in C and an idempotent e in End_{C}(X); a morphism in Ĉ from (X, e) to (Y, f) is a triple (e, s, f) where s : X → Y is a morphism in C satisfying s ∘ e = s = f ∘ s. There is an obvious embedding from C to Ĉ sending an object X in C to (X, Id_{X}). With the previous notation, two objects (X, e) and (Y, f) are isomorphic in Ĉ if there are morphisms s : X → Y and t : Y → X in C satisfying s ∘ e = s = f ∘ s, t ∘ f = t = e ∘ t, t ∘ s = e and s ∘ t = f. In that case, we also say that the idempotents e and f are isomorphic. Note that in that case s ∘ t ∘ s = s and t ∘ s ∘ t = t, and hence if C is an inverse category, we have t = ŝ. In other words, in an inverse category C, two idempotents e ∈ End_{C}(X) and f ∈ End_{C}(Y) are isomorphic if and only if there is a morphism s : X → Y satisfying ŝ ∘ s = e and s ∘ ŝ = f.

Proposition 2.4. Let C be a small category. Then C is an inverse category if and only if the idempotent completion Ĉ of C is an inverse category.

Proof. Let X, Y be objects in C and e ∈ End_{C}(X), f ∈ End_{C}(Y) idempotents. Let (e, s, f) : (X, e) → (Y, f) be a morphism in Ĉ; that is, s : X → Y is a morphism in C satisfying f ∘ s = s = s ∘ e. If C is an inverse category then ŝ ∘ f̂ = ŝ = ê ∘ ŝ, so (f, ŝ, e) is a morphism in Ĉ from (Y, f) to (X, e). An easy verification shows that Ĉ is indeed an inverse category, and a similar argument shows the converse.

By the Preston-Wagner representation theorem (cf. [15, §1.5]), any inverse semigroup can be embedded into an inverse monoid I(M) of partial bijections on a non empty set M. A similar statement holds for inverse categories (cf. [8]). A partition of a non empty set M is a set λ of pairwise disjoint non empty subsets of M whose union is equal to M. Any such partition gives rise to an inverse category I(λ), defined as follows. The object set of I(λ) is equal to λ. For any two subsets U, V of M in λ, the morphism set Hom_{I(λ)}(U, V) is the set of all bijective maps s : U′ ≅ V′ between a subset U′ of U and a subset V′ of V, including the unique bijection between empty subsets; the morphism ŝ is defined as s^{−1} : V′ ≅ U′. Composition of morphisms in I(λ) is induced by composition in the inverse monoid I(M); that is, given U, V, W in λ, a subset U′ of U, subsets V′, V′′ of V and a subset W′ of W, and bijections s : U ≅ V′, t : V′′ ≅ W′, the composition t ∘ s in I(λ) is the induced bijection t|_{V′∩V′′} ∘ s|_{s^{−1}(V′∩V′′)} : s^{−1}(V′ ∩ V′′) ≅ t(V′ ∩ V′′).

Theorem 2.5. Let C be an inverse category. Let M = Mor(C) be the set of all morphisms in C. For any object X in C denote by M_{X} the set of all morphisms in C terminating at X. Then the set of subsets λ = {M_{X} | X ∈ Ob(C)} is a partition of M and there is a functor Φ : C → I(λ) with the following properties.

(i) Φ sends an object X in C to the set M_{X} ∈ λ; (ii) Φ induces an injective map Hom_{C}(X, Y) → Hom_{I(λ)}(Φ(X), Φ(Y)) for any two objects X, Y in C. (iii) Φ(ŝ) = Φ̂(s) for any morphism s in C. (iv) We have s ≤ t if and only if Φ(s) ≤ Φ(t), for any two morphisms s, t in C.

Proof. The sets M_{X}, with X ∈ Ob(C) are clearly pairwise disjoint and their union is the set of all morphisms in C, so this determines a partition λ of M. By 2.3, for any morphism s : X → Y in C we have s ∘ M_{X} = s ∘ ŝ ∘ M_{Y}. We define the functor Φ as follows. On objects, we set Φ(X) = M_{X}, for any X ∈ Ob(C). Let s : X → Y be a morphism in C. Then ŝ ∘ s ∘ M_{X} is a subset of M_{X}, and s ∘ ŝ∘ M_{Y} is a subset of M_{Y}. The map Φ(s) : ŝ∘ s ∘ M_{X} → s ∘ ŝ∘ M_{Y} sending f ∈ ŝ ∘ s ∘ M_{X} to s ∘ f is a bijection, with inverse Φ(ŝ) sending g ∈ s ∘ ŝ ∘ M_{Y} to ŝ ∘ g. Thus, Φ(s) defined in this way is a morphism from M_{X} to M_{Y} in I(λ) satisfying Φ(ŝ) = Φ̂(s). We need to show that this assignment is functorial. Let s : X → Y and t : Y → Z be morphisms in C. Both morphisms Φ(t ∘ s) and Φ(t) ∘ Φ(s) are induced by composing morphism sets with t ∘ s; we only need to check that their domain in M_{X} is equal. The map Φ(t ∘ s) is defined on the set t̂ ∘ s∘t∘s∘M_{X} = ŝ∘t̂∘t∘s∘M_{X}. Since s∘M_{X} = s∘ŝ∘M_{Y} this set is equal to ŝ∘t̂∘t∘s∘ŝ∘M_{Y}. Since idempotents commute, this set is in fact equal to ŝ ∘ s ∘ ŝ ∘ t̂∘ t ∘ M_{Y} = ŝ ∘ t̂∘ t ∘ M_{Y}. Now Φ(s)(ŝ ∘ t̂∘ t ∘ M_{Y}) = s ∘ ŝ ∘ t̂∘ t ∘ M_{Y} = s ∘ ŝ ∘ M_{Y} ∩ t ∘ t̂∘ M_{Y}, where we use again that s ∘ ŝ and t̂∘ t are commuting idempotents. This intersection is precisely the intersection of the image of Φ(s) and the domain of Φ(t), whence the equality Φ(t ∘ s) = Φ(t) ∘ Φ(s). Thus Φ is a functor. By construction, Φ is bijective on objects. An equality Φ(s) = Φ(t) implies s = s ∘ ŝ ∘ s = Φ(s)(ŝ ∘ s) = Φ(t)(ŝ ∘ s) = t ∘ ŝ ∘ s, hence s ≤ t, and exchanging the roles of s and t yields also t ≤ s, hence s = t, which shows that Φ is injective on morphism sets. The remaining two statements (iii) and (iv) are trivial verifications.

Remarks 2.6. (1) The uniqueness of the morphism ŝ in the definition of an inverse category implies that statement (iii) in the above theorem holds automatically; that is, if C, D are inverse categories and if Φ : C → D is a functor then Φ(ŝ) = Φ̂(s) for any morphism s in C.

(2) It is well-known that a monoid S in which all idempotents commute and which admits an involution s → ŝ satisfying s = sŝs is an inverse monoid. A similar statement holds for inverse categories. An involution of a small category C is a bijection on the morphism set of C sending a morphism s : X → Y in C to a morphism ŝ: Y → X such that ˆ ŝ = s and t̂ ∘ s = ŝ ∘ t̂ for any two composable morphisms s, t. In particular, a small category with an involution is isomorphic to its opposite. One can show that if in addition s ∘ ŝ∘ s = s for any morphism s in C and if any two idempotents in End_{C}(X) commute, for any object X in C, then C is an inverse category.

(3) If C is a finite inverse category then every endomorphism monoid End_{C}(X) of an object X of C has a unique minimal idempotent endormophism e_{X}, namely the product of all idempotent endomorphisms in End_{C}(X). Thus, for any morphism s : X → Y the set {t : X → Y | t ≤ s} has a unique minimal element, namely t = s ∘ e_{X} = e_{Y} ∘ s.

(4) Let C be a finite inverse category, D an inverse subcategory of C and Y an object in D. Then End_{D}(Y) contains all idempotents in End_{C}(Y) if and only if End_{D}(Y) is a downwardly closed subposet of End_{C}(Y); that is, if and only if for s, t ∈ End_{C}(Y) satisfying t ≤ s and s ∈ End_{D}(Y) we have t ∈ End_{D}(Y). Indeed, since End_{D}(Y) contains Id_{Y}, if End_{D}(Y) is downwardly closed it contains all idempotents in End_{C}(Y), and the converse follows from 2.1

Second cohomology classes of a category C with coefficients in an abelian group, viewed as a constant functor, correspond to category extensions of C (cf. [1]); we refer to [23, §7] for an exposition and further references on this material. Extensions of inverse categories by constant functors are again inverse categories:

Theorem 2.7. Let C be an inverse category and A an abelian group, viewed as a constant contravariant functor on C. Let D be an extension of C by A. Then D is an inverse category. Moreover, the canonical functor D → C induces an isomorphism E(D) ≅ E(C) between the partially ordered sets E(D), E(C) of idempotent endomorphisms in D and C, respectively.

Proof. An extension category D of C by A has the same object set as C and morphism set Mor(D) × A; the composition of morphisms in D is given by

(t, b) ∘ (s, a) = (t ∘ s, baα(t, s))

for any two composable morphisms s, t in C and a, b ∈ A, where α is a function sending any two composable morphisms s, t in C to an element α(t, s) in A, such that α satisfies for any three composable morphisms s, t, u in C the 2-cocycle identity

α(u ∘ t, s)α(u, t) = α(u, t ∘ s)α(t, s)

This identity is equivalent to the associativity of the composition of morphisms in D. Up to isomorphism of extensions, D depends only on the class of α in H^{2}(C; A). Let s be a morphism in C. The 2-cocyle identity applied to the three composable morphisms ŝ, s ∘ ŝ, s yields the equality

α(ŝ ∘ s ∘ ŝ, s)α(ŝ, s ∘ ŝ) = α(ŝ, s ∘ ŝ ∘ s)α(s ∘ ŝ, s)

Since ŝ ∘ s ∘ ŝ = ŝ and s ∘ ŝ ∘ s = s, this implies

α(ŝ, s ∘ ŝ) = α(s ∘ ŝ, s)

Let σ = (s, a) be a morphism in D. A morphism σ̂ satisfying σ ∘ σ̂ ∘ σ = σ must necessarily be of the form (ŝ, a^{−1}γ(s)) for some γ(s) ∈ A. Then σ ∘ σ̂ ∘ σ = (s ∘ ŝ, α(s, ŝ)γ(s)) ∘ (s, a) = (s, aα(s ∘ ŝ, s)α(s, ŝ)γ(s)), so this forces γ(s) = α(s ∘ ŝ, s)^{−1}α(s, ŝ)^{−1}. By the equality above we also have γ(s) = α(ŝ, s ∘ ŝ)^{−1}α(s, ŝ)^{−1}. Thus σ̂ ∘ σ ∘ σ̂ = (ŝ, a^{−1}γ(s)) ∘ (s ∘ ŝ, α(s, ŝ)γ(s)) = (ŝ, a^{−1}α(ŝ, s ∘ ŝ)α(s, ŝ)γ(s)^{2}) = (ŝ, a^{−1}γ(s)) = σ̂. This shows that D is indeed an inverse category. Any idempotent endomorphism in D is necessarily of the form (e, a) for some idempotent endomorphism e in C and some a ∈ A. The equality (e, a) = (e, a) ∘ (e, a) = (e, a^{2}α(e, e)) forces a = α(e, e)^{−1}. Thus the canonical functor D → C induces a bijection between the sets of idempotent endomorphisms E(D) and E(C) in D and C. To see that this bijection is an isomorphism of posets we need to show that if e, f are idempotent endomorphisms in C satisfying e ≤ f then also (e, α(e, e)^{−1})) ≤ (f, α(f, f)^{−1}). The 2-cocycle identity applied to e, f, f yields α(e, f)α(f, f) = α(ef, f)α(e, f); using the equality ef = e this yields α(e, f) = α(f, f). Thus (e, α(e, e)^{−1})(f, α(f, f)^{−1}) = (ef, α(e, f)α(e, e)^{−1}α(f, f)^{−1}) = (e, α(e, e)^{−1}) as required.

3 The Möbius function for finite inverse categories

The purpose of this section is to verify that results of B. Steinberg in [22, §4] carry over, with no difficulty, to finite inverse categories. As before, we give detailed proofs for the convenience of the reader. The Möbius function μ of a finite partially ordered set (P, ≤) with coefficients in a commutative ring k is defined on the set of pairs (x, y) ∈ P × P satisfying x ≤ y, by μ(x, x) = 1 for x ∈ P and ∑_{x≤u≤y} μ(u, y) = 0 if x < y. The Möbius Inversion Theorem, due to Rota, states that if f, g are functions from P to k such that g(x) = ∑_{u≤x} f(u) for all x ∈ P then f(x) = ∑_{u≤x} μ(u, x)g(u) for all x ∈ P. Given a finite inverse category C and a commutative ring k, we consider the set of morphisms in C endowed with the natural partial order as in the previous section and use the letter μ for the corresponding Möbius function with coefficients in k. For any morphism s in C we define an element \underline{s} in the subspace kHom_{C}(X, Y) of the category algebra kC by setting

\underline{s} = ∑_{u≤s} μ(u, s)u

Möbius inversion implies that s = ∑_{u≤s} \underline{u}, and hence the set of \underline{s}, with s running over all morphisms in C, is again a k-basis of kC. Since u ≤ s if and only if û ≤ ŝ we have μ(û, ŝ) = μ(u, s).

Proposition 3.1. Let C be a finite inverse category and k a commutative ring. For any two composable morphisms s : X → Y and t : Y → Z in C the following hold.

(i) We have t · \underline{s} = \underline{t ∘ s} if s ∘ ŝ ≤ t̂∘ t and t · \underline{s} = 0 otherwise. (ii) We have \underline{t} · \underline{s} = \underline{t ∘ s} if s ∘ ŝ = t̂∘ t and \underline{t} · \underline{s} = 0 otherwise.

The proof is essentially a variation of the arguments in [22, §4]; we break this up in a series of lemmas.

Lemma 3.2. Let C be an inverse category and let s : X → Y, t : Y → Z be two composable morphisms in C. The map sending a morphism u ≤ s to t ∘ u induces a bijection of sets

{u | u ≤ t̂∘ t ∘ s} ≅ {w | w ≤ t ∘ s}

with inverse sending w ≤ t ∘ s to t̂∘ w.

Proof. If u ≤ t̂∘t∘s then u = t̂∘t∘u, and if w ≤ t∘s then w = t∘s∘ŵ∘w, hence t∘t̂∘w = w.

Lemma 3.3. Let C be an inverse category and let s : X → Y, t : Y → Z be two composable morphisms in C. We have s ∘ ŝ ≤ t̂∘ t if and only if s = t̂∘ t ∘ s.

Proof. We have s ∘ ŝ ≤ t̂∘ t if and only if s ∘ ŝ = t̂∘ t ∘ s ∘ ŝ. Precomposing with s shows that this equality implies the equality s = t̂∘ t ∘ s, and precomposing by ŝ shows that the two are indeed equivalent.

Lemma 3.4. Let C be an inverse category and s a morphism in C. We have \underline{s} = s − ∑_{u<s} \underline{u}.

Proof. By Möbius inversion we have s = ∑_{u≤s} \underline{u} = \underline{s} + ∑_{u<s} \underline{u}, whence the result.

Proof of Proposition 3.1. We prove (i) by induction, assuming that the result holds for s′ < s instead of s. Assume first that s ∘ ŝ ≤ t̂∘ t, which by 3.3 is equivalent to s = t̂∘ t ∘ s. Then for u ≤ s we also have u ∘ û ≤ t̂∘ t. By 3.4, we have \underline{s} = s − ∑_{u<s} \underline{u}. By induction, we have

t · \underline{s} = t ∘ s − ∑_{u<s} t · \underline{u} = t ∘ s − ∑_{u<s} \underline{t ∘ u}

Since s = t̂∘ t ∘ s it follows from 3.2 that if u runs over the morphisms satisfying u < s then t ∘ u runs over the morphisms w satisfying w < t ∘ s. Thus

t · \underline{s} = t ∘ s − ∑_{w<t∘s} \underline{w} = \underline{t ∘ s}

Assume next that s ∘ ŝ ≰ t̂∘ t; again by 3.3, this is equivalent to t̂∘ t ∘ s < s. Thus if u ≤ s such that u∘û ≤ t̂∘t then u = u∘û∘s ≤ t̂∘t∘s < s. By induction we get t·\underline{s} = t∘s−∑_{u≤t̂∘t∘s} \underline{t ∘ u} and hence 3.2 implies that

t · \underline{s} = t ∘ s − ∑_{w≤t∘s} \underline{w} = t ∘ s − t ∘ s = 0

which completes the proof of (i). Using (i) and 3.3 we get

\underline{t} · \underline{s} = ∑_{v≤t} μ(v, t)v · \underline{s} = ∑_{v≤t; v̂∘v∘s=s} μ(v, t)\underline{v ∘ s}

If t̂∘t∘s < s then v̂∘v ∘s < s for all v ≤ t, so \underline{t}·\underline{s} = 0 in that case. Assume now that t̂∘t∘s = s. We show first that if v ≤ t such that v̂ ∘ v ∘ s = s then v ∘ s = t ∘ s. Indeed, since v ≤ t we have v = t ∘ v̂ ∘ v, hence v ∘ s = t ∘ v̂ ∘ v ∘ s = t ∘ s. This also implies t ∘ s ∘ ŝ = v ∘ s ∘ ŝ ≤ v. Therfore we have

\underline{t} · \underline{s} = ∑_{v≤t; v̂∘v∘s=s} μ(v, t) \underline{v ∘ s} = ( ∑_{t∘s∘ŝ≤v≤t} μ(v, t)) \underline{t ∘ s}

This sum is zero if t ∘ s ∘ ŝ < s, and equal to \underline{t ∘ s} if t ∘ s ∘ ŝ = t. Since the two conditions t̂∘ t ∘ s = s and t ∘ s ∘ ŝ = t are equivalent to the equality s ∘ ŝ = t̂∘ t, statement (ii) follows.

For the sake of completeness we point out that one can also prove 3.1 by making use of the following observation, which is from the argument used in the proof of [22, 4.1]:

Lemma 3.5. Let C be an inverse category and let s : X → Y, t : Y → Z be two composable morphisms in C. Let w : X → Z be a morphism in C such that w ≤ t ∘ s. Then u = s ∘ w ∘ ŵ and v = w ∘ ŵ ∘ t are the unique morphisms in C satisfying u ≤ s, v ≤ t, v̂ ∘ v = u ∘ û and v ∘ u = w. Moreover, we have u = t̂∘ w and v = w ∘ ŝ.

Proof. Since w ≤ t ∘ s we have w = w ∘ ŵ ∘ t ∘ s. Suppose first that u ≤ s, v ≤ t are morphisms such that v ∘ u = w and v̂ ∘ v = u ∘ û. Then v ∘ v̂ ∘ w = w = w ∘ û ∘ u and v = v ∘ v̂ ∘ v = v ∘ u ∘ û = w ∘ û. Similarly, u = v̂ ∘ w, hence also û = ŵ ∘ v. Since u ≤ s we have u = s ∘ û ∘ u = s ∘ ŵ ∘ v ∘ v̂ ∘ w = s ∘ ŵ ∘ w. Similarly, since v ≤ t we have v = v ∘ v̂ ∘ t = w ∘ û ∘ u ∘ ŵ ∘ t = w ∘ ŵ ∘ t. Conversely, set u = s ∘ w ∘ ŵ and v = w ∘ ŵ ∘ t. Then clearly u ≤ s and v ≤ t. Moreover, v ∘ u = w ∘ ŵ ∘ t ∘ s ∘ ŵ ∘ w = w because t ∘ s = w and w ∘ ŵ ∘ w = w. We show next that û = ŵ ∘ t. Indeed, we have u ∘ (ŵ ∘ t) ∘ u = s ∘ ŵ ∘ w ∘ t ∘ s ∘ ŵ ∘ w = s ∘ ŵ ∘ w = u, and we have (ŵ ∘ t) ∘ u ∘ (ŵ ∘ t) = ŵ ∘ t ∘ s ∘ ŵ ∘ w ∘ ŵ ∘ t = ŵ ∘ t. This shows û = ŵ ∘ t, hence u = t̂∘ w. A similar argument yields v̂ = s ∘ ŵ, and hence v = w ∘ ŝ. Finally, we have u ∘ û = (s ∘ ŵ ∘ w) ∘ (ŵ ∘ t) = (s ∘ ŵ) ∘ (w ∘ ŵ ∘ t) = v̂ ∘ v, which concludes the proof.

Remark 3.6. Let C be a finite inverse category. If s, t are morphisms in C such that s ≤ t and such that t is an idempotent endomorphism of an object X in C, then 2.1 together with the fact that idempotents in End_{C}(X) commute, implies that s is an idempotent endomorphism as well. Thus the set E(C) of all idempotent endomorphisms in C is a downwardly closed subposet of the set Mor(C) of all morphisms in C. In particular, the Möbius function of E(C) is the restriction of that of Mor(C). Moreover, End_{C}(X) has a unique minimal idempotent e_{X}, namely the product of all idempotents in End_{C}(X). Thus, for any idempotent e in End_{C}(X), we have ∑_{f;f≤e} μ(f, e) = ∑_{e_{X}≤f≤e} μ(f, e) = 0, unless e = e_{X}, in which case this sum is equal to 1.

4 Finite inverse category algebras

Let C be an inverse category. The groupoid associated with C is the category, denoted G(C), defined as follows. The objects of G(C) are the pairs (X, e) with X an object in C and e an idempotent in End_{C}(X). A morphism in G(C) from (X, e) to (Y, f) is a triple (e, s, f), where s : X → Y is a morphism in C satisfying ŝ ∘ s = e and s ∘ ŝ = f. Note that in that case s = s∘e = f ∘s. In other words, G(C) is the subcategory of the idempotent completion Ĉ of C having the same objects as Ĉ and all isomorphisms in Ĉ as morphisms; in particular, G(C) is indeed a groupoid (and this definition of G(C) makes sense for any small category C). As mentioned earlier, the idempotents e, f are called isomorphic if the objects (X, e), (Y, f) are isomorphic in G(C). By the above this is equivalent to the existence of a morphism (X, e) → (Y, f) in G(C) and also equivalent to the existence of a morphism s : X → Y in C satisfying ŝ∘s = e and s∘ŝ = f. The group C_{e} considered in the introduction is canonically isomorphic to Aut_{G(C)}(X, e) via the map sending s ∈ End_{C}(X) satisfying s ∘ ŝ = e = ŝ ∘ s to (e, s, e). The category algebra of the groupoid G(C) turns out to be isomorphic to the category algebra of the category C itself; this extends [22, 4.2] to inverse categories:

Theorem 4.1. Let C be a finite inverse category and k a commutative ring. The map sending a morphism (e, s, f) : (X, e) → (Y, f) in G(C) to the element \underline{s} in kC, where X, Y are objects in C and e ∈ End_{C}(X), f ∈ End_{C}(Y) are idempotents, is an isomorphism of k-algebras kG(C) ≅ kC.

Proof. The map sending (e, s, f) to \underline{s} is multiplicative by 3.1 (ii). This map is also a k-linear isomorphism because, by Möbius inversion, the set of \underline{s}, with s ∈ Mor(C), form again a k-basis of kC.

Finite inverse categories provide thus examples of non-equivalent categories with isomorphic category algebras. Even though there is a canonical forgetful functor G(C) → C the isomorphism kG(C) ≅ kC is in general not induced by any functor. In particular a constant functor on C need not correspond to a constant functor on G(C). A groupoid algebra over a commutative ring k is well-known to be isomorphic to a direct product of matrix algebras over the group algebras of automorphism groups of objects, one factor for each isomorphism class of objects, and the size of that isomorphism class is equal to the size of the involved matrices. Translated to the situation of 4.1 this yields the following statement:

Corollary 4.2. Let C be a finite inverse category, k a commutative ring and E a set of representatives of the isomorphism classes of idempotent endomorphisms in C. For e ∈ E denote by n(e) the number of idempotents in Mor(C) isomorphic to e. We have an isomorphism of k-algebras kC ≅ ∏_{e∈E} M_{n(e)}(kC_{e}).

This isomorphism shows that if k is a field of characteristic zero or prime characteristic not dividing the order of any of the groups C_{e}, where e runs over the idempotents in Mor(C), then kC is semi-simple. This generalises well-known facts about inverse semi-groups [19], [20]. This isomorphism shows further that the category of functors from C to Mod(k) is equivalent to the product of the module categories of the group algebras kC_{e}; thus any block of kC is Morita equivalent to a block of one of the finite group algebras kC_{e}. In particular, every functor F : C → Mod(k) decomposes naturally as direct sum of functors F = ⊕_{e∈E}F_{e}. Using the isomorphism from 4.1, one can describe the functors F_{e} and their cohomological properties more explicitly. An idempotent endomorphism e of an object X in C, when viewed as the morphism (e, e, e) in G(C), is mapped under the isomorphism kG(C) ≅ kC from 4.1 to the idempotent \underline{e} = ∑_{f≤e} μ(f, e)f in kEnd_{C}(X). If F : C → Mod(k) is a functor, setting F(\underline{e}) = ∑_{f≤e} μ(f, e)F(f), we get that F(\underline{e}) is an idempotent endomorphism of the k-module F(X), and its image F(\underline{e})(F(X)) is a kC_{e}-module. If F is a constant functor on C then F(\underline{e}) = ∑_{f≤e} μ(f, e)Id_{X}, and hence, by 3.6 we have F(\underline{e}) = 0 unless e is a minimal idempotent with respect to the canonical partial order on the morphism set of C. Combining these observations with the isomorphism 4.1 yields immediately the following statements:

Corollary 4.3. Let C be a finite inverse category, k a commutative ring and E a set of representatives of the isomorphism classes of idempotent endormophisms in C.

(i) Every functor F : C → Mod(k) can be written uniquely as direct sum F = ⊕_{e∈E} F_{e} of functors F_{e} : C → Mod(k) satisfying F_{e}(\underline{f}) = F(\underline{f}) if e, f are isomorphic idempotent endomorphisms in C, and F_{e}(\underline{f}) = 0, otherwise. (ii) For any two functors F, G : C → Mod(k) and e ∈ E we have

Ext_{kC}^{∗}(F_{e}, G_{e}) ≅ Ext_{kC_{e}}^{∗}(F(\underline{e})(F(X)), G(\underline{e})(G(X)))

and Ext_{kC}^{∗}(F_{e}, G_{f}) = {0} if e, f ∈ E are different.

(iii) If F : C → Mod(k) is a constant functor then F = ⊕_{e} F_{e}, where e runs over a set of representatives of the isomorphism classes of minimal idempotent endomorphisms in C; in particular, F(\underline{e}) = 0 if e is an idempotent endomorphism which is not minimal. (iv) We have H^{∗}(C; k) ≅ ∏_{e} H^{∗}(C_{e}; k), where e runs over a set of representatives of the isomorphism classes of minimal idempotent endomorphisms in C.

(v) We have HH^{∗}(kC) ≅ ∏_{e∈E} HH^{∗}(kC_{e}).

The following statement on symmetrising forms of groupoid algebras is well-known and easily verified (we leave the proof to the reader):

Lemma 4.4. Let k be a commutative ring and G a finite groupoid. The category algebra kG is symmetric. More precisely, there is a bimodule isomorphism (kG)^{∗} ≅ kG sending μ ∈ (kG)^{∗} to the element ∑_{s∈Mor(G)} μ(s^{−1})s in kG. The symmetrising form τ ∈ (kG)^{∗} corresponding to 1_{kG} under this isomorphism is satisfies τ(s) = 1 if s = Id_{X} for some X ∈ Ob(G) and τ(s) = 0 otherwise.

Proof of 1.1. We have already noted in 4.2 that kC is a direct product of matrix algebras over finite group algebras. In order to prove the statement on the symmetrising form, we need to follow the canonical symmetrising forms on the finite groupoid algebra kG(C) from 4.4 through the isomorphism in 4.1. The identity morphism of an object (X, e) in the groupoid G(C) is of the form (e, e, e), which under the isomorpism from 4.1 is sent to \underline{e}. Thus kC inherits a canonical symmetrising form sending \underline{s} to 1 if s is an idempotent and to zero otherwise. Since s = ∑_{u≤s} \underline{u} by Möbius inversion, the formula for τ follows.

Proof of 1.2. By a result of Gerstenhaber [7], the Hochschild cohomology of an algebra is graded commutative. The Hochschild cohomology of a finite group algebra over a commutative Noetherian ring is finitely generated. Since Hochschild cohomology is invariant under Morita equivalences and compatible with direct products of algebras, the finite generation of HH^{∗}(kC) follows. By [24, Theorem A], H^{∗}(C; k) is a quotient of HH^{∗}(kC), whence the result.

Proof of 1.3. This follows from 4.3 (iv) and the standard fact [10, (11.15)].

Lemma 4.5. Let k be a commutative ring, G a finite groupoid and H a subgroupoid of G. Set j = 1_{kH} = ∑_{Y ∈Ob(H)} Id_{Y}. Then kGj is finitely generated projective as a right kH-module and jkG is finitely generated as a left kH-module.

Proof. Projectivity of a module is invariant under Morita equivalences, and hence we may assume that the objects of H are pairwise non-isomorphic. Thus H is a direct product of groups. The result follows from the fact that the action of Aut_{H}(Y) on Hom_{G}(X, Y) and Hom_{G}(Y, X) is free, for any object Y in H and any object X in G.

Under certain circumstances, the map in 4.1 is functorial, proving in particular theorem 1.4:

Theorem 4.6. Let k be a commutative ring, C a finite inverse category and D an inverse subcategory of C such that for any object Y in D the endomorphism monoid End_{D}(Y) contains all idempotent endomorphisms in End_{C}(Y). Then the following diagram of non-unitary algebra homomorphisms is commutative:

\xymatrix{kG(D) \ar[r]^{≅} \ar[d] & kD \ar[d] \\ kG(C) \ar[r]_{≅} & kC}

where the horizontal maps are the isomorphisms from 4.1 and the vertical maps are the injective non-unitary algebra homomorphisms induced by the corresponding inclusions of categories. In particular, setting j = 1_{kD} = ∑_{Y ∈Ob(D)} Id_{Y}, the kC-kD-bimodule kCj and the kD-kC-bimodule jkC are both finitely generated projective as left and right modules. Moreover, the restriction to kD of the canonical symmetrising form of kC is the canonical symmetrising form of kD.

Proof. Since Mor(D) contains all idempotents in endomorphism monoids in C of objects in D it follows from 2.1 (ii) that if s, t are morphisms in C satisfying s ≤ t and if t is a morphism in D then s is a morphism in D as well. In other words, Mor(D) is a downwardly closed subposet of Mor(C), and hence the Möbius function on Mor(D) is the restriction of the Möbius function on Mor(C). This means that for a morphism s in D the expression \underline{s} = ∑_{u≤s} μ(u, s)u does not depend on whether we regard s as a morphism in C or in D. This shows the commutativity of the diagram in the statement. The statement on projectivity follows from 4.5. The description of the canonical symmetrising forms in 1.1 implies the last statement.

Example 4.7. Let λ be a partition of a finite set M and k a commutative ring. Any idempotent endomorphism in the inverse category I(λ) defined in §2 is of the form Id_{V} for a subset V of a set U ∈ λ. Given subsets V ⊆ U, V′ ⊆ U′, where U, U′ ∈ λ, the idempotents Id_{V} and Id_{V′} are isomorphic if and only if V, V′ have the same cardinality. Thus, setting m = |M| and i(n) = ∑_{U∈λ} \binom{|U|}{n} for any nonnegative integer n, we have an isomorphism of k-algebras

kI(λ) ≅ ∏_{0≤n≤m} M_{i(n)}(kS_{n})

where S_{n} denotes as usual the symmetric group of degree n, with the convention i_{0} = |λ| and S_{0} = {1}; the factor for n = 0 corresponds to the isomorphism class of idempotents associated with the empty subsets of the subsets of M in λ.

5 Adjunction morphisms for groupoid algebras

Throughout this section, k is a commutative ring. If A is a k-algebra, i an idempotent in A and B a unitary subalgebra of iAi then the ‘truncated restriction’ functor sending an A-module U to the B-module iU is isomorphic to the functor iA⊗_{A} −, where iA is considered as B-A-bimodule. This functor has as left adjoint the ‘truncated induction’ functor Ai ⊗_{B} −. There is a canonical choice of an adjunction isomorphism (this is a special case of the standard adjunction between Hom-functors and the tensor product - the proof is left to the reader):

Lemma 5.1. Let A be a k-algebra, i an idempotent in A and B a unitary subalgebra of iAi. The functor Ai ⊗_{B} − is left adjoint to the functor iA ⊗_{A} −, and there is a canonical adjunction isomorphism whose unit and counit are as follows:

(i) The adjunction unit is represented by the inclusion B → iAi ≅ iA ⊗_{A} Ai, viewed as B-B-bimodule homomorphism. (ii) The adjunction counit is represented by the A-A-bimodule homomorphism Ai ⊗_{B} iA → A induced by multiplication in A.

If A, B are symmetric and Ai is finitely generated projective as right B-module then Ai⊗_{B} − is also right adjoint to iA ⊗_{A} −. More precisely, any choice of symmetrising forms on A, B determines an adjunction isomorphism whose adjunction unit and counit are obtained from dualising the adjunction unit and counit from the left adjunction of Ai ⊗_{B} − to iA − ⊗_{A}−; see e.g. for some background material. In that case we get relativly projective elements π_{Ai} ∈ Z(A) and π_{iA} ∈ Z(B) defined as follows (cf. [16, 3.1]): the composition A → Ai ⊗_{B} iA → A of the appropriate adjunction unit and counit yields an endomorphism of A as A-A-bimodule, which is hence induced by left (or right) multiplication by an element π_{Ai} ∈ Z(A); similarly, the composition B → iA ⊗_{A} Ai → B of the appropriate adjunction unit and counit is an endomorphism of B as B-B-bimodule, which is hence induced by left or right multiplication by a an element π_{iA} ∈ Z(B). The purpose of this section is to calculate these elements when A is a finite groupoid algebra and B a subgroupoid algebra; for finite group algebras these calculations are well-known.

For the remainder of this section, let G be a finite groupoid, H a subgroupoid of G and j = 1_{kH} = ∑_{Y ∈Ob(H)} Id_{Y}, the sum taken in kG. As mentioned in 4.4, the algebra kG is symmetric with a canonical symmetrising form τ ∈ (kG)^{∗} satisfying τ(s) = 1 if s = Id_{X} for some X ∈ Ob(G) and τ(s) = 0 otherwise. The restriction of τ to kH yields the corresponding symmetrising form for kH. We consider kG and kH endowed with their canonical symmetrising forms. Denote by Mor(G/H) the set of all morphisms s in G which start at an object in H, or equivalently, which satisfy s · j = s in kG. Two morphisms s, s′ in Mor(G/H) are called H-equivalent if there is a morphism t in H such that s′ = s ∘ t. Note that H-equivalent morphisms have necessarily the same target. We denote by G/H a set of representatives of the H-equivalence classes of morphisms in Mor(G/H). We denote by π_{G/H} the element in kGj ⊗_{kH} jkG defined by

π_{G/H} = ∑_{s∈G/H} s ⊗ s^{−1}

This element does not depend on the choice of G/H and we have s · π_{G/H} = π_{G/H} · s for any morphism s in G.

Lemma 5.2. With the notation above, the functor kGj ⊗_{kH} − is right adjoint to the functor jkG ⊗_{kG} − and there is a canonical choice of an adjunction isomorphism whose unit and counit are as follows:

(i) The adjunction unit is represented by the kG-kG-bimodule homomorphism kG → kGj⊗_{kH}jkG sending 1_{kG} to π_{G/H}. (ii) The adjunction counit is represented by the kH-kH-bimodule homomorphism jkG ⊗_{kG} kGj ≅ jkGj → kH induced by the map sending s ∈ Mor(H) to s and s ∈ Mor(G) \ Mor(H) to 0.

Proof. One way to prove this is by dualising the adjunction unit and counit of the left adjunction of kGe ⊗_{kH} − to ekG ⊗_{kG} −; this is a standard argument (see e.g. [16, §6]). Alternatively, an easy verification shows that the appropriate compositions kGj → kGj ⊗_{kH} jkG ⊗_{kG} kGj → kGj and jkG → jkG ⊗_{kG} kGj ⊗_{kH} jkG → jkG of the given maps in the statement are the identity bimodule endomorphisms on kGj and jkG, respectively.

For any object X in G define an integer n(X) by

n(X) = ∑_{Y} [Aut_{G}(Y) : Aut_{H}(Y)]

where Y runs over a set of representatives of the H-isomorphism classes of objects contained in the G-isomorphism class of X, with the usual convention that this is zero if the sum is empty.

Lemma 5.3. With the notation above, for any object X in G, the integer n(X) is equal to the number of morphisms in G/H ending at X; in particular, we have |G/H| = ∑_{X} n(x), where X runs over a set of representatives of the isomorphism classes of objects in G.

Proof. If s : Y → X and s′ : Y′ → X are two morphisms in G ending at X then Y, Y′ belong to the same G-isomorphism class as X. Moreover, s, s′ are H-equivalent if and only if there is a morphism t : Y → Y′ in H satisfying s′ = s ∘ t, which in particular is only possible if Y, Y′ are isomorphic objects in H. The number of pairwise inequivalent morphisms from Y to X is clearly equal to the number of pairwise inequivalent automorphisms of Y, hence equal to [Aut_{G}(Y) : Aut_{H}(Y)]. The result follows.

We use this to calculate the relatively projective elements π_{kGj} ∈ Z(kG) and π_{jkG} ∈ Z(kH).

Lemma 5.4. With the notation above, we have

π_{jkG} = j = 1_{kH}

π_{kGj} = ∑_{X∈Ob(G)} n(X) · Id_{X}

Proof. The composition kH → jkG⊗_{kG} kGj → kH of the appropriate adjunction unit and counit sends 1_{kH} to 1_{kH}, where we use the first statement of 5.1 and the second statement of 5.2. The composition kG → kGj ⊗_{kH} jkG → kG of the remaining adjunction unit and counit sends 1_{kG} to ∑_{s∈G/H} s ∘ s^{−1}. The lemma follows from 5.3.

Proof of 1.5. The existence of a transfer map tr_{D}^{C} with the property stated in (i) is a purely formal consequence of 1.4 and the general reciprocity result in [18, 4.8]. Similarly, (ii) follows from [17, 5.1]. We need to calculate htr_{D}^{C}(1). This element is equal to the relative projective element π_{kCi} in Z(kC), where i = 1_{kD}. We calculate this using the results of this section applied to the groupoids associated with C , D and then applying 4.6. By 4.6, the isomorphism kG(C) ≅ kC from 4.1 maps j = 1_{kG(D)} to i = 1_{kD}. Thus π_{D}^{C} is the image under this isomorphism of the relatively projective element π_{kG(C)j}. Note that Aut_{G(C)}(X, e) = C_{e} for any object X in C and any idempotent endomorphism e of X in C. Thus, by 5.4, we have π_{kG(C)j} = ∑_{e} n(e)e, the sum taken in kG(C), where n(e) = ∑_{f} [C_{f} : D_{f}], with f running over a set of representatives of the D-isomorphism classes of idempotent endomorphisms in D contained in the C-isomorphism class of e. The isomorphism kG(C) ≅ kC from 4.1 maps this element to the element π_{D}^{C} = ∑_{e} n(e)\underline{e}. Using \underline{e} = ∑_{f≤e} μ(f, e)f yields the formula for π_{D}^{C} as in the paragraph preceding 1.5. Clearly π_{kG(C)j} is invertible if and only if all n(e) are invertible in k, hence the same is true for π_{D}^{C}. This proves (iii). In order to prove (iv) we need to calculate τ_{D}^{C} more explicitly. It follows from statement (ii) applied to η = 1 ∈ HH^{0}(kD) that tr_{D}^{C}(1) is indeed equal to the image τ_{D}^{C} of π_{D}^{C} under the canonical map HH^{0}(kC) = Z(kC) → H^{0}(C; k) = lim_{C}(k). This map sends π_{D}^{C} to the family τ_{D}^{C} = (τ_{X})_{X∈Ob(C)} in lim_{C}(k) where τ_{X} = ∑_{(e,f)} n(e)μ(f, e), whith (e, f) running over the pairs of idempotents in End_{C}(X) satisfying f ≤ e. If e is not a minimal idempotent then, by 3.6, we have ∑_{f;f≤e} n(e)μ(f, e) = 0. Thus τ_{X} = n(e_{X}) for all objects X in C. This shows (iv), and (v) follows from applying (i) with θ = 1.

6 Embeddings into inverse categories

Example 6.1. Let n be a non negative integer and view the totally ordered set n = {0, 1, 2, ⋯, n} as a category. Then n can be embedded into an inverse category n̂ defined as follows: the objects of n̂ are those of n, and for 0 ≤ i, j ≤ n, the morphism set Homn̂(i, j) consists of all triples (i, j, a) such that 0 ≤ a ≤ min{i, j}. The canonical involution on the morphism set is given by (i, j, a)^{}ˆ = (j, i, a), and the composition in n̂ is defined by (k, j, b) ∘ (i, j, a) = (i, k, min{a, b}). There is a canonical functor n → n̂ which is the identity on objects and which sends a morphism i → j in n to the morphism (i, j, i) in n̂, where 0 ≤ i ≤ j ≤ n. Idempotents in n̂ are of the form (i, i, a). Two idempotents (i, i, a), (j, j, b) are isomorphic if and only if a = b, via the morphisms (i, j, a) and (j, i, a). Thus the isomorphism class of the idempotent (a, a, a) contains the n − a + 1 idempotents (i, i, a), a ≤ i ≤ n. If a runs from 0 to n then b = n − a + 1 runs from n + 1 to 1. Moreover, the automorphism group associated with any idempotent is trivial. It follows from 4.1 that if k is a commutative ring, the algebra kn̂ is a direct product of matrix algebras ∏_{b=1}^{n+1} M_{b}(k).

The above example is a special case of more general embeddings of categories into inverse categories. This is based on a well-known construction principle for categories, described, for instance, in [21, §1]. Given a small category C in which pullbacks exist we define a category Ĉ as follows. We set Ob(Ĉ) = Ob(C). A morphism in Ĉ from X to Y is an equivalence class [U; σ, φ] of triples (U, σ, φ) consisting of an object U in C and two morphisms σ : U → X, φ : U → Y, where two such triples (U, σ, φ), (U′, σ′, φ′) are equivalent if there is an isomorphism α : U ≅ U′ in C such that σ = σ′ ∘ α and φ = φ′ ∘ α. The composition of a morphism [U; σ, φ] from X to Y and a morphism [V; τ, ψ] from Y to Z is the morphism [W; σ ∘ μ, ψ ∘ ν] such that the square in the following diagram is a pullback:

\xymatrix{X &  & Y &  & Z \\  & U \ar[ul]^{σ} \ar[ur]^{φ} &  & V \ar[ul]_{τ} \ar@{-->}[ur]_{ψ} \\  &  & W \ar[ul]^{μ} \ar[ur]_{ν}}

Using the universal property of pullbacks one checks that this composition is associative. If s = [U; σ, φ] is a morphism in Ĉ we set ŝ = [U; φ, σ]. Clearly ˆ ŝ = s and t̂ ∘ s = ŝ ∘ t̂ for two composable morphisms s and t in Ĉ. In other words, the map sending s to ŝ is an involution on Ĉ; in particular, Ĉ is isomorphic to its opposite category. There is a canonical covariant functor from C to Ĉ defined as identity on objects and sending a morphism φ : X → Y in Hom_{C}(X, Y) to the morphism [X; Id_{X}, φ] in HomĈ(X, Y). The canonical functor C → Ĉ is an isomorphism of categories if and only if C is a groupoid. There is also a canonical contravariant functor sending φ to [X; φ, Id_{X}]; the two embeddings ‘differ’ by the involution of Ĉ. The following theorem contains a restatement of 1.6.

Theorem 6.2. Let C be a small category in which any morphism is a monomorphism and in which pullbacks exist. Let k be a commutative ring and A a k-module.

(i) The category Ĉ as defined above is an inverse category, the canonical functor C → Ĉ is injective on morphisms and Ĉ is idempotent complete. (ii) If C is finite then Ĉ is finite and the category algebra kĈ is Morita equivalent to ∏_{U} kAut_{C}(U), where U runs over a set of representatives of the isomorphism classes of objects in C. (iii) Restriction induces an isomorphism H^{∗}(Ĉ; A) ≅ H^{∗}(C; A). (iv) Let Φ be a covariant functor from C to an inverse category D with the property that if

\xymatrix{ & Y \\ U \ar[ur]^{φ} &  & V \ar[ul]_{τ} \\  & W \ar[ul]^{μ} \ar[ur]_{ν}}

is a pullback square in C then the diagram

\xymatrix{ & Φ(Y) \ar[dl]_{{ \atop Φ(φ)}} \\ Φ(U) &  & Φ(V) \ar[ul]_{Φ(τ)} \ar[dl]^{{ \atop Φ(μ)}} \\  & Φ(W) \ar[ul]^{Φ(μ)}}

in D is commutative. Then Φ extends uniquely to a functor Φ̂ from Ĉ to D.

Proof. Let s = [U; σ, φ] and t = [V; τ, ψ] be morphisms in Ĉ satisfying s ∘ t ∘ s = s and t ∘ s ∘ t = t. The morphism s ∘ t ∘ s is represented by a commutative diagram of the form

\xymatrix{X &  & Y &  & X &  & Y \\  & U \ar[ul]_{σ} \ar[ur]^{φ} &  & V \ar[ul]_{τ} \ar@{-->}[ur]^{ψ} &  & U \ar[ul]_{σ} \ar[ur]^{φ} \\  &  & W \ar[ul]_{μ} \ar[ur]_{ν} \\  &  &  & T \ar[ul]_{δ} \ar[uurr]^{η}}

Since this represents also s there is an isomorphism α : T ≅ U such that σ ∘ α = σ ∘ μ ∘ δ and φ ∘ σ = φ ∘ η. Using that σ and φ are monomorphisms this implies α = η = μ ∘ δ. Thus we may assume T = U and η = μ ∘ δ = Id_{U}. Precomposing the last identity with μ yields μ ∘ δ ∘ μ = μ, hence δ∘μ = Id_{W}, again because μ is a monomorphism. Thus μ, δ are inverse isomorphisms, and we hence may assume W = U and μ = δ = Id_{U}. Note that the morphism ν : U → V satisfies therefore φ = τ ∘ ν and σ = ψ ∘ ν. Similarly, the equality t ∘ s ∘ t ∘ t = t yields the existence of a morphism λ : V → U satisfying ψ = σ ∘ λ and τ = φ ∘ λ. Thus ψ = ψ ∘ ν ∘ λ, and since ψ is a monomorphism we get ν ∘ λ = Id_{V}. Similarly, since φ = τ ∘ ν = φ ∘ λ ∘ ν we get λ ∘ ν = Id_{U}. Thus we may assume V = U and and λ = ν = Id_{U}, which implies that ψ = σ and τ = φ. This shows that Ĉ is an inverse category. The functor C → Ĉ in the statement is clearly injective on morphisms. Using the fact that every morphism in C is a monomorphism one verifies that any idempotent in Ĉ is of the form [U; φ, φ] for some morphism φ : U → X. We have [U; φ, φ] = [U; Id_{U}, φ]∘ [U; φ, Id_{U}] and [U; Id_{U}, Id_{U}] = [U; φ, Id_{U}]∘ [U; Id_{U}, φ], hence [U; φ, φ] splits. This proves (i). Given morphisms φ : U → X and ψ : V → Y, the idempotents e = [U; φ, φ] and f = [V; ψ, ψ] are isomorphic if and only if U ≅ V. More precisely, if μ : U ≅ V is an isomorphism in C then the morphism s = [U; φ, ψ ∘ μ] in Ĉ satisfies ŝ ∘ s = e and s ∘ ŝ = f. We have an automorphism of groups Aut_{C}(U) ≅ Ĉ_{e} sending μ ∈ Aut_{C}(U) to [U; φ, φ∘μ], and now (ii) follows from 4.1. In order to prove (iii), we need to calculate the right Kan extensions of the canonical functor C → Ĉ. For Y an object in Ĉ, the undercategory, denoted C^{Y}, has as objects the pairs (X, s) consisting of an object X in C and a morphism s : X → Y in Ĉ. A morphism, in C^{Y}, from (X, s) to (X′, s′) is a morphism α : X → X′ in C satisfying s = s′ ∘ α, where we abusively denote the image of α in Ĉ by α again. Write s = [U; σ, φ] and s′ = [U′; σ′, φ′]. The equality s = s′ ∘ α means that s′ ∘ α is represented by a diagram of the form

\xymatrix{X &  & X′ &  & Y \\  & X \ar[ul]^{_{Id_{X}}} \ar[ur]^{α} &  & U′ \ar[ul]_{σ′} \ar[ur]_{φ′} \\  &  & U \ar[ul]^{σ} \ar[ur]_{ν}}

satisfying φ′ ∘ ν = φ; this determines ν uniquely as φ′ is a monomorphism. This shows that if s is in fact a morphism in C (that is, if U = X and σ = Id_{X}) then there is at most one morphism, in C^{Y}, from (X, s) to (X′, s′). Denote by I the full subcategory of C^{Y} consisting of those (X, s) for which s is a morphism in C. By the previous paragraph, I is in fact a partially ordered set, and it has a terminal object, namely (Y, Id_{Y}). The assignment sending an object (X, s) in C^{Y} as in the previous diagram to (U, φ) and the morphism α : (X, s) → (X′, s′) to ν : (U, φ) → (U′, φ′) determines a functor Ψ : C^{Y} → I. This functor is easily seen to be right adjoint to the inclusion functor I → C^{Y}. By standard properties of functor cohomology, we get that H^{∗}(C^{Y}; A) ≅ H^{∗}(I; A) ≅ A, concentrated in degree zero (cf. cite[5.1]JMOb or [11, 3.1] for the first isomorphism and [11, 3.4] for the second). Thus the base change spectral sequence (cf. [13, 5.3] or [6, Appendix 2, Theorem 3.6] for the homology version) associated with the functor C → Ĉ collapses to the isomorphism as stated in (iii). Finally, if Φ : C → D is a functor as in (iii), where D is an inverse category, then the unique extension of Φ to a functor Φ̂ of inverse categories from Ĉ to D is the functor sending a morphism [U; σ, φ] in Ĉ to the morphism Φ(φ) ∘ { \atop Φ(σ)}; this is functorial thanks to the assumptions on Φ.

Example 6.3. Let C be an EI-category; that is, C is a small category such that any endomorphism of an object X in C is an automorphism of X. Let S(C) be the subdivision category of C; that is, the objects of S(C) are faithful functors σ : m → C, with m ≥ 0; a morphism in S(C) from σ : m → C to τ : n → C is a pair (α, φ) consisting of an injective order preserving map α : m → n and an isomorphism of functors φ : σ ≅ τ ∘ α. Denote by S(C)_{+} the category obtained from adding to S(C) the empty chain ∅ as initial object. All morphisms in S(C)_{+} are monomorphisms and pullbacks exist in S(C)_{+}; indeed, the pullback of two morphisms (α, φ) and (α′, φ′) from σ : m → C and σ′ : m′ → C to τ : n → C, respectively, is ∅ if Im(α), Im(α′) are disjoint, and otherwise equal to the obvious object k ≅ Im(α) ∩ Im(α′) → C obtained from restricting τ, where k + 1 is the cardinality of this intersection.

Remark 6.4. Let C be a small category in which pullbacks exist. One of the crucial properties of a Mackey functor M from C to an abelian category A in the sense of [5, Part I], [11, §5] is that M = M^{∗} is the contravariant part of a pair (M_{∗}, M^{∗}) consisting of a covariant functor M_{∗} and a contravariant functor M^{∗} from C to A which coincide on objects and which send a pullback diagram

\xymatrix{ & Y \\ U \ar[ur]^{φ} &  & V \ar[ul]_{τ} \\  & W \ar[ul]^{μ} \ar[ur]_{ν}}

in C to a commutative diagram

\xymatrix{ & M(Y) \ar[dl]_{M^{∗}(φ)} \\ M(U) &  & M(V) \ar[ul]_{M_{∗}(τ)} \ar[dl]^{M^{∗}(μ)} \\  & M(W) \ar[ul]^{M_{∗}(μ)}}

in A. This means exactly that M can be viewed as the restriction to C of a contravariant functor M̂ : Ĉ → A sending X to M(X) and a morphism [U; σ, φ] : X → Y in Ĉ to the morphism M_{∗}(σ) ∘ M^{∗}(φ) : Y → X in A.

References

[1] H.-J. Baues, G. Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), 187–211.

[2] D. J. Benson, Representations and Cohomology, Vol. II, Cambridge Studies in Advanced Mathematics 31, Cambridge University Press (1990)

[3] C. Broto, R. Levi, B. Oliver, Homotopy equivalences of p-completed classifying spaces of finite groups, Invent. Math. 151 (2003), 611–664.

[4] M. Broué, Higman’s criterion revisited, Michigan Math. J. 58 (2009), 125–179.

[5] A. Dress, Contributions to the theory of induced representations, Lecture Notes in Mathematics 342 (1973), Springer Verlag, 183–240.

[6] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Erg. der Math. 35, Springer Verlag Berlin, Heidelberg, New York (1967).

[7] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963), 267–288.

[8] M. Grandis, Concrete representations for inverse and distributive exact categories, Rend. Acad. Naz. Scienze XL, Mem. di Mat. VIII (1984), 99-120.

[9] M. Grandis, Cohesive categories and manifolds, Annali di Mat. pura appl., CLVII (1990), 199–244.

[10] I. M. Isaacs, Character Theory of Finite Groups, Dover Publications Inc. , New York (1994).

[11] S. Jackowski, J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), 113–132.

[12] S. Jackowski, J. McClure, R. Oliver, Homotopy classifications of self-maps of BG via G-actions II, Ann. Math. 135 (1992), 227–270.

[13] S. Jackowski, J. Słomińska, G-functors, G-posets and homotopy decompositions of Gspaces, Fund. Math. 169 (2001), 249–287.

[14] J. Kastl, Inverse categories, in: Algebraische Modelle, Kategorien und Gruppoide (ed. H.-J. Hoehnke), Akademie Verlag, Berlin (1979), 51–60.

[15] M. V. Lawson, Inverse Semigroups, World Scientific Publishing Company, Singapore (1998)

[16] M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras Representation Theory 2 (1999), 107–135.

[17] M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.

[18] M. Linckelmann, On graded centers and block cohomology, Proc. Edinburgh Math. Soc. 52 (2009), 489–514.

[19] W. D. Munn, Matrix representations of semi-groups, Proc. Cambridge Philos. Soc. 53 (1957), 5–12.

[20] I. S. Ponizovskii, On matrix representations of associative systems, Mat. Sb. N. S. 38 (1956), 241–260.

[21] E. Robinson, G. Rosolini, Categories of Partial Maps, Information and Computation 79 (1988), 95–130.

[22] B. Steinberg, Möbius functions and semigroup representation theory II: Character formulas and multiplicities, Advances in Math. 217 (2008), 1521–1557.

[23] P. J. Webb, An introduction to the representations and cohomology of categories, in: Group Representation Theory, M. Geck, D. Testerman, J. Thévenaz (eds.), EPFL Press, Lausanne (2007), 149–173.

[24] F. Xu, Hochschild and ordinary cohomology rings of small categories, Advances in Math. 219 (2008), 1872–1893.