A Pirashvili-type theorem for functors on non-empty finite sets
aa r X i v : . [ m a t h . A T ] S e p A PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTYFINITE SETS
GEOFFREY POWELL AND CHRISTINE VESPA
Abstract.
Pirashvili’s Dold-Kan type theorem for finite pointed sets follows from the iden-tification in terms of surjections of the morphisms between the tensor powers of a functorplaying the role of the augmentation ideal; these functors are projective.We give an unpointed analogue of this result: namely, we compute the morphisms betweenthe tensor powers of the corresponding functor in the unpointed context. We also calculate theExt groups between such objects, in particular showing that these functors are not projective;this is an important difference between the pointed and unpointed contexts.This work is motivated by our functorial analysis of the higher Hochschild homology of awedge of circles. Introduction
The aim of this paper is to study the category of functors from the category
Fin of finitenon-empty sets and all morphisms to the category of modules over a fixed commutative ring k .We denote this category of functors by F ( Fin ; k ); more generally, F ( C ; k ) (or the category of C -modules) denotes the category of functors from an essentially small category C to k -modules.Our main motivation for studying F ( Fin ; k ) comes from our functorial approach to higherHochschild homology to a wedge of circles [PV18], as indicated later in this introduction.In particular, this motivates our introduction in Definition 3.17 of the objects k [ − ] of F ( Fin ; k ):for X a finite non-empty set, k [ X ] is the kernel of the natural ‘augmentation’ k [ X ] → k . Theseare fundamental building blocks of F ( Fin ; k ).The main result of the paper is the following: Theorem 1 (Theorem 7.1) . For a, b ∈ N , there are isomorphisms Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = (cid:26) k ⊕ Rk( b,a ) b ≥ a > b < a ;Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = (cid:26) k b = a + 10 otherwise,where Rk( b, a ) is a combinatorially-defined function of b, a (cf. Proposition 6.18). This leads to the following:
Corollary 2. (Corollary 7.2) For a ∈ N , k [ − ] ⊗ a is not projective in F ( Fin ; k ) . These results should be compared with the pointed situation studied previously by Pirashvili,where the category
Fin should be replaced by the category Γ of finite pointed sets. In [Pir00b],Pirashvili gave an equivalence of categories between the categories F (Γ; k ) and F ( Ω ; k ) where Ω Mathematics Subject Classification.
Key words and phrases. functors on non-empty finite sets; category of finite sets and surjections; comonad onΓ-modules; Koszul complex; FI op -cohomology.This work was partially supported by the ANR Project ChroK , ANR-16-CE40-0003 . is the category of finite sets and surjections. The proof of this result is based on using a certainfamily of projective objects ( t ∗ ) ⊗ n in F (Γ; k ), for n ∈ N . More precisely, the equivalence followsfrom the identification, for a, b ∈ N [Pir00b, (1.10.2)](1.1) Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) ∼ = k Hom Ω ( b , a )where, for m ∈ N , m := { , . . . , m } . We refer to this here as Pirashvili’s theorem.The forgetful functor ϑ : Γ → Fin (i.e., forgetting the chosen basepoint) induces an exactfunctor ϑ ∗ : F ( Fin ; k ) → F (Γ; k ) satisfying ϑ ∗ k [ − ] ∼ = t ∗ and, more generally, ϑ ∗ ( k [ − ] ⊗ a ) ∼ =( t ∗ ) ⊗ a , for a ∈ N .Hence the first part of Theorem 1 can be considered as the unpointed version of the iso-morphism (1.1). However, Corollary 2 exhibits a major difference between ( t ∗ ) ⊗ a and the Fin -modules k [ − ] ⊗ a , for a ∈ N : the former are projective whereas the latter are not.We make the link with Pirashvili’s result more explicit in Corollary 7.4, where we identifyHom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) as a submodule of k Hom Ω ( b , a ), more precisely as the kernel ofan explicit map given by further structure on k Hom Ω ( − , a ). Theorem 1 determines the rank ofthis kernel.The results are refined by taking into account the natural action of the group S a × S op b onHom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ). For instance: Theorem 3. (Proposition 7.5) For a < b ∈ N ∗ and k = Q there is an isomorphism of (virtual) S a × S op b -representations Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = ( − b − a +1 [sgn S a ⊗ sgn S b ] + b − a X t =0 ( − t (cid:2)(cid:0) k Hom Ω ( b − t , a ) ⊗ sgn S t (cid:1) ↑ S op b S op b − t × S op t (cid:3) . As indicated above, this project was motivated by our work on higher Hochschild homologyof a wedge of circles [PV18]. For coefficients taken in F (Γ; k ), Hochschild homology gives aclose relationship with F ( gr ; k ), functors on the category gr of finitely-generated free groups; inparticular, it leads to interesting representations of the automorphism groups of free groups. Forcoefficients taken in F ( Fin ; k ), this Hochschild homology is related to representations of outer automorphism groups of free groups [TW19] and corresponds, in the functorial context, to outerfunctors [PV18], which are functors on which inner automorphisms of free groups act trivially.The study of the category of outer functors F Out ( gr ; k ) and its rˆole in the study of the non-pointed version of higher Hochschild homology is of significant interest, as witnessed by therelationship with hairy graph homology [TW19]. To understand higher Hochschild homologyin the unpointed context, one of our goals is to understand the functor cohomology groupsExt i F Out ( gr ; k ) ( a ⊗ a k , a ⊗ b k ), where a k is induced by abelianization. There are natural morphismsExt i F Out ( gr ; k ) ( a ⊗ a k , a ⊗ b k ) → Ext i F ( gr ; k ) ( a ⊗ a k , a ⊗ b k )where the codomain is given by the main result of [Ves18]:Ext i F ( gr ; k ) ( a ⊗ a k , a ⊗ b k ) ∼ = (cid:26) k Hom Ω ( b , a ) i = b − a gr explained before. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 3
We expect that the image of the above morphism, for i = b − a , should be closely related toHom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ). As evidence for this, in [PV20] we prove that, if k is a field,Ext F Out ( gr ; k ) ( a ⊗ a k , a ⊗ a +1 k ) ∼ = Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ a +1 ) . This result explains why we focus here on the computation of Hom and Ext ; the case ofhigher Ext groups, which involves further material, will be presented elsewhere. Strategy of proof and organization of the paper.
We denote by FI the category offinite sets and injections and by Σ the category of finite sets and bijections.(1) We begin by giving a comonadic description of morphisms in F ( Fin ; k ) using the Barr-Beck theorem. This identifies Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) as the equalizer of a pair ofmaps, or equivalently, as the kernel of an explicit map (see Remark 5.5).Unfortunately, this is inaccessible to direct study. To get around this difficulty, weintroduce, in Section 5, a cosimplicial object, extending the previous equalizer diagram.This arises via the general cosimplicial object that is constructed from a comonad inAppendix B.2.The 0th and 1st cohomology groups of this cosimplicial object correspond respectivelyto Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) and Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ). The rest of the paperis largely devoted to computing the cohomology of this cosimplicial object.(2) The main result of Section 5 is an isomorphism between the above cosimplicial objectand another one, that is defined in terms of the category Ω . The latter is given by acobar-type cosimplicial construction (see Section B.1).More precisely, in Section 4, for F an FI op -module we introduce the cosimplicial Σ op -module C • F . For F = k Hom Ω ( − , a ) (which is an FI op -module by Proposition 5.21) weobtain the cosimplicial object C • k Hom Ω ( − , a ), which is the second cosimplicial objectabove.Section 5 provides the technical underpinnings of the paper.(3) In Section 4, for F an FI op -module we introduce the Koszul complex Kz F in Σ op -modules and we prove that this complex is quasi-isomorphic to the normalized cochaincomplex associated to the opposite of the cosimplicial object C • F .(4) The heart of the paper is Section 6, in which we compute the cohomology of the Koszulcomplex of k Hom Ω ( − , a ), for a ∈ N ∗ . The key result is that its homology is concentratedin its top and bottom degrees (see Theorem 6.15), so that the explicit calculation followsby considering the Euler-Poincar´e characteristic of the complex.(5) These results are combined in Section 7 to prove Theorem 1.Section 4 relates our calculations to FI op -cohomology, which is introduced in Section 4.7.This is the dual notion of the FI -homology introduced in [CE17]. The Koszul complex Kz F isthe FI op -version of the Koszul complex considered in [Gan16, Theorem 2] for FI -modules andTheorem 4.1 is the analogue of [Gan16, Theorem 2]. Thus Theorem 6.15 can be interpreted asthe computation of the FI op -cohomology of the FI op -module k Hom Ω ( − , a ) as follows: Theorem 4.
For a, b, n ∈ N , H n FI op ( k Hom Ω ( − , a ))( b ) = k ⊕ Rk( b,a ) if b ≥ a > and n = 0 k if ( b > a ≥ and n = b − a ) or ( a = b = n = 0)0 otherwise. GEOFFREY POWELL AND CHRISTINE VESPA
Contents
1. Introduction 12. Categories of sets 53. Functor categories on categories of sets 64. The Koszul complex 145. Coaugmented cosimplicial objects 266. The cohomology of the Koszul complex of k Hom Ω ( − , a ) 387. Computation of Hom Fin and Ext Fin
Notation. • k denotes a commutative ring with unit. • N denotes the non-negative integers and N ∗ the positive integers. • For n ∈ N , S n denotes the symmetric group on n letters and sgn S n the sign representa-tion, with underlying module k and σ ∈ S n acting via the signature sign ( σ ). • For n ∈ N , n denotes the set { , , . . . , n } (by convention = ∅ ); we draw the reader’sattention to the fact that n + m denotes the set { , . . . , n + m } (respectively n − m denotes { , . . . , n − m } if n > m ). • The notation F : C ⇄ D : G for an adjunction always indicates that F is the left adjointto G (i.e., F ⊣ G in category-theoretic notation). Remark . The set b is equipped with the canonical total order. Arbitrary subsets of b occurin the text, denoted variously by b ′ , b ′′ , b ( t ) i . . . ; these inherit a total order from b . (We neverconsider the finite set associated to a natural number b ′ , for example, so this notation should notlead to confusion.) Notation . Let ∆ aug denote the skeleton of the category of finite ordinals with objects [ n ]indexed by integers n ≥ −
1, so that the object indexed by − ∅ and the usualcategory of non-empty finite ordinals ∆ is the full subcategory on objects indexed by N .Denote by δ i and σ j the face and degeneracy morphisms of ∆ opaug and d i , s j the coface andcodegeneracy morphisms of ∆ aug .Recall that there is an involution of ∆ aug given by reversing the order on finite ordinals, whichrestricts to an involution of ∆ (cf. [Wei94, 8.2.10]). Notation . For C a category, let op : ∆ op C → ∆ op C op : ∆ C → ∆ C denote the opposite structure functors on simplicial (respectively cosimplicial) objects, inducedby the composition with the involution of ∆ .Explicitly, if A • is a simplicial object in C with face operators δ i : A n → A n − and degeneracyoperators σ i : A n → A n +1 for i ∈ { , , . . . , n } , ( A • ) op is the simplicial object in C with faceoperators ˜ δ i : A n → A n − and degeneracy operators ˜ σ i : A n → A n +1 , where ˜ δ i = δ n − i and˜ σ i = σ n − i .This notation is also applied for augmented objects. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 5
Notation . For C ∗ a chain complex (respectively C ∗ a cochain complex), we use the conventionfor shifting given in [Wei94, 1.2.8], i.e., C [ p ] n = C n + p (respectively C [ p ] n = C n − p ) . Categories of sets
Let
Set denote the category of sets and
Set ∗ that of pointed sets. Consider the commutativediagram of categories and forgetful functors Σ mM | | ①①①①①①①①① (cid:17) q ●●●●●●●●● Ω (cid:17) q " " ❊❊❊❊❊❊❊❊ FI mM { { ①①①①①①①①① Fin (cid:31) (cid:127) / / Fin (cid:31) (cid:127) / / Set Γ ϑ O O (cid:31) (cid:127) / / Set ∗ forget O O where(1) Fin ⊂ Set is the full subcategory of finite sets;(2)
Fin ⊂ Fin the full subcategory of non-empty finite sets;(3) Γ ⊂ Set ∗ the full subcategory of finite pointed sets;(4) Ω the category of finite sets and surjections;(5) FI the category of finite sets and injections;(6) Σ ⊂ Fin the subcategory of finite sets and bijections;(7) the functor ϑ forgets the marking of the basepoint. Remark . The functor ϑ : Γ → Fin is essentially surjective and faithful but is not full.
Remark . By definition,(1) Hom Ω ( , ) = {∗} , Hom Ω ( a , ) = ∅ = Hom Ω ( , a ) for a > Ω ( b , ) = (cid:26) {∗} b > ∅ b = 0 , where, for b >
0, the morphism is given by the unique surjective map b ։ ;(2) the symmetric monoidal structure on Fin given by disjoint union of finite sets induces asymmetric monoidal structure on Ω . Remark . In [Pir00a], the category of non-empty finite sets and surjections is denoted by Ωwhereas, in [Pir00b], Ω denotes the category of finite sets and surjections. Here we adopt thesecond convention, denoting this category Ω . Lemma 2.4.
There is an adjunction ( − ) + : Fin ⇄ Γ : forget , that restricts to an adjunction ( − ) + : Fin ⇄ Γ : ϑ , where the functor ( − ) + sends a finite set X to X + , with + as basepoint.The adjunction unit Id Fin → ϑ (cid:0) ( − ) + (cid:1) for a finite set X ∈ Ob Fin is the natural inclusion
X ֒ → X + . For a finite pointed set ( Z, z ) ∈ Ob Γ , the adjunction counit ( ϑ ) + → id Γ is the natural mor-phism ( ϑZ ) + → Z in Γ that extends the identity on Z by sending + to z .Remark . The categories
Fin and Γ have small skeleta with objects n ∈ Ob Fin and n + ∈ Ob Γ, for n ∈ N . GEOFFREY POWELL AND CHRISTINE VESPA
The following underlines the fact that Hom Ω ( b , a ) can be understood in terms of the fibresof maps: Lemma 2.6.
For a > and any non-empty subset b ′ ⊂ b , there is a bijection between Hom Ω ( b ′ , a ) and the set of ordered partitions of b ′ into | a | = a disjoint, non-empty subsets,via ( f : b ′ ։ a ) ( f − (1) , f − (2) , . . . , f − ( a )) . In particular, the ordered sets: f − (1) , f − (2) , . . . , f − ( a )(2.1) f − (1) , f − (2) , . . . , f − ( a ) , b \ b ′ (2.2) define permutations of b ′ and b respectively, where each fibre f − ( i ) , i ∈ a , and b \ b ′ inherittheir order from b . Functor categories on categories of sets
In this section, we first review functors on Γ, before considering the corresponding structuresfor
Fin . Then we compare the functor categories on Γ and on
Fin , obtaining the comonad ⊥ Γ : F (Γ; k ) → F (Γ; k ) in Notation 3.21. Finally, we recall the relationship between (contravariant)functors on FI and on Σ , obtaining the comonad ⊥ Σ : F ( Σ op ; k ) → F ( Σ op ; k ) introduced inNotation 3.30. The comonads ⊥ Γ and ⊥ Σ play a crucial role in the paper.We begin with some recollections concerning functor categories in order to fix notation.For C an (essentially) small category, F ( C ; k ) denotes the category of functors from C to k -modules. (This category may also be referred to as the category of C -modules when thecodomain is clear from the context.) This is a Grothendieck abelian category with structureinherited from k -modules; moreover the tensor product over k induces a symmetric monoidalstructure ⊗ on F ( C ; k ) with unit the constant functor k .If ψ : C → D is a functor between small categories, the functor given by precomposition with ψ is denoted ψ ∗ : F ( D ; k ) → F ( C ; k ). This is an exact functor, which is symmetric monoidal. Notation . For X an object of a small category C , P C X denotes the projective of F ( C ; k )given by k Hom C ( X, − ), the composite of the corepresentable functor Hom C ( X, − ) with the free k -module functor.The projectives P C X , as X ranges over a set of representatives of the isomorphism classes ofobjects of C , form a family of projective generators of F ( C ; k ).3.1. The functor category F (Γ; k ) . The functor category F (Γ; k ) has been studied by Pi-rashvili and his co-authors (see for example [Pir00b, Pir00a]). We begin this section by somebasic facts concerning this category.Since Γ is a pointed category, one has the splitting: Lemma 3.2.
There is an equivalence of categories F (Γ; k ) ∼ = F (Γ; k ) × k − mod , where F (Γ; k ) is the full subcategory of reduced objects (i.e., those vanishing on + ). The projectives given in Notation 3.1 have the following property:
Proposition 3.3.
For m, n ∈ N , there is an isomorphism P Γ m + n + ∼ = P Γ m + ⊗ P Γ n + . In particular,for n ∈ N , one has P Γ n + ∼ = ( P Γ + ) ⊗ n .Proof. Use that the object m + n + of Γ is isomorphic to the coproduct m + W n + in Γ. (cid:3) One of the most important results concerning the category F (Γ; k ) is Pirashvili’s Dold-Kantype theorem for Γ-modules (see Theorem 3.7). To recall this, we introduce the following: PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 7
Definition 3.4. (Cf. [Pir00b].) Let t ∗ ∈ Ob F (Γ; k ) be the quotient of P Γ + given by t ∗ ( n + ) := k [ n + ] / k [ + ].One has the following, in which the image of the generator [ x ] ∈ k [ n + ] (corresponding to x ∈ n + ) in t ∗ ( n + ) is written [[ x ]], which is zero if and only if x = +. Lemma 3.5. [Pir00b]
The functor t ∗ is projective in F (Γ; k ) and there is a canonical splitting P Γ + ∼ = t ∗ ⊕ k .Explicitly, for ( Z, z ) a finite pointed set, the inclusion t ∗ ( Z ) ∼ = k [ Z ] / k [ z ] ֒ → P Γ + ( Z ) ∼ = k [ Z ] isinduced on generators by [[ y ]] [ y ] − [ z ] , for y ∈ Z \{ z } , and the projection P Γ + ( Z ) ∼ = k [ Z ] → k ∼ = k [ + ] is induced by the morphism Z → + to the terminal object of Γ .Proof. The canonical splitting is provided by Lemma 3.2 and the remaining statements followby explicit identification of the functors. (cid:3)
Lemma 3.5 together with Proposition 3.3 lead directly to the following Proposition, in which( t ∗ ) ⊗ = k : Proposition 3.6. [Pir00b]
The set { ( t ∗ ) ⊗ a | a ∈ N } is a set of projective generators of F (Γ; k ) .Restricting to a ∈ N ∗ gives projective generators of F (Γ; k ) . The category F (Γ; k ) is symmetric monoidal for the structure induced by ⊗ on k -modulesand Ω has a symmetric monoidal structure (see Remark 2.2).In the following result, k Ω denotes the k -linearization of the category Ω . Theorem 3.7. [Pir00b]
There is a k -linear, symmetric monoidal embedding ( k Ω ) op → F (Γ; k ) that is induced by t ∗ . This is fully faithful and induces an equivalence of categories between F (Γ; k ) and F ( Ω ; k ) .In particular, for a, b ∈ N , there is a natural isomorphism Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) ∼ = k Hom Ω ( b , a ) . In Section 5 it is necessary to have an explicit description of the natural isomorphism given inthe previous Theorem. So, the rest of this section is devoted to make explicit the isomorphism k Hom Ω ( b , a ) ∼ = → Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) . For a = 0, we have: Hom F (Γ; k ) ( k , ( t ∗ ) ⊗ b ) ∼ = (cid:26) k b = 00 b > , where, for b = 0, a generator is given by the identity Id k : k → k .For a >
0, understanding the morphisms Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b ) is a key ingredient. By Lemma3.5, for ( Z, z ) a finite pointed set, { [ y ] − [ z ] | y ∈ Z \{ z }} is a basis t ∗ ( Z ). We use this basis inthe following: Lemma 3.8.
For b ∈ N , Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b ) ∼ = (cid:26) k b > b = 0 , where, for b > , a generator is given by the morphism ξ b : t ∗ → ( t ∗ ) ⊗ b given upon evaluation on ( Z, z ) by [ y ] − [ z ] ([ y ] − [ z ]) ⊗ b , where y ∈ Z \{ z } .In particular, the action of S b on Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b ) induced by the place permutationaction on ( t ∗ ) ⊗ b is trivial. GEOFFREY POWELL AND CHRISTINE VESPA
Proof. If b = 0, since t ∗ is a reduced Γ-module, the statement follows from Lemma 3.2. For b >
0, the inclusion t ∗ ⊂ P Γ + ∼ = t ∗ ⊕ k (where the isomorphism is given by Lemma 3.5) inducesan isomorphism Hom F (Γ; k ) ( P Γ + , ( t ∗ ) ⊗ b ) ∼ = Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b )by Lemma 3.2. Yoneda’s Lemma gives that the left hand side is k , with trivial S b -action, wherethe given generator corresponds to the element ([1] − [+]) ⊗ b ∈ t ∗ ( + ) ⊗ b . (cid:3) The symmetric monoidal structure of Ω (see Remark 2.2) gives the following: Lemma 3.9.
For a ∈ N ∗ and b ∈ N the symmetric monoidal structure of Ω induces a S op b -equivariant isomorphism a ( b i ) ∈ ( N ∗ ) a P ai =1 b i = b (cid:0) a Y i =1 Hom Ω ( b i , ) (cid:1) ↑ S b Q ai =1 S bi ∼ = → Hom Ω ( b , a ) , in which Hom Ω ( b i , ) ∼ = {∗} .Proof. For b = 0, the two sides of the bijection are the empty set. For b = 0, the component ofthe morphism indexed by ( b i ) ∈ ( N ∗ ) a is induced by the set map a Y i =1 Hom Ω ( b i , ) → Hom Ω ( b , a )( ξ i ) b ∼ = ∐ ai =1 b i ∐ ai =1 ξ i −→ ∐ ai =1 ∼ = a . This induces the given isomorphism, since a surjection f : b ։ a is uniquely determined bythe partition of b given by the fibres b i := f − ( i ) (cf. Lemma 2.6). (cid:3) This allows the following definition to be given.
Definition 3.10.
Let DK a , b : Hom Ω ( b , a ) → Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ), for a, b ∈ N , be the S a × S op b -equivariant map determined by the following:(1) DK , (Id ) := Id k ;(2) for a > b > b i ) ∈ ( N ∗ ) a such that P ai =1 b i = b , the restriction of DK a , b to Q ai =1 Hom Ω ( b i , ) ⊂ Hom Ω ( b , a ) (using the inclusion given by Lemma 3.9), a Y i =1 Hom Ω ( b i , ) → Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b )sends the one point set on the left to the morphism N ai =1 ( t ∗ → ( t ∗ ) ⊗ b i ), the tensorproduct of the generators ξ b i given by Lemma 3.8. Proposition 3.11.
For a, b ∈ N , the k -linear extension of DK a , b induces a S a × S op b -equivariantisomorphism k DK a , b : k Hom Ω ( b , a ) ∼ = → Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) . Proof.
The construction of the map DK a , b ensures that the k -linearization is S a × S op b -equivariant.The fact that it is an isomorphism follows from Pirashvili’s result, Theorem 3.7. (cid:3) Remark . We will see in Proposition 5.23 that the maps DK a , b define an isomorphism of Σ × Σ op -modules. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 9
The functor category F ( Fin ; k ) . The aim of this section is to give an analysis of thecategory F ( Fin ; k ) parallel to that of F (Γ; k ) given in the previous section.Whilst ∅ is the initial object of Fin , it is not final, so that
Fin is not pointed and there is nosplitting analogous to that of Lemma 3.2. Instead one has:
Proposition 3.13.
The restriction functor F ( Fin ; k ) → F ( Fin ; k ) induced by Fin ⊂ Fin isexact and symmetric monoidal and has exact left adjoint given by extension by zero (i.e., for F ∈ Ob F ( Fin ; k ) , defining F ( ∅ ) := 0 ).Via these functors, F ( Fin ; k ) is equivalent to the full subcategory of reduced objects of F ( Fin ; k ) (functors G such that G ( ∅ ) = 0 ).Notation . Let k denote the constant functor of F ( Fin ; k ) taking value k . (This notationis introduced so as to avoid potential confusion with the constant functor of F ( Fin ; k ), which isdenoted by k .) Proposition 3.15.
The following conditions are equivalent: (1) k is injective as a k -module; (2) the constant functor k is injective in F ( Fin ; k ) ; (3) the constant functor k is injective in F ( Fin ; k ) .Proof. Since is the terminal object of both Fin and
Fin , in both cases, the constant functor isisomorphic to the maps from Hom( − , ) to k , k Hom( − , ) , where Hom is calculated respectivelyin Fin and
Fin . The equivalence of the statements then follows by using Yoneda’s Lemma. (cid:3)
Proposition 3.3 has the following analogue for F ( Fin ; k ); the proof is similar. Proposition 3.16.
For m, n ∈ N ∗ , there is an isomorphism P Finm + n ∼ = P Finm ⊗ P Finn . In particular,for n ∈ N ∗ , one has P Finn ∼ = ( P Fin1 ) ⊗ n . We introduce the functor k [ − ] ∈ Ob F ( Fin ; k ), which is the analogue of the functor t ∗ ∈ Ob F (Γ; k ) introduced in Definition 3.4. Definition 3.17.
Let k [ − ] be the object of F ( Fin ; k ) defined as the kernel of the surjection P Fin1 ։ k in F ( Fin ; k ) which is induced by the map to the terminal object of Fin . (Explicitly, P Fin1 ( n ) ∼ = k [ n ] and the canonical map n → induces k [ n ] ։ k [ ] = k .)This gives the defining short exact sequence:0 → k [ − ] → P Fin1 → k → . (3.1)The functor k [ − ] plays an essential rˆole in analysing the category of Fin -modules. Theprincipal difference between t ∗ ∈ Ob F (Γ; k ) and k [ − ] ∈ Ob F ( Fin ; k ) is that the functor k [ − ]is not projective (see Corollary 7.2). We record immediately the following important observation: Proposition 3.18.
The short exact sequence (3.1) does not split, hence (1) k is not projective in F ( Fin ; k ) ; (2) the short exact sequence represents a non-trivial class in Ext F ( Fin ; k ) ( k , k [ − ]) .Proof. The morphism P Fin1 ։ k is an isomorphism when evaluated on hence, were a sectionto exist, it would send 1 ∈ k to the generator [ ]. However, this morphism cannot extend toa natural transformation from the constant functor k to P Fin1 , since [ ] is not invariant undermorphisms of Fin . The remaining statements follow immediately. (cid:3)
Comparison of the functor categories F (Γ; k ) and F ( Fin ; k ) . In this section, we relatethe categories F (Γ; k ) and F ( Fin ; k ) using the functors ϑ and ( − ) + introduced in Section 2.Recall that a functor ψ is conservative if a morphism f is an isomorphism if and only if ψ ( f )is. Proposition 3.19.
The adjunction ( − ) + : Fin ⇄ Γ : ϑ induces an adjunction ϑ ∗ : F ( Fin ; k ) ⇄ F (Γ; k ) : ( − ) ∗ + of exact functors that are symmetric monoidal. Moreover: (1) ϑ ∗ : F ( Fin ; k ) → F (Γ; k ) is faithful and conservative; (2) ϑ ∗ sends projectives to projectives. (Explicitly, for n ∈ N ∗ , ϑ ∗ P Finn ∼ = P Γ n + .)Proof. The first statement is general: precomposition with a functor induces an exact, symmetricmonoidal functor; precomposition with an adjunction yields an adjunction, with the rˆole of theadjoints reversed.The fact that ϑ ∗ is faithful and conservative follows from the fact that ϑ is essentially surjectiveand faithful. (This is the reason for restricting attention to Fin .)Finally, it follows formally from the adjunction that ϑ ∗ sends projectives to projectives. Theexplicit identification follows from the adjunction isomorphism Hom Fin ( n , ϑ ( X )) ∼ = Hom Γ ( n + , X ),for X a finite pointed set. (cid:3) Proposition 3.20. (1)
Applying ϑ ∗ to the short exact sequence (3.1) yields the canonical (split) short exactsequence in F (Γ; k ) : → t ∗ → P Γ + → k → for a ∈ N , there is an isomorphism ϑ ∗ ( k [ − ] ⊗ a ) ∼ = ( t ∗ ) ⊗ a .Proof. It is clear that, with respect to the identification of the action of ϑ ∗ on projectives givenin Proposition 3.19, ϑ ∗ ( P Fin1 ։ k ) is the projection P Γ + ։ k . Since ϑ ∗ is exact, this gives thefirst statement, together with the isomorphism ϑ ∗ ( k [ − ]) ∼ = t ∗ .The second statement then follows from the fact that ϑ ∗ is symmetric monoidal. (cid:3) In the rest of the section we establish the comonadic description of F ( Fin ; k ) using the Barr-Beck theorem (see Theorem 3.24). Notation . Let ⊥ Γ : F (Γ; k ) → F (Γ; k ) be the comonad given by ⊥ Γ = ϑ ∗ ◦ ( − ) ∗ + .One has the following immediate consequence of Proposition 3.19: Corollary 3.22.
The functor ⊥ Γ : F (Γ; k ) → F (Γ; k ) is exact and symmetric monoidal. The structure maps of the comonad ⊥ Γ are described in the following Proposition. Proposition 3.23.
Let F ∈ Ob F (Γ; k ) and ( Y, y ) be a finite pointed set. (1) The counit ε Γ F : ⊥ Γ F ( Y ) = F ( Y + ) → F ( Y ) is given by F ( α ) where α extends the identityon Y by sending + to y . (2) Iterating ⊥ Γ , labelling the functors ⊥ Γ and their associated basepoint, one has ⊥ Γ ⊥ Γ F ( Y ) = F (( Y + ) + ) . (3) The diagonal ∆ : ⊥ Γ F ( Y ) → ⊥ Γ ⊥ Γ F ( Y ) is induced by the pointed map Y + → ( Y + ) + given by the identity on Y and sending + to + . (4) The counit ⊥ Γ ε Γ F : ⊥ Γ ⊥ Γ F ( Y ) → ⊥ Γ F ( Y ) is given by sending + to + and the counit ε Γ ⊥ Γ2 F : ⊥ Γ ⊥ Γ F ( Y ) → ⊥ Γ F ( Y ) by sending + to y . PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 11
Proof.
By Proposition 3.19, the adjunction giving rise to ⊥ Γ is induced by the adjunction ( − ) + : Fin ⇄ Γ : ϑ , hence the structure morphisms of the comonad ⊥ Γ can be deduced from Lemma2.4.Note that, by definition of ⊥ Γ , we have: ⊥ Γ ⊥ Γ F ( Y ) = F ◦ ( − ) + ◦ ϑ ◦ ( − ) + ◦ ϑ ( Y ) = F (( Y + ) + ) . (cid:3) The Barr-Beck theorem (see, for example, [KS06, Theorem 4.3.8] for the dual monadic state-ment) then gives the following:
Theorem 3.24.
The category F ( Fin ; k ) is equivalent to the category F (Γ; k ) ⊥ Γ of ⊥ Γ -comodulesin F (Γ; k ) .Remark . The Barr-Beck correspondence can be understood explicitly as sketched below.Suppose that G ∈ Ob F ( Fin ; k ). Let ψ ϑ ∗ G := ϑ ∗ ( η G ) : ϑ ∗ G → ⊥ Γ ϑ ∗ G where η : Id → ( − ) ∗ + ◦ ϑ ∗ is the unit of the adjunction given in Proposition 3.19. Explicitly, for X ∈ Fin ,( η G ) X = G ( i X ) : G ( X ) → G ( X ∐ { + } ) where i X : X → X ∐ { + } is the canonical inclusion.Then ( ϑ ∗ G, ψ ϑ ∗ G ) is a ⊥ Γ -comodule and the correspondence G ( ϑ ∗ G, ψ ϑ ∗ G ) defines a functor F ( Fin ; k ) → F (Γ; k ) ⊥ Γ .Suppose that F ∈ Ob F (Γ; k ) ⊥ Γ ; the associated functor on Fin is given for n ∈ N ∗ by n F (( n − ) + ). This corresponds to choosing a basepoint for each object of the skeleton of Fin .The ⊥ Γ -comodule structure allows the full functoriality to be recovered. For X ∈ Ob Γ a finitepointed set, one has the morphisms F ( X ) → ⊥ Γ F ( X ) ∼ = F ( X + ) → F ( X ) , where the first is the comodule structure map and the second is induced by sending the addedbasepoint + to the basepoint of X . The middle term is independent of the basepoint of X and these morphisms induce a canonical isomorphism between F ( X, x ) and F ( X, y ) for anybasepoints x, y ∈ X .3.4. Comparison of the functor categories F ( FI op ; k ) and F ( Σ op ; k ) . In this section werelate the categories F ( FI op ; k ) and F ( Σ op ; k ).We begin by exhibiting the right adjoint of the restriction functor ↓ : F ( FI op ; k ) → F ( Σ op ; k )induced by Σ op ⊂ FI op . Consider the functor in F ( Σ op × FI ; k ) given by ( b , n ) k Hom FI ( b , n ).(Observe that k Hom FI ( b , n ) = 0 for b > n .) Hence, for G a Σ op -module, n Hom F ( Σ op ; k ) ( k Hom FI ( · , n ) , G ( · ))is an FI op -module and the right hand side identifies as M b ≤ n Hom S op b ( k Hom FI ( b , n ) , G ( b )) ⊂ M b ≤ n Hom k ( k Hom FI ( b , n ) , G ( b )) . (3.2) Definition 3.26.
Let ↑ : F ( Σ op ; k ) → F ( FI op ; k ) be the functor given by ↑ G := (cid:16) n Hom F ( Σ op ; k ) ( k Hom FI ( · , n ) , G ( · )) (cid:17) for G ∈ Ob F ( Σ op ; k ). Proposition 3.27.
The functor ↑ : F ( Σ op ; k ) → F ( FI op ; k ) is the right adjoint of the re-striction functor ↓ : F ( FI op ; k ) → F ( Σ op ; k ) . In other words, for F ∈ Ob F ( FI op ; k ) and G ∈ Ob F ( Σ op ; k ) there is a natural isomorphism: Hom F ( Σ op ; k ) ( ↓ F, G ) ∼ = Hom F ( FI op ; k ) ( F, ↑ G ) . Proof.
We analyse Hom F ( FI op ; k ) ( F ( − ) , Hom F ( Σ op ; k ) ( k Hom FI ( · , − ) , G ( · ))), based upon the iden-tification in equation (3.2), which gives:Hom F ( FI op ; k ) ( F ( − ) , Hom F ( Σ op ; k ) ( k Hom FI ( · , − ) , G ( · ))) ∼ = M b ≤ n Hom F ( FI op ; k ) ( F ( − ) , Hom S op b ( k Hom FI ( b , − ) , G ( b ))) . Fixing b ∈ N and neglecting the S b -action, we consider:Hom F ( FI op ; k ) ( F ( − ) , Hom k ( k Hom FI ( b , − ) , G ( b ))) . Fixing n ∈ N and neglecting the functoriality with respect to FI , one has the standard natural adjunction isomorphisms for k -modules:Hom k ( F ( n ) , Hom k ( k Hom FI ( b , n ) , G ( b ))) ∼ = Hom k ( F ( n ) ⊗ k Hom FI ( b , n ) , G ( b )) ∼ = Hom k ( k Hom FI ( b , n ) , Hom k ( F ( n ) , G ( b ))) . Then, taking into account the functoriality with respect to FI gives the first isomorphismbelow:Hom F ( FI op ; k ) ( F ( − ) , Hom k ( k Hom FI ( b , − ) , G ( b ))) ∼ = Hom F ( FI ; k ) ( k Hom FI ( b , − ) , Hom k ( F ( − ) , G ( b ))) ∼ = Hom k ( F ( b ) , G ( b )) , the second isomorphism being given by Yoneda’s lemma.Finally, taking into account the naturality with respect to Σ op , one arrives at the requiredisomorphism. (cid:3) One can identify the values taken by the functor ↑ : F ( Σ op ; k ) → F ( FI op ; k ) by the following: Proposition 3.28.
For n ∈ N and G a Σ op -module, (1) there is a natural isomorphism: ↑ G ( n ) ∼ = M n ′ ⊆ n G ( n ′ );(2) with respect to these isomorphisms, the morphism ↑ G ( n ) →↑ G ( l ) induced by f : l → n in FI has restriction to the factor G ( n ′ ) ⊂↑ G ( n ) indexed by n ′ ⊆ n given by (a) zero if n ′ f ( l ) ; (b) if n ′ ⊆ f ( l ) , the composite G ( n ′ ) G ( f | f − n ′ ) ) / / G ( f − ( n ′ )) (cid:31) (cid:127) / / ↑ G ( l ) , where f | f − ( n ′ ) : f − ( n ′ ) ∼ = → n ′ .Proof. Consider the first statement. We have ↑ G ( n ) = Hom F ( Σ op ; k ) ( k Hom FI ( − , n ) , G ( − )) (cid:17) ∼ = M b ≤ n Hom S op b ( k Hom FI ( b , n ) , G ( b )) . For b, n ∈ N , Hom FI ( b , n ) is a free S op b -set on the set of subsets n ′ ⊆ n of cardinal b . Moreprecisely, Hom FI ( b , n ) ∼ = a n ′ ⊆ n | n ′ | = b Hom FI ( b , n ′ ) PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 13 where Hom FI ( b , n ′ ) ∼ = S b as a right S op b -set. Hence: ↑ G ( n ) ∼ = M b ≤ n M n ′ ⊆ n | n ′ | = b G ( b ) ∼ = M b ≤ n M n ′ ⊆ n | n ′ | = b G ( n ′ ) ∼ = M n ′ ⊆ n G ( n ′ ) . The second statement follows by considering the inclusion Hom FI ( − , l ) ֒ → Hom FI ( − , n ) in-duced by f . Evaluated on b , this corresponds to the inclusion a l ′ ⊆ l | l ′ | = b Hom FI ( b , l ′ ) ֒ → a n ′ ⊆ f ( l ) | n ′ | = b Hom FI ( b , n ′ ) ⊆ a n ′ ⊆ n | n ′ | = b Hom FI ( b , n ′ ) , where the first map sends Hom FI ( b , l ′ ) to Hom FI ( b , f ( l ′ )) by postcomposition with f | l ′ .The result follows. (cid:3) Remark . For simplicity of notation, in the rest of the paper, for F ∈ F ( FI op ; k ), therestriction of F to Σ op will be denoted by F instead of ↓ F . Notation . Let ⊥ Σ : F ( Σ op ; k ) → F ( Σ op ; k ) be the comonad associated to the adjunctionof Proposition 3.27 (i.e., ⊥ Σ = ↓ ◦ ↑ ), with structure morphisms ∆ : ⊥ Σ → ⊥ Σ ⊥ Σ and counit ε : ⊥ Σ → Id.
Proposition 3.31.
Let G ∈ Ob F ( Σ op ; k ) and b ∈ N . (1) There is a natural isomorphism: ⊥ Σ G ( b ) ∼ = M b ′ ⊆ b G ( b ′ ) . (3.3)(2) The counit ε Σ G : ⊥ Σ G ( b ) → G ( b ) identifies as the projection L b ′ ⊆ b G ( b ′ ) ։ G ( b ) ontothe summand indexed by b . (3) There is a natural isomorphism ⊥ Σ ⊥ Σ G ( b ) ∼ = L b ′′ ⊆ b ′ ⊆ b G ( b ′′ ) . (4) ∆ : ⊥ Σ G ( b ) → ⊥ Σ ⊥ Σ G ( b ) identifies as the morphism M l ′ ⊆ b G ( l ′ ) → M b ′′ ⊆ b ′ ⊆ b G ( b ′′ ) with component G ( l ′ ) → G ( b ′′ ) , the identity if l ′ = b ′′ and zero otherwise.Proof. The first two statements follow from Proposition 3.28.Since ⊥ Σ G ∈ Ob F ( Σ op ; k ), we can apply (3.3) to this functor to obtain ⊥ Σ ( ⊥ Σ G )( b ) ∼ = M b ′ ⊆ b ( ⊥ Σ G )( b ′ ) . Applying (3.3) to each term ⊥ Σ G ( b ′ ) gives the expression for ⊥ Σ ⊥ Σ G ( b ).The natural morphism ∆ : ⊥ Σ G → ⊥ Σ ⊥ Σ G is induced by the FI op -module structure of ↑ G ,which is given by Proposition 3.28 (2). This leads to the stated identification. (cid:3) Remark . The decomposition of ⊥ Σ ⊥ Σ G (we label the functors for clarity) given in Propo-sition 3.31 (3) can be reindexed as follows: ⊥ Σ ( ⊥ Σ G )( b ) ∼ = M b = b (1)Σ ∐ b Σ0 ⊥ Σ G ( b (1)Σ ) ∼ = M b =( b (2)Σ ∐ b Σ1 ) ∐ b Σ0 G ( b (2)Σ ) . The sum is indexed by ordered decompositions of b into three subsets (possibly empty). Withrespect to the decomposition given in Proposition 3.31 (3), we have b (2)Σ := b ′′ , b Σ1 := b ′ \ b ′′ and b Σ0 := b \ b ′ . The indices in b Σ0 , b Σ1 record from which application of ⊥ Σ the subset b Σ i arises.More generally, we have:(3.4) ⊥ Σ ⊥ Σ . . . ⊥ Σ ℓ G ( b ) ∼ = M b =( ... (( b ( ℓ +1)Σ ∐ b Σ ℓ ) ∐ b Σ ℓ − )) ... ∐ b Σ1 ) ∐ b Σ0 G ( b ( ℓ +1)Σ ) . The general theory of comonads associated to an adjunction implies that, if F ∈ Ob F ( FI op ; k ),the underlying object F ∈ Ob F ( Σ op ; k ) has a canonical ⊥ Σ -comodule structure. Proposition 3.33.
For F ∈ Ob F ( FI op ; k ) , the ⊥ Σ -comodule structure of the underlying object ↓ F ∈ Ob F ( Σ op ; k ) (denoted below simply by F ) identifies with respect to the isomorphism (3.3)of Proposition 3.31 as follows. For b ∈ N , the structure morphism ψ F : F → ⊥ Σ F is given by ψ F : F ( b ) → M b ′ ⊆ b F ( b ′ ) with component F ( b ) → F ( b ′ ) given by the FI op structure of F for b ′ ⊂ b . The Barr-Beck theorem implies:
Theorem 3.34.
The category F ( FI op ; k ) is equivalent to the category F ( Σ op ; k ) ⊥ Σ of ⊥ Σ -comodules in F ( Σ op ; k ) . Proposition 3.35 below shows that the comonad ⊥ Σ is equipped with a natural transformation η : Id → ⊥ Σ , which can be considered as a form of coaugmentation. This form of additionalstructure is considered in Appendix B.1; it is important that this satisfies the Hypothesis B.5. Proposition 3.35.
There is a natural transformation η : Id → ⊥ Σ of endofunctors of F ( Σ op ; k ) ,where the morphism η G : G ( b ) → ⊥ Σ G ( b ) , for G ∈ Ob F ( Σ op ; k ) and b ∈ N , is given by G ( b ) → M b ′ ⊆ b G ( b ′ ) , the inclusion of the factor indexed by b ′ = b .Hence the following diagrams commute: G η G / / η G (cid:15) (cid:15) ⊥ Σ G ∆ G (cid:15) (cid:15) G η G / / G ! ! ❉❉❉❉❉❉❉❉ ⊥ Σ G ε Σ G (cid:15) (cid:15) ⊥ Σ G η ( ⊥ Σ G ) / / ⊥ Σ ⊥ Σ G G.
Thus, the structure ( F ( Σ op ; k ); ⊥ Σ , ∆ , ε, η ) satisfies Hypothesis B.5.Proof. This is proved by direct verification, using Proposition 3.31. (cid:3)
Remark . The constructions of this Section apply verbatim to functors from Σ op to anyabelian category. 4. The Koszul complex
The purpose of this section is to introduce the Koszul complex Kz F in Σ op -modules, where F is an FI op -module, and to identify its cohomology. For this, we apply the general constructiongiven in Appendix B in our specific situation. More precisely, Proposition 3.35 shows that thecomonad ( ⊥ Σ , ∆ , ε ) on F ( Σ op ; k ) is equipped with a coaugmentation η : Id → ⊥ Σ satisfyingHypothesis B.5 and Theorem 3.34 shows that the FI op -module F can be considered equivalently PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 15 as a ⊥ Σ -comodule in F ( Σ op ; k ). Hence, one can form the cosimplicial Σ op -module C • F , as inProposition B.6, which has the following form:(4.1) F d = η F / / d = ψ F / / ⊥ Σ F s = ε F w w d = η ⊥ F / / d =∆ F / / d = ⊥ ψ F / / ⊥ Σ ⊥ Σ F s = ε ⊥ F u u s = ⊥ ε F u u / / ⊥ Σ ⊥ Σ ⊥ Σ F . . . . t t In order to compare this cosimplicial object to the Koszul complex, we consider its opposite( C • F ) op . Considering the normalized subcomplex associated to a cosimplicial object (see Ap-pendix A), the main result of the Section is the following Theorem, in which Kz F denotes theKoszul complex introduced in Section 4.1: Theorem 4.1.
For F ∈ Ob F ( FI op ; k ) , there is a natural surjection of complexes of Σ op -modules N ( C • F ) op ։ Kz F which is a quasi-isomorphism, where ( C • F ) op is the opposite cosimplicial Σ op -module to C • F (cf. Notation 1.3).Remark . The complex Kz F is the FI op -version of the Koszul complex considered in [Gan16,Theorem 2] for FI -modules, where it is shown that this has the same cohomology as N ( C • F ) op by using [CE17].To keep this paper self-contained, we give a direct proof of Theorem 4.1, rather than derivingit from [Gan16, Theorem 2]. The relation between the Koszul complex and FI op -cohomology isexplained in Section 4.7.4.1. The Koszul complex Kz F . In this section we construct, for an FI op -module F , thecomplex Kz F in Σ op -modules. To keep track of the signs in the construction, we introduce thefollowing notation: Notation . For S a finite set,(1) denote by Or ( S ) the orientation module associated to S , i.e., the free, rank one k -moduleΛ | S | ( k S ) (the top exterior power), where k S denotes the free k -module generated by S ;(2) Or ( S ) is considered as having cohomological degree | S | ;(3) if S is ordered with elements x < . . . < x | S | , let ω ( S ) ∈ Or ( S ) denote the element: ω ( S ) := x ∧ . . . ∧ x | S | . Remark . (1) By construction, Or ( S ) is a S | S | -module; it is isomorphic to the signature representationsgn S | S | . In particular, we may consider it either as a left or a right S | S | -module withoutambiguity.(2) The orientation modules Or ( b ), b ∈ N , assemble to give a Σ op -module.(3) If S is ordered, then ω ( S ) gives a canonical generator of Or ( S ). Definition 4.5.
For t ∈ N and b ∈ N ,(1) let ( Kz F ) t ( b ) be the S op b -module M b ( t ) ⊂ b | b ( t ) | = b − t F ( b ( t ) ) ⊗ Or ( b \ b ( t ) ) , where g ∈ S b sends the summand indexed by b ( t ) to that indexed by g − ( b ( t ) ) via themorphism F ( b ( t ) ) ⊗ Or ( b \ b ( t ) ) → F ( g − ( b ( t ) )) ⊗ Or ( g − ( b \ b ( t ) )) given by the action of g | g − ( b ( t ) ) on the first tensor factor and that of g | g − ( b \ b ( t ) ) on thesecond factor.(2) Let d : ( Kz F ) t ( b ) → ( Kz F ) t +1 ( b ) be the morphism of k -modules given on Y ⊗ α , where Y ∈ F ( b ( t ) ) and α ∈ Or ( b \ b ( t ) ), by d (cid:0) Y ⊗ α (cid:1) = X x ∈ b ( t ) ι ∗ x Y ⊗ ( x ∧ α ) , where ι x : b ( t ) \{ x } ֒ → b ( t ) is the inclusion.This construction is compatible with the action of the group S b : Lemma 4.6.
For t ∈ N and b ∈ N (1) there is an isomorphism of S op b -modules: ( Kz F ) t ( b ) ∼ = (cid:0) F ( b − t ) ⊗ Or ( t ′ ) (cid:1) ↑ S op b S op b − t × S op t , for b − t ⊂ b the canonical inclusion with complement t ′ ; (2) the morphism d : ( Kz F ) t ( b ) → ( Kz F ) t +1 ( b ) is a morphism of S op b -modules.Proof. The inclusion F ( b − t ) ⊗ Or ( t ′ ) ֒ → ( Kz F ) t ( b ) corresponding to the summand indexedby b − t is ( S b − t × S t ) op -equivariant, thus induces up to a morphism of S op b -modules. It isstraightforward to check that this is an isomorphism, as required.It remains to check that the morphism d is equivariant with respect to the S op b action. Thisfollows from the explicit formula for d given in Definition 4.5. (cid:3) Proposition 4.7.
The construction ( Kz F, d ) is a cochain complex in F ( Σ op ; k ) . Hence, H ∗ ( Kz F ) takes values in N -graded Σ op -modules.Proof. It suffices to show that d is a differential. Using the notation of Definition 4.5, d (cid:0) Y ⊗ α (cid:1) = X { x,y }⊂ b ( t ) ι ∗ x,y Y ⊗ (cid:0) ( y ∧ x ∧ α ) + ( x ∧ y ∧ α ) (cid:1) , where ι x,y denotes the inclusion b ( t ) \{ x, y } ⊂ b ( t ) . This is zero, as required. (cid:3) The following Proposition will be used in Section 6.
Proposition 4.8.
The Koszul construction defines an exact functor Kz : F ( FI op ; k ) → coCh( F ( Σ op ; k )) , where coCh( F ( Σ op ; k )) is the category of cochain complexes in F ( Σ op ; k ) .Proof. The functoriality is clear from the construction. Moreover, the functor F ( Kz F ) t ( b )as in Definition 4.5 is exact, since the underlying k -module of each Or ( b \ b ( t ) ) is flat (moreprecisely, free of rank one). This implies that Kz is exact considered as a functor to cochaincomplexes. (cid:3) The cosimplicial Σ op -module ( C • F ) op and its normalized cochain complex. In thissection, we identify the cosimplicial Σ op -module ( C • F ) op and its associated normalized cochaincomplex N ( C • F ) op .In order to describe ( C • F ) op , we will use the following decompositions, given by Remark 3.32(3.4), distinguishing the indices of ( ⊥ Σ ) ℓ F ( b ) by a ˜:( ⊥ Σ ) ℓ +1 F ( b ) = ⊥ Σ ⊥ Σ . . . ⊥ Σ ℓ F ( b ) ∼ = M b =( ... (( b ( ℓ +1)Σ ∐ b Σ ℓ ) ∐ b Σ ℓ − )) ... ∐ b Σ1 ) ∐ b Σ0 F ( b ( ℓ +1)Σ ) PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 17 ( ⊥ Σ ) ℓ F ( b ) = ⊥ Σ ⊥ Σ . . . ⊥ Σ ℓ − F ( b ) ∼ = M b =( ... ((˜ b ( ℓ )Σ ∐ ˜ b Σ ℓ − ) ∐ ˜ b Σ ℓ − )) ... ∐ ˜ b Σ1 ) ∐ ˜ b Σ0 F (˜ b ( ℓ )Σ )The structure of C • F is described in (4.1) and, by Notation 1.3, ( C • F ) op is the cosimplicial ob-ject with coface operators ˜ d i : ( ⊥ Σ ) ℓ F → ( ⊥ Σ ) ℓ +1 F and codegeneracy operators ˜ s j : ( ⊥ Σ ) ℓ +1 F → ( ⊥ Σ ) ℓ F satisfying ˜ s j = s ℓ − j and ˜ d j = d ℓ +1 − j . Proposition 4.9.
The structure morphisms of the cosimplicial Σ op -module ( C • F ) op are givenby the following: (1) for ≤ j ≤ ℓ , the restriction of ˜ s j : ( ⊥ Σ ) ℓ +1 F → ( ⊥ Σ ) ℓ F to the factor indexed by b = b ( ℓ +1)Σ ∐ ` ℓi =0 b Σ i is: (a) 0 if b Σ j = ∅ ; (b) if b Σ j = ∅ , the projection onto the direct summand of ( ⊥ Σ ) ℓ F such that ˜ b ( ℓ )Σ = b ( ℓ +1)Σ ˜ b Σ i = (cid:26) b Σ i i < j b Σ i +1 j ≤ i ≤ ℓ ;(2) for ≤ i ≤ ℓ + 1 , the restriction of ˜ d i : ( ⊥ Σ ) ℓ F → ( ⊥ Σ ) ℓ +1 F to the factor indexed by b = ˜ b ( ℓ )Σ ∐ ` ℓ − i =0 ˜ b Σ i is given as follows: (a) for i = 0 , it maps to the direct summand of ( ⊥ Σ ) ℓ +1 F such that ˜ b ( ℓ )Σ = b ( ℓ +1)Σ ∐ b Σ0 and ˜ b Σ i = b Σ i +1 ( ≤ i ≤ ℓ − ) via the injection b ( ℓ +1)Σ ֒ → ˜ b ( ℓ )Σ . (b) for < i ≤ ℓ , is the identity map to each direct summand of ( ⊥ Σ ) ℓ +1 F such that ˜ b ( ℓ )Σ = b ( ℓ +1)Σ and ˜ b Σ j = b Σ j j < i b Σ i ∐ b Σ i +1 j = i b Σ j +1 i < j ≤ ℓ − it is zero to the other summands of ( ⊥ Σ ) ℓ +1 F ; (c) for i = ℓ + 1 , is the identity map to the direct summand of ( ⊥ Σ ) ℓ +1 F correspondingto b ( ℓ +1)Σ = ˜ b ( ℓ )Σ and b Σ i = ˜ b Σ i for ≤ i < ℓ and b Σ ℓ = ∅ ; it is zero to the othersummands of ( ⊥ Σ ) ℓ +1 F .Proof. The general forms of s j : ( ⊥ Σ ) ℓ +1 F → ( ⊥ Σ ) ℓ F and d i : ( ⊥ Σ ) ℓ F → ( ⊥ Σ ) ℓ +1 F are givenin Proposition B.6. Using Notation 1.3 (recalled before the statement), we deduce the explicitforms given of ˜ s j and ˜ d i using Proposition 3.31 (2) and (4), Proposition 3.33 and Proposition3.35. (cid:3) For the version of the normalized cochain complex used in the following, see Definition A.2.
Proposition 4.10.
For F ∈ Ob F ( FI op ; k ) , the associated normalized cochain complex N ( C • F ) op evaluated on b , for b ∈ N , has terms: ( N ( C • F ) op ) t ( b ) = M b = b ( t ) ∐ ` t − i =0 b ( t ) i b ( t ) i = ∅ F ( b ( t ) ) . In particular, ( N ( C • F ) op ) t ( b ) = 0 if t < or if t > | b | .The differential d : ( N ( C • F ) op ) t → ( N ( C • F ) op ) t +1 is given by P ti =0 ( − i ˜ d i = ( − t +1 P t +1 i =1 ( − i d i . Proof.
The identification of the normalized subcomplex follows from the form of the codegen-eracies ˜ s j given in Proposition 4.9. The vanishing statement follows immediately.By Definition A.2, the differential on N ( C • F ) op is the restriction of that on the cochaincomplex associated to ( C • F ) op , i.e., P t +1 i =0 ( − i ˜ d i . By Proposition 4.9 (2c) the restriction of ˜ d t +1 to N ( C • F ) op is zero. (cid:3) Relating the Koszul complex and the normalized cochain complex.
In this section,we define a natural surjection from N ( C • F ) op to the Koszul complex Kz F . The proof that thissurjection is a quasi-isomorphism is given in the subsequent sections.To define this natural surjection, we use the following direct summand of the underlying N -graded object of the normalized complex N ( C • F ) op ; we stress that it is not in general a directsummand as a complex. Notation . For F ∈ Ob F ( FI op ; k ) and t, b ∈ N , let ( N ( C • F ) op ) t ( b ) be the direct summandof ( N ( C • F ) op ) t ( b ) consisting of terms such that | b ( t ) i | = 1 for each 0 ≤ i ≤ t − | b ( t ) | = b − t ).The S op b -action on ( N ( C • F ) op ) t ( b ) is described as follows, in a form suitable for comparisonwith the Koszul complex: Proposition 4.12.
For F ∈ Ob F ( FI op ; k ) and b, t ∈ N , (1) ( N ( C • F ) op ) t ( b ) is a direct summand of ( N ( C • F ) op ) t ( b ) as S op b -modules; (2) there is an isomorphism of S op b -modules: (cid:0) F ( b − t ) ⊗ k Aut( t ′ ) (cid:1) ↑ S op b S op b − t × S op t ∼ = → ( N ( C • F ) op ) t ( b ) , for b − t ⊂ b the canonical inclusion with complement t ′ .Proof. The first statement is clear.The inclusion b − t ⊂ b induces a S op b − t -equivariant inclusion F ( b − t ) ֒ → ( N ( C • F ) op ) t ( b )corresponding to the summand indexed by taking b ( t ) i = { b − t + 1 + i } for 0 ≤ i ≤ t −
1. This induces a morphism of S op b -modules, as in the statement. By inspection, this is anisomorphism. (cid:3) Proposition 4.13.
There is a surjection N ( C • F ) op ։ Kz F of cochain complexes in F ( Σ op ; k ) given, for t, b ∈ N , as the composite ( N ( C • F ) op ) t ( b ) ։ ( N ( C • F ) op ) t ( b ) ։ ( Kz F ) t ( b ) , where the first morphism is the projection of Proposition 4.12 and the second is ( N ( C • F ) op ) t ( b ) ∼ = (cid:0) F ( b − t ) ⊗ k Aut( t ′ ) (cid:1) ↑ S op b S op b − t × S op t ։ (cid:0) F ( b − t ) ⊗ Or ( t ′ ) (cid:1) ↑ S op b S op b − t × S op t ∼ = ( Kz F ) t ( b ) , induced by the surjection k Aut( t ′ ) ։ Or ( t ′ ) of S op t -modules that sends the generator [id t ′ ] to ω ( t ′ ) .Explicitly, an element Y ∈ F ( b ( t ) ) that represents a class of ( N ( C • F ) op ) t ( b ) in the summandindexed by ( b ( t ) ; b ( t )0 , . . . , b ( t ) t − ) in the decomposition given in Proposition 4.10 is sent to zero,unless | b ( t ) i | = 1 for all i , ≤ i ≤ t − , when it is sent to Y ⊗ ( b ( t )0 ∧ . . . ∧ b ( t ) t − ) ∈ F ( b ( t ) ) ⊗ Or ( b \ b ( t ) ) ⊂ ( Kz F ) t ( b ) . Proof.
Lemma 4.6 together with Proposition 4.12 imply that the morphism N ( C • F ) op ։ Kz F is a surjection of N -graded Σ op -modules. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 19
It remains to prove that this surjection is compatible with the respective differentials. Firstobserve that, in general, the kernel of the projection( N ( C • F ) op ) ∗ ( b ) ։ ( N ( C • F ) op ) ∗ ( b )is not stable under the differential of N ( C • F ) op . This is due to the fact that a coface mapof ( C • F ) op ( b ) can split a set b ( t ) i with two elements { u, v } into the ordered decompositions assingletons ( { u } , { v } ) and ( { v } , { u } ). However, these contributions will vanish on passing to Kz F ,due to the relation u ∧ v = − v ∧ u . Using the previous observation, one verifies that the kernel(denoted here simply by ker) of the morphism ( N ( C • F ) op ) ∗ ( b ) ։ ( Kz F ) ∗ ( b ) is stable underthe differential.The quotient differential on ( N ( C • F ) op ) ∗ ( b ) / ker is induced by the coface operator ˜ d on( C • F ) op ( b ).By inspection, this quotient complex is isomorphic to ( Kz F ) ∗ ( b ) via the given morphism. (cid:3) The complex of ordered partitions.
The key to proving Theorem 4.1, i.e., that N ( C • F ) op ։ Kz F is a quasi-isomorphism, is the case F = ∈ Ob F ( FI op ; k ), where is the following FI op -module: Notation . Denote by the FI op -module given by ( ) = k and ( b ) = 0 for b >
0, so thatthe only morphism that acts non-trivially on is the identity on .The aim of this section is to give an explicit description of the complex N ( C • ) op in terms ofthe complex given by ordered partitions (see Proposition 4.23) described below.Henceforth in this section, X denotes a non-empty finite set. Notation . For t ∈ N ∗ , let Part t X be the set of ordered partitions of X into t non-emptysubsets.Consider the refinement of ordered partitions as follows: Definition 4.16.
Let p = ( p , . . . , p t ) ∈ Part t ( X ) and q = ( q , . . . , q s ) ∈ Part s ( X ), then q is anordered refinement of p , denoted q ≤ p , if there exists an order-preserving surjection α : s ։ t such that p i = [ j ∈ α − ( i ) q j . (In particular, this requires that s ≥ t , with equality if and only if p = q .)The following is clear: Lemma 4.17.
The set
Part ∗ ( X ) := ∐ t ∈ N ∗ Part t ( X ) is a poset under refinement of orderedpartitions. Example 4.18.
For X = { , , } the poset Part ∗ ( X ) is given by: ( { } , { } , { } ) ( ( PPPPPPPPPPPP v v ♥♥♥♥♥♥♥♥♥♥♥♥ ( { } , { } , { } ) / / ( ( PPPPPPPPPPPP ( { } , { , } ) ( ( PPPPPPPPPPPP ( { , } , { } ) v v ♥♥♥♥♥♥♥♥♥♥♥♥ ( { } , { } , { } ) o o v v ♥♥♥♥♥♥♥♥♥♥♥♥ ( { , } , { } ) / / ( { , , } ) ( { } , { , } ) o o ( { } , { } , { } ) / / ♥♥♥♥♥♥♥♥♥♥♥♥ ( { } , { , } ) ♥♥♥♥♥♥♥♥♥♥♥♥ ( { , } , { } ) h h PPPPPPPPPPPP ( { } , { } , { } ) h h PPPPPPPPPPPP o o ( { } , { } , { } ) h h PPPPPPPPPPPP ♥♥♥♥♥♥♥♥♥♥♥♥ The following Lemma highlights properties of the poset (
Part ∗ ( X ) , ≤ ) that are exploited inthe proof of Proposition 4.27. Lemma 4.19. (1)
For p = ( p , . . . , p | X | ) , p ′ = ( p ′ , . . . , p ′| X | ) ∈ Part | X | ( X ) , there exists q ∈ Part | X |− ( X ) such that p ≤ q and p ′ ≤ q if and only if there exists i ∈ { , . . . , | X | − } such that p i = p ′ i +1 , p i +1 = p ′ i and p j = p ′ j for j
6∈ { i, i + 1 } . (2) Let p = ( p , . . . , p t ) ∈ Part t ( X ) , for ≤ t ≤ | X | − , and r ∈ Part t +2 ( X ) such that r ≤ p . Then |{ q ∈ Part t +1 ( X ) | r ≤ q ≤ p }| = 2 . Proof.
For the forward implication of the first point, suppose that there exists q ∈ Part | X |− ( X )such that p ≤ q and p ′ ≤ q . By Definition 4.16, we have order-preserving surjections α : X →| X | − α ′ : X → | X | − q k = a j ∈ α − ( k ) p j and q k = a j ∈ α ′− ( k ) p ′ j . Since α and α ′ are order preserving surjections there is i, j ∈ { , . . . , | X | − } such that α − ( i ) = { i, i + 1 } and α ′− ( j ) = { j, j + 1 } with all other fibres of cardinal one. We deduce from (4.2)that i = j and q i = p i ∐ p i +1 = p ′ i ∐ p ′ i +1 so p i = p ′ i +1 , p i +1 = p ′ i and p j = p ′ j for j
6∈ { i, i + 1 } .For the converse, pick p and p ′ as in the statement. The following partition q ∈ Part | X |− ( X )satisfies the inequalities of the statement: q j = p j for 1 ≤ j ≤ i − , q i = p i ∐ p i +1 , q j = p j − for i + 1 ≤ j ≤ | X | − . For the second point, by hypothesis, r ≤ p . By Definition 4.16, this corresponds to giving r together with an order-preserving surjection α : t + ։ t . To prove the result, it suffices toshow that there are precisely two possible factorizations of α via order-preserving surjections: t + α ′ ։ t + α ′′ ։ t . The factorization condition implies that α ′′ is uniquely determined by α ′ .There are two possibilities:(1) there exists i ∈ t such that | α − ( i ) | = 3, all other fibres having cardinal one;(2) there exists i < j ∈ t such that | α − ( i ) | = | α − ( j ) | = 2, with all other fibres of cardinalone.In the first case, one has either α ′− ( i ) = { i, i + 1 } or α ′− ( i + 1) = { i + 1 , i + 2 } , with allother fibres of cardinal one. In the second case, either α ′− ( i ) = { i, i + 1 } or α ′− ( j ) = { j, j + 1 } ,again with all other fibres of cardinal one. (cid:3) Definition 4.20.
Let C perm ( X ) denote the cochain complex with C perm ( X ) t := (cid:26) k [ Part t X ] 1 ≤ t ≤ | X | . The differential d : C perm ( X ) t → C perm ( X ) t +1 is the alternating sum P ti =1 ( − i δ i , where δ i sends a generator [ p ] corresponding to the ordered partition p = ( p , . . . , p i , . . . , p t ) to the sumof ordered partitions of the form ( p , . . . , p ′ i , p ′′ i , . . . , p t ) where p i = p ′ i ∐ p ′′ i . Remark . The element d [ p ], for p ∈ Part t ( X ), is a signed sum (i.e., a linear combinationwith coefficients in {± } ) of the generators [ q ] where q ∈ Part t +1 ( X ) and q ≤ p . Example 4.22. (1) C perm ( ∅ ) = k in cohomological degree zero, by convention; PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 21 (2) C perm ( { } ) ∼ = k in cohomological degree one;(3) C perm ( { , } ) is the complex k diag → k ⊕ concentrated in cohomological degrees one andtwo; this has cohomology k concentrated in degree 2;(4) C perm ( { , , } ) has the form k diag → k ⊕ → k ⊕ in cohomological degrees 1 , ,
3; this hascohomology k concentrated in degree 3. Here, the generators are given by the orderedpartitions: degree 1 degree 2 degree 3( { } ) ( { } , { } ) ( { } , { } , { } )( { } , { } ) ( { } , { } , { } )( { } , { } ) ( { } , { } , { } )( { } , { } ) ( { } , { } , { } )( { } , { } ) ( { } , { } , { } )( { } , { } ) ( { } , { } , { } )and, for instance, the differential on ( { } , { , } ) is given by the operator δ which sendsthe generator to the sum of ( { } , { } , { } ) and ( { } , { } , { } ). Proposition 4.23.
For X a non-empty finite set, there is an isomorphism of complexes ( N ( C • ) op )( X ) ∼ = C perm ( X ) . Proof.
By Proposition 4.10 and Notation 4.14, since vanishes on non-empty sets, we have:( N ( C • ) op ) t ( X ) = M X = b ( t ) ∐ ` t − i =0 b ( t ) i b ( t ) i = ∅ ( b ( t ) ) = M X = ` t − i =0 b ( t ) i b ( t ) i = ∅ k ∼ = k [ Part t X ]and the coface ˜ d : ( ⊥ Σ ) ℓ → ( ⊥ Σ ) ℓ +1 is zero. So, using Proposition 4.10, the differential of( N ( C • ) op ) is given by P ti =1 ( − i ˜ d i .For 1 ≤ i ≤ t , comparing the definition of δ i given in Definition 4.20 and the explicit descriptionof ˜ d i : ( ⊥ Σ ) ℓ → ( ⊥ Σ ) ℓ +1 given in Proposition 4.9, we obtain that ˜ d i corresponds, via the aboveisomorphism, to δ i . (cid:3) Cohomology of the complex of ordered partitions.
Let X be a non-empty finiteset. To calculate the cohomology of C perm ( X ), we relate it to a cellular complex as follows. Thepermutohedron Π X is an (abstract) ( | X |− k -faces in bijection with Part | X |− k X (see [Zie95, Example 0.10]). In particular, the vertices (i.e., 0-faces) are indexed byelements of Part | X | ( X ) (i.e., by permutations of the set X ).We record the necessary information on the face inclusions of Π X : Lemma 4.24.
Let p = ( p , . . . , p t ) , q = ( q , . . . , q t +1 ) ∈ Part ∗ ( X ) , for ≤ t < | X | , be orderedpartitions of the finite non-empty set X . Then q is a face of p if and only if q ≤ p . Lemma 4.19 can be rephrased as follows:
Lemma 4.25. (1)
Two vertices of Π X are linked by a -face if and only if they differ by a transposition ofadjacent elements in the corresponding ordered lists of elements of X . (2) Let p be a n -face of Π X with ≤ n ≤ | X | − ; a ( n − -face r of p lies in precisely two ( n − -faces of p . The polytope Π X has a geometric realization | Π X | as the convex hull in R | X | of the vectors withpairwise distinct coordinates from { , , . . . , | X |} ; in particular, | Π X | is contractible. Moreover,by the above, the geometric polytope | Π X | is equipped with a cellular structure with the cells of dimension k indexed by the elements of Part | X |− k ( X ), for 0 ≤ k ≤ | X | −
1. The cellularstructure is understood by using Lemma 4.24.
Example 4.26.
The geometric realization of the permutohedron Π { , , } is a hexagon withvertices indexed by the elements of S :(1 , , ( { , } , , { , } ) ttttttttt (2 , , (2 , { , } ) ❏❏❏❏❏❏❏❏❏ (1 , , ( { , } , ❏❏❏❏❏❏❏❏❏ ( { , , } ) (2 , , ( { , } , ttttttttt (3 , , (3 , { , } ) (3 , , C perm ( X ) is considered with homological degree and [ −| X | ] de-notes the shift of homological degree (i.e., C perm ( X )[ −| X | ] t = C perm ( X ) t −| X | by Notation 1.4). Proposition 4.27.
Let X be a non-empty finite set, then there is an isomorphism of chaincomplexes γ X : C perm ( X )[ −| X | ] ∗ ∼ = → C cell ∗ ( | Π X | ; k ) , where C cell ∗ ( | Π X | ; k ) is the cellular complex of | Π X | with coefficients in k .In particular, C perm ( X ) has cohomology k concentrated in degree | X | .Proof. The case | X | = 1 is clear, hence we suppose that | X | ≥ C cell ∗ ( | Π X | ; k ) has underlying graded k -module which is free on Part ∗ ( X ),so that there is an isomorphism of k -modules(4.3) C perm ( X ) | X |− t ∼ = C cell t ( | Π X | ; k ) . Moreover, for [ p ] cell the generator corresponding to p ∈ Part t ( X ), d [ p ] cell is a signed sum ofthe generators [ q ] cell , where q ∈ Part t +1 ( X ) and q ≤ p . (Here and in the following, the subscriptsindicate in which complex the generators live.) With the regrading used here, this is analogous tothe behaviour observed in Remark 4.21; the only difference being in the signs which may occur.In homological degree zero, the isomorphism (4.3) is given by C perm ( X ) | X | → C cell0 ( | Π X | ; k )[ σ ] perm sign ( σ )[ σ ] cell , using the identification of ordered partitions of X into singletons with the symmetric groupAut( X ) to define sign ( σ ).To prove the statement, we consider the cellular filtration of | Π X | . More precisely, we prove,by induction on n , that we have an isomorphism of chain complexes τ ≤ n C perm ( X )[ −| X | ] ∗ γ X ∼ = / / τ ≤ n C cell ∗ ( | Π X | ; k )such that, at each degree, γ X [ p ] perm = ± [ p ] cell and where τ ≤ n C ∗ is the truncation of the complex C ∗ defined by ( τ ≤ n C ∗ ) i = 0 if i > n and ( τ ≤ n C ∗ ) i = C i if i ≤ n .To prove the initial step, note that the 1-skeleton of | Π X | can be given the structure of anoriented graph, where each edge is oriented from the vertex with negative signature to that of PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 23 positive signature; this relies crucially upon the fact that edges are indexed by transpositions(see Lemma 4.25 (1)). This serves to make explicit the signs appearing in the differential of theassociated chain complex.It follows that the identity on
Part | X |− ( X ) induces an isomorphism of chain complexes γ X : τ ≤ C perm ( X )[ −| X | ] ∗ → τ ≤ C cell ∗ ( | Π X | ; k ): C perm ( X ) | X |− ∼ = [ p ] perm [ p ] cell (cid:15) (cid:15) d / / C perm ( X ) | X |∼ = [ σ ] perm sign ( σ )[ σ ] cell (cid:15) (cid:15) C cell1 ( | Π X | ; k ) d / / C cell0 ( | Π X | ; k ) . For n ≥ γ X : τ ≤ n − C perm ( X )[ −| X | ] ∗ → τ ≤ n − C cell ∗ ( | Π X | ; k ) is defined and isan isomorphism. To prove the inductive step, choose p ∈ Part | X |− n ( X ); as observed above,the differential of the generator [ p ] cell ∈ C cell n ( | Π X | ; k ) is a signed sum of the generators [ q ] cell corresponding to its ( n − d [ p ] cell = X q ∈ Part | X |− n +1 ( X ) q ≤ p η q [ q ] cell , with η q ∈ {± } . Now, by the inductive hypothesis, we have an isomorphism γ X : C perm ( X ) | X |− n +1 ∼ = → C cell n − ( | Π X | ; k )that is compatible with the differential and which has the form [ q ] perm
7→ ± [ q ] cell . Hence, byRemark 4.21: γ X d [ p ] perm = X q ∈ Part | X |− n +1 ( X ) q ≤ p α q [ q ] cell , with α q ∈ {± } . ; moreover, γ X d [ p ] perm is a cycle.We claim that γ X d [ p ] perm = ε p d [ p ] cell , for some ε p ∈ {± } . If γ X d [ p ] perm = d [ p ] cell , take ε p = 1. Otherwise, consider γ X d [ p ] perm + d [ p ] cell = X q ∈ Part | X |− n +1 ( X ) q ≤ p κ q [ q ] cell , where κ q = α q + η q .By the hypothesis, there is some q ∈ Part | X |− n +1 ( X ) such that κ q = 0.If γ X d [ p ] perm + d [ p ] cell = 0, there is some q ′ such that κ q ′ = 0. Using Lemma 4.25 (2), wecan find such a pair ( q , q ′ ) such that q and q ′ have a ( n − q ∩ q ′ ;moreover, q , q ′ are the only ( n − q ∩ q ′ is a face. We deduce that d ( q ∩ q ′ ) = 0:this contradicts the fact that γ X d [ p ] perm + d [ p ] cell is a cycle, therefore γ X d [ p ] perm + d [ p ] cell = 0.Taking ε p = − d [ p ] cell explained above, it follows that γ X d [ p ] perm = ε p d [ p ] cell , where ε p ∈ {± } . Thus, an extension of γ X compatible with the differential and of therequired form is given by defining γ X [ p ] perm := ε p [ p ] cell . Applying this argument for all cells p of dimension n completes the inductive step.The isomorphism of chain complexes γ X induces an isomorphism in homology. Now | Π X | iscontractible, hence has homology k concentrated in degree zero. Returning to the cohomological,unshifted grading, this implies that C perm ( X ) has cohomology isomorphic to k concentrated incohomological degree | X | . (cid:3) Corollary 4.28.
There is an isomorphism of cohomologically-graded Σ op -modules H ∗ ( N ( C • ) op ) ∼ = Or . Proof.
This follows from Propositions 4.23 and 4.27, noting that the result holds evaluated on by inspection. For X a non-empty finite set, the isomorphism in cohomology is induced by themorphism of k -modules C perm ( X ) | X | → k given by [ σ ] sign ( σ ). (cid:3) Proof of Theorem 4.1.
In this section we pass from the case treated in Corollary 4.28to that of an arbitrary FI op -module F .The complex ( N ( C • F ) op ) ∗ ( b ) has a finite filtration given as follows: Lemma 4.29.
Let F be an FI op -module and b ∈ N . For s ∈ N , let f s ( N ( C • F ) op ) ∗ ( b ) ⊂ ( N ( C • F ) op ) ∗ ( b ) be the graded k -submodule : f s ( N ( C • F ) op ) t ( b ) = M b = b ( t ) ∐ ` t − i =0 b ( t ) i b ( t ) i = ∅| b ( t ) |≤ b − s F ( b ( t ) ) . Then f s ( N ( C • F ) op ) ∗ ( b ) is a subcomplex and there is a finite filtration ⊂ f b ( N ( C • F ) op ) ∗ ( b ) ⊂ f b − ( N ( C • F ) op ) ∗ ( b ) ⊂ . . . ⊂ f ( N ( C • F ) op ) ∗ ( b ) = ( N ( C • F ) op ) ∗ ( b ) . Moreover, the quotient complex f s ( N ( C • F ) op ) ∗ ( b ) / f s +1 ( N ( C • F ) op ) ∗ ( b ) decomposes as the di-rect sum of complexes M b ′ ⊂ b | b ′ | = b − s F ( b ′ ) ⊗ ( N ( C • ) op ) ∗ ( b \ b ′ ) . Proof.
From the construction of C • F , it is clear that f s ( N ( C • F ) op ) ∗ ( b ) is stable under thedifferential and that one has a finite filtration as given.The subquotient f s / f s +1 only has contributions from terms with | b ′ | = b − s . Moreover, byconstruction, the differential behaves as though all non-isomorphisms of FI op act as zero. Byinspection, the subquotient identifies as stated. (cid:3) Proof of Theorem 4.1.
The natural surjection N ( C • F ) op ։ Kz F of cochain complexes in F ( Σ op ; k )is given by Proposition 4.13. It suffices to show that ( N ( C • F ) op ) ∗ ( b ) ։ ( Kz F ) ∗ ( b ) is a quasi-isomorphism, for each b ∈ N . This follows from the spectral sequence associated to the filtration f • ( N ( C • F ) op ) ∗ ( b ) that calculates the cohomology of ( N ( C • F ) op ) ∗ ( b ). By Lemma 4.29, the E -page is given by E s,t = f s ( N ( C • F ) op ) s + t ( b ) / f s +1 ( N ( C • F ) op ) s + t ( b ) ∼ = M b ′ ⊂ b | b ′ | = b − s F ( b ′ ) ⊗ ( N ( C • ) op ) s + t ( b \ b ′ ) . The E -page is calculated by using Corollary 4.28 to identify the cohomology of each complex( N ( C • ) op ) ∗ ( b \ b ′ ) and by applying universal coefficients. In particular, for | b ′ | = b − s thecohomology of ( N ( C • ) op ) ∗ ( b \ b ′ ) is concentrated in degree s .It follows that the E -page is concentrated in the line t = 0. This line, equipped with thedifferential d , identifies with the complex ( Kz F ) ∗ ( b ). The spectral sequence degenerates at the E -page, since there is no space for differentials. The result follows. (cid:3) PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 25
Relating to FI op -cohomology. In [CE17, Gan16], the authors define the FI -homology ofa FI -module F as the left derived functors of a right exact functor H FI . The functor H FI canbe defined as the left adjoint of the functor F ( Σ ; k ) → F ( FI ; k ) given by extension by zero onmorphisms (see Definition 4.30 for a precise definition in the dual case).In this section, we consider the dual situation to define the FI op -cohomology of a FI op -module F . This allows us to rephrase Theorem 4.1 in Corollary 4.35, using FI op -cohomology. Theorem6.15 can also be reinterpreted as the computation of the FI op -cohomology of the FI op -module k Hom Ω ( − , a ) considered in Proposition 5.21 (see Theorem 4).By Theorem 3.34, a FI op -module F can be considered equivalently as a ⊥ Σ -comodule in F ( Σ op ; k ), with structure morphism ψ F : F → ⊥ Σ F . Moreover, by Proposition 3.35, one has thenatural coaugmentation η F : F → ⊥ Σ F .The following constructions should be compared with the general framework that is given inAppendix B.2, working with the category of FI op -modules, which is abelian and has enoughinjectives (see below for the latter). Definition 4.30. (Cf. Example B.8.) For F ∈ Ob F ( FI op ; k ), let H FI op ( F ) ∈ Ob F ( Σ op ; k ) bethe equalizer of the diagram F η F / / ψ F / / ⊥ Σ F, which defines a functor H FI op : F ( FI op ; k ) → F ( Σ op ; k ). Definition 4.31.
Let Z : F ( Σ op ; k ) → F ( FI op ; k ) be the exact functor given by extensionby zero on morphisms (i.e., an Σ op -module is considered as an FI op -module by specifying thatnon-isomorphisms of FI op act as zero). Proposition 4.32.
The functor H FI op : F ( FI op ; k ) → F ( Σ op ; k ) is right adjoint to Z : F ( Σ op ; k ) →F ( FI op ; k ) . In particular, it is left exact.Proof. One checks that, for F an FI op -module, H FI op F is the largest sub FI op -module that liesin the image of Z , hence H FI op is the right adjoint to Z . This implies, in particular, that H FI op is left exact. (cid:3) It is a standard fact that the category F ( FI op ; k ) has enough injectives. (This follows fromYoneda’s Lemma, which shows that, for I an injective k -module and n ∈ Ob FI , the FI op -moduleMap(Hom FI ( n , − ) , I ) represents the functor F Hom k ( F ( n ) , I ).)Thus one can define FI op -cohomology as follows: Definition 4.33.
For s ∈ N ,(1) let H s FI op denote the s -th right derived functor of H FI op ;(2) H s FI op ( F ) ∈ F ( Σ op ; k ) is the s th FI op -cohomology of the FI op -module F .Proposition 3.35 shows that the coaugmentation η of ⊥ Σ satisfies Hypothesis B.5, hence onecan form the cosimplicial object C • F , as in Proposition B.6, with respect to the structure ( ⊥ Σ , η )on F ( Σ op ; k ). Proposition 4.34.
For F ∈ Ob F ( FI op ; k ) , there is a natural isomorphism H ∗ FI op ( F ) ∼ = π ∗ ( C • F ) , where the right hand side is the cohomotopy of the cosimplicial object C • F of F ( Σ op ; k ) .Proof. By definition, H FI op ( F ) = π ( C • F ). Moreover, Proposition B.6 implies that F C • F defines an exact functor to cosimplicial objects, so F π ∗ ( C • F ) forms a cohomological δ -functor.The usual argument of homological algebra (cf. [Wei94, Exercise 2.4.5], for example), using theeffacability property established in Corollary B.10, then provides the isomorphism. (cid:3) This allows Theorem 4.1 to be restated using FI op -cohomology: Corollary 4.35.
For F ∈ Ob F ( FI op ; k ) , there is a natural isomorphism H ∗ FI op ( F ) ∼ = H ∗ ( Kz F ) . Proof.
The normalized complex N ( C • F ) op is canonically isomorphic to N C • F , in particularhas canonically isomorphic cohomology. Thus the statement follows immediately from Theorem4.1. (cid:3) Remark . This result is an FI op -cohomology version of [Gan16, Theorem 2], which concerns FI -homology. 5. Coaugmented cosimplicial objects
The main objective of this paper is to calculate Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) and, more gener-ally, Ext i F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ), for a, b ∈ N and i ∈ N . As a step in this direction, Theorem 5.4establishes that, for i ∈ { , } , these Ext-groups are given by the cohomology of the cosimplicialabelian group Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b ) introduced in Proposition 5.3.In this section we focus principally upon understanding this cosimplicial object. We give analternative, highly explicit description of this by using the Dold-Kan equivalence of Theorem 3.7.More precisely, as k Hom Ω ( − , a ) is an FI op -module (see Proposition 5.21), we can consider thecosimplicial Σ op -module C • k Hom Ω ( − , a ) described in (4.1). The main result of this section isthe following: Theorem 5.1.
For a ∈ N , there is an isomorphism of cosimplicial objects in F ( Σ op ; k ) : Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • t ∗⊗ ) ∼ = [ C • k Hom Ω ( − , a )] op where − op is the functor introduced in Notation 1.3 and t ∗⊗ is the functor in F (Γ × Σ op ; k ) givenby t ∗⊗ ( b ) := ( t ∗ ) ⊗ b (cf. Definition 5.13). Using Theorem 4.1 and Corollary 4.35, this has the following immediate consequence:
Corollary 5.2.
For a ∈ N , there is an isomorphism of graded Σ op -modules: H ∗ (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • t ∗⊗ )) ∼ = H ∗ ( Kz k Hom Ω ( − , a )) ∼ = H ∗ FI op ( k Hom Ω ( − , a )) . Combining Corollary 5.2, Theorem 5.4 and the computation of the cohomology of Kz k Hom Ω ( − , a )that is carried out in Section 6, leads to the computation of Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) andExt F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) given in Theorem 7.1.5.1. The cosimplicial abelian group
Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b ) . In this section we applythe general constructions given in Appendix B.2 for C = F (Γ; k ) and ⊥ Γ : F (Γ; k ) → F (Γ; k ) thecomonad introduced in Notation 3.21.By Proposition B.12, for a, b ∈ N , there is an equalizer:(5.1)Hom F (Γ; k ) ⊥ Γ (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) / / Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) d / / d / / Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ ( t ∗ ) ⊗ b )where, for f ∈ Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ), d f := ( ⊥ Γ f ) ◦ ψ ( t ∗ ) ⊗ a , d f := ψ ( t ∗ ) ⊗ b ◦ f and F (Γ; k ) ⊥ Γ is the category of ⊥ Γ -comodules in F (Γ; k ).The comonad ⊥ Γ provides the usual augmented simplicial object ⊥ Γ • +1 ( t ∗ ) ⊗ b (cf. PropositionB.3) and hence an augmented simplicial object Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • +1 ( t ∗ ) ⊗ b ) (cf. LemmaB.13). PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 27
Proposition 5.3.
The augmented simplicial object
Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • +1 ( t ∗ ) ⊗ b ) given byLemma B.13 extends to a cosimplicial abelian group Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b ) . by using the structure morphisms d and d and their generalizations.Proof. This is a direct application of Proposition B.14. It is a cosimplicial abelian group because F (Γ; k ) is abelian and ⊥ Γ is additive. (cid:3) The interest of this cosimplicial object is established by the following:
Theorem 5.4.
For a, b ∈ N there are isomorphisms: Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = H (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b ))Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = H (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b )) . Proof.
The identification of the cohomology groups H i , for i = 0 ,
1, is given by Theorem B.23,using the Barr-Beck theorem (Theorem 3.24 and Remark 3.25) to translate from ⊥ Γ -comodulesto F ( Fin ; k ). The fact that the functor ( t ∗ ) ⊗ a is projective in F (Γ; k ) (by Proposition 3.6)implies that Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) = Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) , using the notation of Theorem B.23. (cid:3) Remark . By the Barr-Beck theorem (Theorem 3.24 and Remark 3.25) and Proposition 3.20,for a, b ∈ N , we have:Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = Hom F (Γ; k ) ⊥ Γ (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b )so, by diagram (5.1), the calculation of Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) reduces to the calculationof the kernel of d − d . Unfortunately, this kernel is inaccessible to direct computation.The above isomorphism shows that Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) can be calculated using mor-phisms of ⊥ Γ -comodules in F (Γ; k ). This motivates the introduction of the simplicial objectHom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • +1 ( t ∗ ) ⊗ b ).5.2. Structure morphisms for the simplicial object ( ⊥ Γ ) • +1 ( t ∗ ) ⊗ b . In this section we givethe explicit description of the structure of the augmented simplicial object ( ⊥ Γ ) • +1 ( t ∗ ) ⊗ b obtainedfrom Proposition B.3 applied to the comonad ⊥ Γ .We begin by the cases b = 0 and b = 1, exploiting the following basic result, which followsfrom Theorem 3.7: Lemma 5.6.
The identity induces canonical isomorphisms
Hom F (Γ; k ) ( k , k ) ∼ = k and Hom F (Γ; k ) ( t ∗ , t ∗ ) ∼ = k . Moreover,
Hom F (Γ; k ) ( t ∗ , k ) = 0 = Hom F (Γ; k ) ( k , t ∗ ) . Recall from Section 3.3 the identifications ϑ ∗ k ∼ = k and ϑ ∗ P Fin1 ∼ = P Γ + , together with ϑ ∗ k [ − ] ∼ = t ∗ . In particular, since k and t ∗ lie in the image of ϑ ∗ , they are ⊥ Γ -comodules (see Remark 3.25).For b = 0, ( t ∗ ) ⊗ = k and ⊥ Γ k = k . The structure morphisms ε Γ k : ⊥ Γ k → k and ψ : k → ⊥ Γ k are the respective identity natural transformations.For b = 1, the structure morphisms ε Γ t ∗ and ψ are identified by the following: Lemma 5.7.
There are isomorphisms ( − ) ∗ + t ∗ ∼ = P Fin1 ⊥ Γ t ∗ ∼ = P Γ + ∼ = t ∗ ⊕ k . The ⊥ Γ -comodule structure morphism ψ : t ∗ → ⊥ Γ t ∗ ∼ = P Γ + ∼ = t ∗ ⊕ k is the canonical inclusionand the counit ε Γ t ∗ : ⊥ Γ t ∗ → t ∗ is the canonical projection.Explicitly, for ( Z, z ) a finite pointed set, t ∗ ( Z ) has generators [ y ] − [ z ] , for y ∈ Z \{ z } and ⊥ Γ t ∗ ( Z ) has generators [ y ] − [ z ] , for y ∈ Z \{ z } , and [ z ] − [+] . The morphism ψ sends [ y ] − [ z ] to [ y ] − [ z ] and ε Γ t ∗ sends [ y ] − [ z ] to [ y ] − [ z ] and [ z ] − [+] to .Proof. As in Proposition 3.20, we consider t ∗ as a sub-functor of P Γ + . Since ( − ) ∗ + is exact (byProposition 3.19), for X ∈ Ob Fin we have:( − ) ∗ + t ∗ ( X ) = t ∗ ( X + ) ⊂ ( − ) ∗ + P Γ + ( X ) = P Γ + ( X + ) ∼ = k [ X + ]and t ∗ ( X + ) identifies as the submodule generated by the elements [ x ] − [+], for x ∈ X , bythe explicit description given in Lemma 3.5. By the Yoneda Lemma, there is a morphism P Fin1 → ( − ) ∗ + t ∗ determined by the element [1] − [+] ∈ t ∗ ( + ). This identifies as the morphism P Fin1 ( X ) ∼ = k [ X ] → t ∗ ( X + ) ⊂ k [ X + ] that sends a generator [ x ] to [ x ] − [+], for x ∈ X ; this givesthe isomorphism ( − ) ∗ + t ∗ ∼ = P Fin1 . The identification of ⊥ Γ t ∗ then follows from the isomorphism ϑ ∗ P Fin1 ∼ = P Γ + given by Proposition 3.19; the second isomorphism is given by Lemma 3.5.By Lemma 5.6, since the composite ε ⊥ Γ ψ is the identity on t ∗ , to identify these structuremorphisms it suffices to show that, under the isomorphism ⊥ Γ t ∗ ∼ = t ∗ ⊕ k , ε ⊥ Γ is the canonicalprojection. This follows from the description of ε ⊥ Γ given in Proposition 3.23, as seen explicitlyas follows.By Lemma 3.5, t ∗ ( Z ) ⊂ k [ Z ] has generators [ y ] − [ z ], for y ∈ Z \{ z } , thus t ∗ ( Z + ) has generators:[ y ] − [+], y as above, and [ z ] − [+]. Writing ([ y ] − [ z ]) + ([ z ] − [+]) = [ y ] − [+], one also has that t ∗ ( Z + ) has generators [ y ] − [ z ] and [ z ] − [+], corresponding to generators of t ∗ ( Z ) ⊂ t ∗ ( Z + ) and k respectively. The element [ z ] − [+] generates the kernel of ε ⊥ Γ and [ y ] − [ z ] is sent by ε ⊥ Γ to[ y ] − [ z ] ∈ t ∗ ( Z ), corresponding to the projection to t ∗ . (cid:3) Consider the beginning of the augmented simplicial object ( ⊥ Γ ) • +1 t ∗ : ⊥ Γ ⊥ Γ t ∗ δ = ε ⊥ Γ t ∗ / / δ = ⊥ Γ ε t ∗ / / ⊥ Γ t ∗ δ = ε t ∗ / / σ =∆ u u t ∗ . The structure morphisms are identified by the following, in which we implicitly use Lemma5.6:
Proposition 5.8. (1)
There is an isomorphism ⊥ Γ ⊥ Γ t ∗ ∼ = t ∗ ⊕ k ⊕ k , where k i is the rank one free k -module given by k i := ker( δ i ) for i ∈ { , } . (2) The diagonal ∆ : ⊥ Γ t ∗ ∼ = t ∗ ⊕ k → ⊥ Γ ⊥ Γ t ∗ ∼ = t ∗ ⊕ k ⊕ k is given by the canonicalinclusion of t ∗ and the diagonal map k → k ⊕ k . (3) The morphism ⊥ Γ ψ : ⊥ Γ t ∗ ∼ = t ∗ ⊕ k → ⊥ Γ ⊥ Γ t ∗ ∼ = t ∗ ⊕ k ⊕ k is the inclusion of the directsummand t ∗ ⊕ k .Proof. The isomorphism ⊥ Γ t ∗ ∼ = t ∗ ⊕ k of Lemma 5.7 induces an isomorphism ⊥ Γ ⊥ Γ t ∗ ∼ = t ∗ ⊕ k ⊕ .The operators δ and δ give a canonical decomposition, as follows. For ( Z, z ) a finite pointedset, by Lemma 5.7, t ∗ ( Z ) has generators [ y ] − [ z ], for y ∈ Z \{ z } and ⊥ Γ t ∗ ( Z ) has generators[ y ] − [ z ], for y ∈ Z \{ z } , and [ z ] − [+]. By Proposition 3.23, we have: ⊥ Γ ⊥ Γ t ∗ ( Z ) = t ∗ (( Z + ) + ) , PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 29 so that ⊥ Γ ⊥ Γ t ∗ ( Z ) is generated by [ y ] − [+ ], for y ∈ Z \{ z } , [ y ] − [+ ], and [+ ] − [+ ]. Usingthe equalities[ y ] − [+ ] = ([ y ] − [ z ]) + ([ z ] − [+ ]); [ z ] − [+ ] = ([ z ] − [+ ]) + ([+ ] − [+ ])we obtain that ⊥ Γ ⊥ Γ t ∗ ( Z ) is generated by: (cid:8) ([ y ] − [ z ]) , ([ z ] − [+ ]) , ([+ ] − [+ ]) | y ∈ Y \{ z } (cid:9) . By the description of the counits given in Proposition 3.23, ([ z ] − [+ ]) generates the kernel of ε ⊥ Γ t ∗ (i.e., k ) and ([+ ] − [+ ]) generates the kernel of ⊥ Γ ε t ∗ (i.e., k ).By Proposition 3.23, the diagonal ∆ : ⊥ Γ t ∗ → ⊥ Γ ⊥ Γ t ∗ sends ([ y ] − [ z ]) to ([ y ] − [ z ]) and([ z ] − [+]) to ([ z ] − [+ ]) = ([ z ] − [+ ]) + ([+ ] − [+ ]); this identifies ∆. (An alternative argumentis to use the fact that the composites ε ⊥ Γ ∆ and ( ⊥ Γ ε )∆ are both the identity on ⊥ Γ t ∗ .)The identification of ⊥ Γ ψ is similar, by using Lemma 5.7. (cid:3) For the case b >
1, we use the fact that the functor ⊥ Γ is symmetric monoidal, by Corollary3.22, to obtain the following: Proposition 5.9.
Let b ∈ N . (1) The canonical isomorphism ⊥ Γ t ∗ ∼ = t ∗ ⊕ k induces isomorphisms: ⊥ Γ ( t ∗ ) ⊗ b ∼ = P Γ b + ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ ; ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b ∼ = M b ′′ ⊆ b ′ ⊆ b ( t ∗ ) ⊗ b ′′ where b ′ = | b ′ | and b ′′ = | b ′′ | . (2) The comodule structure morphism ψ ( t ∗ ) ⊗ b : ( t ∗ ) ⊗ b → ⊥ Γ ( t ∗ ) ⊗ b ∼ = L b ′ ⊆ b ( t ∗ ) ⊗ b ′ is theinclusion of the direct summand indexed by b ′ = b . (3) The counit ε Γ( t ∗ ) ⊗ b : ⊥ Γ ( t ∗ ) ⊗ b ∼ = L b ′ ⊆ b ( t ∗ ) ⊗ b ′ → ( t ∗ ) ⊗ b is the canonical projection. (4) The diagonal ∆ : ⊥ Γ ( t ∗ ) ⊗ b ∼ = L b ′ ⊆ b ( t ∗ ) ⊗ b ′ → ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b ∼ = L b ′′ ⊆ ˜ b ′ ⊆ b ( t ∗ ) ⊗ b ′′ withcomponent for b ′ ⊆ b and b ′′ ⊆ ˜ b ′ ⊆ b , ( t ∗ ) ⊗ b ′ → ( t ∗ ) ⊗ b ′′ , the identity if b ′′ = b ′ andzero otherwise. (Here the notation ˜ b ′ is introduced for the indexing set of ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b toavoid confusion.)Proof. Using that ⊥ Γ is symmetric monoidal (see Corollary 3.22), we have ⊥ Γ ( t ∗ ) ⊗ b ∼ = ( ⊥ Γ t ∗ ) ⊗ b . The decomposition of ⊥ Γ ( t ∗ ) ⊗ b follows by considering the isomorphism ⊥ Γ ( t ∗ ) ⊗ b ∼ = ( t ∗ ⊕ k ) ⊗ b that is given by using the decomposition of Lemma 5.7. The tensor factors of the tensor productare indexed by the elements of b : the direct summand indexed by b ′ corresponds to the term inwhich t ∗ is a tensor factor for i ∈ b ′ and is k otherwise.The decomposition of ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b is obtained by iterating the previous argument, since ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ⊥ Γ ( t ∗ ) ⊗ b ′ . The descriptions of ψ ( t ∗ ) ⊗ b and ε Γ( t ∗ ) ⊗ b follow from the descriptions of ψ : t ∗ → ⊥ Γ t ∗ ∼ = P Γ + ∼ = t ∗ ⊕ k and ε Γ t ∗ : ⊥ Γ t ∗ → t ∗ given in Lemma 5.7 and the previous decomposition.Since ⊥ Γ is monoidal, ∆ : ⊥ Γ ( t ∗ ) ⊗ b → ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b is induced by the diagonal ∆ : ⊥ Γ t ∗ → ⊥ Γ ⊥ Γ t ∗ identified in Proposition 5.8. Namely, this corresponds to( t ∗ ⊕ k ) ⊗ b → ( t ∗ ⊕ k ⊕ k ) ⊗ b induced by the canonical inclusion of t ∗ and the diagonal k → k ⊕ k . Developing the tensorproducts, one obtains the description of the component given in the statement. (cid:3) Recall that, for b ∈ N ∗ , ξ b : t ∗ → ( t ∗ ) ⊗ b is the standard generator given in Lemma 3.8. Proposition 5.10. If b ∈ N ∗ , the morphism ⊥ Γ ξ b : ⊥ Γ t ∗ ∼ = t ∗ ⊕ k → ⊥ Γ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ sends k to the copy of k indexed by ∅ ⊂ b and, for any ∅ 6 = b ′ ⊆ b , the component t ∗ → ( t ∗ ) ⊗| b ′ | is the standard generator ξ | b ′ | .Proof. By Lemma 5.7, for (
Z, z ) a finite pointed set, ⊥ Γ t ∗ ( Z ) has generators [ y ] − [ z ], for y ∈ Z \{ z } , corresponding to t ∗ ( Z ) in the decomposition, and [ z ] − [+], corresponding to k in thedecomposition. We have ⊥ Γ ξ b ([ z ] − [+]) = ([ z ] − [+]) ⊗ b which treats the factor k . For [ y ] − [ z ] we have ⊥ Γ ξ b ([ y ] − [ z ]) = ⊥ Γ ξ b (([ y ] − [+]) − ([ z ] − [+])) = ⊥ Γ ξ b ([ y ] − [+]) − ⊥ Γ ξ b ([ z ] − [+])= ([ y ] − [+]) ⊗ b − ([ z ] − [+]) ⊗ b = (([ y ] − [ z ]) + ([ z ] − [+])) ⊗ b − ([ z ] − [+]) ⊗ b . Developing the tensor product we obtain that the component t ∗ → ( t ∗ ) ⊗| b ′ | indexed by b ′ = ∅ is the map ([ y ] − [ z ]) ([ y ] − [ z ]) ⊗| b ′ | , i.e., the standard generator ξ | b ′ | of Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗| b ′ | ), as required. (cid:3) Remark . (1) Above, we gave an explicit description of ⊥ Γ ( t ∗ ) ⊗ b and ⊥ Γ ⊥ Γ ( t ∗ ) ⊗ b for b ∈ N . The reader’sattention is drawn to the fact that, in general, we have no such description of ⊥ Γ F for F ∈ Ob F (Γ; k ). This should be compared with Proposition 3.31, where we give anexplicit description of ⊥ Σ G and ⊥ Σ ⊥ Σ G for all G ∈ Ob F ( Σ op ; k ).(2) The explicit description of the structure morphisms in simplicial degree 0 and 1 forthe simplicial object ( ⊥ Γ ) • +1 ( t ∗ ) ⊗ b are used to establish the isomorphism of augmentedsimplicial objects, ( ⊥ Γ ) • +1 t ∗⊗ ∼ = (cid:0) ( ⊥ Σ ) • +1 t ∗⊗ (cid:1) op in Proposition 5.20.5.3. Comparing comonads.
In this section we work with functors in the category F (Γ × Σ op ; k ); this is equivalent to the category of functors from Γ to F ( Σ op ; k ) and also to thecategory of functors from Σ op to F (Γ; k ). Fixing one of the variables, we can consider thefollowing comonads on F (Γ × Σ op ; k ): • ⊥ Γ : F (Γ × Σ op ; k ) → F (Γ × Σ op ; k ) obtained from the comonad introduced in Notation3.21. • ⊥ Σ : F (Γ × Σ op ; k ) → F (Γ × Σ op ; k ) obtained from the comonad introduced in Notation3.30.Here we establish two results concerning these comonads: in Proposition 5.12 we prove thatthese two comonads commute and in Proposition 5.14 we construct a natural isomorphism ⊥ Γ t ∗⊗ ∼ = ⊥ Σ t ∗⊗ , compatible with the counits, where t ∗⊗ is the bifunctor introduced in Defi-nition 5.13 that encodes the functors ( t ∗ ) ⊗ b for b ∈ N . Proposition 5.12.
The comonads ⊥ Γ , ⊥ Σ : F (Γ × Σ op ; k ) → F (Γ × Σ op ; k ) commute up tonatural isomorphism. Explicitly, for F ∈ Ob F (Γ × Σ op ; k ) , there is a natural isomorphism ⊥ Γ ⊥ Σ F ∼ = → ⊥ Σ ⊥ Γ F. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 31
This is compatible with the respective counits, in that the following diagrams commute: ⊥ Γ ⊥ Σ F ∼ = / / ε Γ ⊥ Σ F $ $ ■■■■■■■■■ ⊥ Σ ⊥ Γ F ⊥ Σ ε Γ F z z ✉✉✉✉✉✉✉✉✉ ⊥ Γ ⊥ Σ F ∼ = / / ⊥ Γ ε Σ F $ $ ■■■■■■■■■ ⊥ Σ ⊥ Γ F ε Σ ⊥ Γ F z z ✉✉✉✉✉✉✉✉✉ ⊥ Σ F ⊥ Γ F. Proof.
By Proposition 3.31 part (1), for X ∈ Ob Γ and b ∈ Ob Σ , ⊥ Σ F ( X, b ) = M b ′ ⊆ b F ( X, b ′ ) . Hence, by the Definition of ⊥ Γ (see Notation 3.21), ⊥ Γ ⊥ Σ F ( X, b ) = ⊥ Σ F ( X + , b ) = M b ′ ⊆ b F ( X + , b ′ )and the right hand side is canonically isomorphic to L b ′ ⊆ b ⊥ Γ F ( X, b ′ ) which, in turn, is canon-ically isomorphic to ⊥ Σ ⊥ Γ F ( X, b ). These isomorphisms are natural with respect to X and b .The compatibility with the counits is checked by using their explicit identifications, given inProposition 3.23 and Proposition 3.31 respectively. (cid:3) We will apply the comonads ⊥ Γ , ⊥ Σ to the following functor. Definition 5.13.
Let t ∗⊗ be the functor in F (Γ × Σ op ; k ) (considered here as functors from Σ op to F (Γ; k )) given by t ∗⊗ ( b ) := ( t ∗ ) ⊗ b , where b ∈ Ob Σ op and the symmetric group S b acts on the right via place permutations.For b ∈ Ob Σ op , by Proposition 3.31 and Proposition 5.9, there are natural isomorphisms: ⊥ Σ t ∗⊗ ( b ) ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ ; ⊥ Γ t ∗⊗ ( b ) ∼ = ⊥ Γ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ . This leads to the following:
Proposition 5.14.
There is an isomorphism α : ⊥ Γ t ∗⊗ → ⊥ Σ t ∗⊗ in F (Γ × Σ op ; k ) defined, for b ∈ N , by the isomorphisms in F (Γ; k ) α b : ⊥ Γ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ → ⊥ Σ t ∗⊗ ( b ) ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ given by the identity on each factor ( t ∗ ) ⊗ b ′ . Moreover, this isomorphism is compatible with therespective counits, in that the following diagram commutes: ⊥ Γ t ∗⊗ ∼ = α / / ε Γ t ∗⊗ ●●●●●●●●● ⊥ Σ t ∗⊗ ε Σ t ∗⊗ { { ✇✇✇✇✇✇✇✇✇ t ∗⊗ . Proof.
For the first part, one checks that the isomorphisms α b are compatible with the actionof the symmetric group S b . Here, the action of S b on ⊥ Σ t ∗⊗ ( b ) is given by Proposition 3.28 (2)and on ⊥ Γ ( t ∗ ) ⊗ b ∼ = ( ⊥ Γ t ∗ ) ⊗ b is given by place permutations.The compatibility with the counits follows from the explicit description of the counits givenin Proposition 3.31 (2) and Proposition 5.9 (3). (cid:3) An isomorphism of augmented simplicial objects.
The purpose of this section is toconstruct an isomorphism of augmented simplicial objects ⊥ Γ • +1 t ∗⊗ ∼ = (cid:0) ⊥ Σ • +1 t ∗⊗ (cid:1) op extendingthe natural isomorphism ⊥ Γ t ∗⊗ ∼ = ⊥ Σ t ∗⊗ given in Proposition 5.14. Here op denotes the oppositesimplicial structure (see Notation 1.3). Definition 5.15.
For ℓ ∈ N , let α ℓ : ( ⊥ Γ ) ℓ t ∗⊗ ∼ = → ( ⊥ Σ ) ℓ t ∗⊗ be the isomorphism in F (Γ × Σ op ; k )defined recursively by:(1) α = Id t ∗⊗ ;(2) for ℓ > α ℓ is the composite:( ⊥ Γ ) ℓ t ∗⊗ = ( ⊥ Γ ) ℓ − ⊥ Γ t ∗⊗ ( ⊥ Γ ) ℓ − α −→ ( ⊥ Γ ) ℓ − ⊥ Σ t ∗⊗ ∼ = ⊥ Σ ( ⊥ Γ ) ℓ − t ∗⊗ ⊥ Σ α ℓ − −→ ⊥ Σ ( ⊥ Σ ) ℓ − t ∗⊗ = ( ⊥ Σ ) ℓ t ∗⊗ , in which the middle isomorphism is given by Proposition 5.12.In particular, α = α .In order to give an explicit description of α ℓ in Proposition 5.17, we need the following rein-dexation of iterates of the comonads ⊥ Σ and ⊥ Γ . Remark . Recall that by Remark 3.32 (3.4) we have(5.2) ⊥ Σ ⊥ Σ . . . ⊥ Σ ℓ G ( b ) ∼ = M b =( ... (( b ( ℓ +1)Σ ∐ b Σ ℓ ) ∐ b Σ ℓ − )) ... ∐ b Σ1 ) ∐ b Σ0 G ( b ( ℓ +1)Σ ) . On the other hand, the decomposition given in Proposition 5.9 can be reindexed as follows: ⊥ Γ ⊥ Γ t ∗⊗ ( b ) ∼ = ⊥ Γ ( M b = b (1)Γ ∐ b Γ1 ( t ∗ ) ⊗ b (1)Γ ) ∼ = M b =( b (2)Γ ∐ b Γ0 ) ∐ b Γ1 t ∗⊗ ( b (2)Γ ) . With respect to the decomposition given in Proposition 5.9 we have b (2)Γ := b ′′ , b Γ0 := b ′ \ b ′′ and b Γ1 := b \ b ′ . More generally, we have:(5.3) ⊥ Γ ⊥ Γ . . . ⊥ Γ ℓ t ∗⊗ ( b ) ∼ = M b =( ... (( b ( ℓ +1)Γ ∐ b Γ0 ) ∐ b Γ1 )) ... ∐ b Γ ℓ − ) ∐ b Γ ℓ t ∗⊗ ( b ( ℓ +1)Γ ) . Note the reversal of the order of the indices in the decomposition of b between (5.2) and (5.3).This explains why it is the opposite structure for the augmented simplicial object ⊥ Σ • +1 t ∗⊗ thatarises in Proposition 5.20 and justifies why we consider ⊥ Σ ℓ ⊥ Σ ℓ − . . . ⊥ Σ t ∗⊗ in the followingProposition. Proposition 5.17.
For ℓ ∈ N , the isomorphism α ℓ : ( ⊥ Γ ) ℓ t ∗⊗ ∼ = → ( ⊥ Σ ) ℓ t ∗⊗ in F (Γ × Σ op ; k ) isdefined, for b ∈ N , by the isomorphisms in F (Γ; k ) ⊥ Γ . . . ⊥ Γ ℓ − t ∗⊗ ( b ) α ℓ b / / ⊥ Σ ℓ − . . . ⊥ Σ t ∗⊗ ( b ) ∼ = (cid:15) (cid:15) L b =(( b ( ℓ )Γ ∐ b Γ0 ) ∐ ... ∐ b Γ ℓ − ) t ∗⊗ ( b ( ℓ )Γ ) ∼ = O O L b =(( b ( ℓ )Σ ∐ b Σ0 ) ... ∐ b Σ ℓ − ) t ∗⊗ ( b ( ℓ )Σ ) sending the component indexed by b = ( . . . (( b ( ℓ )Γ ∐ b Γ0 ) ∐ b Γ1 )) . . . ∐ b Γ ℓ − ) ∐ b Γ ℓ − to the componentwith b ( ℓ )Σ = b ( ℓ )Γ and b Σ i = b Γ i for i ∈ { , . . . , ℓ − } , where the vertical isomorphisms are givenby (5.3). PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 33
To prove this Proposition we need to make explicit the natural isomorphism given in Propo-sition 5.12 for F = t ∗⊗ . First note that there are isomorphisms ⊥ Γ ( ⊥ Σ t ∗⊗ )( b ) ∼ = ⊥ Γ ( M b = b (1)Σ ∐ b Σ1 ( t ∗ ) ⊗ b (1)Σ ) ∼ = M b =( b (2)Γ ∐ b Γ0 ) ∐ b Σ1 ( t ∗ ) ⊗ b (2)Γ and ⊥ Σ ( ⊥ Γ t ∗⊗ )( b ) ∼ = M b = b (1)Σ ∐ b Σ1 ( ⊥ Γ t ∗⊗ )( b (1)Σ ) ∼ = M b =( b (2)Γ ∐ b Γ0 ) ∐ b Σ1 ( t ∗ ) ⊗ b (2)Γ . The proof of the following is left to the reader:
Lemma 5.18.
The natural isomorphism ⊥ Γ ⊥ Σ t ∗⊗ ∼ = → ⊥ Σ ⊥ Γ t ∗⊗ in F (Γ × Σ op ; k ) of Proposi-tion 5.12 is given, for b ∈ N , by the isomorphisms in F (Γ; k ) : ⊥ Γ ( ⊥ Σ t ∗⊗ )( b ) ∼ = M b =( b (2)Γ ∐ b Γ0 ) ∐ b Σ1 ( t ∗ ) ⊗ b (2)Γ → ⊥ Σ ( ⊥ Γ t ∗⊗ )( b ) ∼ = M b =( b (2)Γ ∐ b Γ0 ) ∐ b Σ1 ( t ∗ ) ⊗ b (2)Γ determined by the identity on each component indexed by b = ( b (2)Γ ∐ b Γ0 ) ∐ b Σ1 .Proof of Proposition 5.17. We prove the result by induction on ℓ . For ℓ = 1, the result is trueby Proposition 5.14. To prove the inductive step, consider the component of ⊥ Γ . . . ⊥ Γ ℓ − t ∗⊗ ( b )indexed by ( . . . ( b ( ℓ )Γ ∐ b Γ0 ) ∐ . . . ∐ b Γ ℓ − ) ∐ b Γ ℓ − . By Proposition 5.14, ( ⊥ Γ ) ℓ − α sends thiscomponent to the component indexed by( . . . ( b ( ℓ )Γ ∐ b Γ0 ) ∐ . . . ∐ b Γ ℓ − ) ∐ b Σ ℓ − . The result follows from Lemma 5.18 and the inductive hypothesis. (cid:3)
Proposition 5.17 and Lemma 5.18 yield the following useful result:
Lemma 5.19.
For ℓ, m ∈ N , there is a commutative diagram of isomorphisms: ( ⊥ Γ ) ℓ + m t ∗⊗ α ℓ + m (cid:15) (cid:15) ( ⊥ Γ ) ℓ ( ⊥ Γ ) m t ∗⊗ ( ⊥ Γ ) ℓ α m / / ( ⊥ Γ ) ℓ ( ⊥ Σ ) m t ∗⊗∼ = (cid:15) (cid:15) ( ⊥ Σ ) ℓ + m t ∗⊗ ( ⊥ Σ ) m ( ⊥ Σ ) ℓ t ∗⊗ ( ⊥ Σ ) m ( ⊥ Γ ) ℓ t ∗⊗ , ( ⊥ Σ ) m α ℓ o o in which the right hand vertical morphism is induced by the interchange isomorphism of Propo-sition 5.12. By Proposition B.3, we have the augmented simplicial objects ( ⊥ Γ ) • +1 t ∗⊗ and ( ⊥ Σ ) • +1 t ∗⊗ in F (Γ × Σ op ; k ) associated to the comonads ⊥ Γ and ⊥ Σ respectively. In the following, op denotesthe opposite augmented simplicial structure, as in Notation 1.3. Proposition 5.20.
The isomorphisms α ℓ +1 : ( ⊥ Γ ) ℓ +1 t ∗⊗ ∼ = → ( ⊥ Σ ) ℓ +1 t ∗⊗ for ℓ ≥ − give anisomorphism of augmented simplicial objects in F (Γ × Σ op ; k )( ⊥ Γ ) • +1 t ∗⊗ ∼ = (cid:0) ( ⊥ Σ ) • +1 t ∗⊗ (cid:1) op extending the identity on t ∗⊗ in degree − .Proof. By Proposition 5.14 and the construction of the morphisms α ℓ +1 , these give an isomor-phism of the underlying graded objects. It remains to check compatibility with the respectiveaugmented simplicial structures. We begin by proving the compatibility with the face maps by induction on the simplicialdegree.The compatibility in simplicial degrees − ⊥ Γ ⊥ Γ t ∗⊗ ∼ = α / / δ Γ0 = ε ⊥ Γ t ∗⊗ (cid:15) (cid:15) ⊥ Σ ⊥ Σ t ∗⊗ e δ = δ Σ1 = ⊥ Σ ε t ∗⊗ (cid:15) (cid:15) ⊥ Γ ⊥ Γ t ∗⊗ ∼ = α / / δ Γ1 = ⊥ Γ ε ∗⊗ t (cid:15) (cid:15) ⊥ Σ ⊥ Σ t ∗⊗ e δ = δ Σ0 = ε ⊥ Σ t ∗⊗ (cid:15) (cid:15) ⊥ Γ t ∗⊗ ∼ = α / / ⊥ Σ t ∗⊗ ⊥ Γ t ∗⊗ ∼ = α / / ⊥ Σ t ∗⊗ , where Notation 1.3 is used for the opposite simplicial structure maps.For the left hand square, using the definition of α , this follows from the commutativity ofthe following: ⊥ Γ ⊥ Γ t ∗⊗ ε ⊥ Γ t ∗⊗ (cid:15) (cid:15) ⊥ Γ α / / ⊥ Γ ⊥ Σ t ∗⊗ ε ⊥ Σ t ∗⊗ (cid:15) (cid:15) ∼ = / / ⊥ Σ ⊥ Γ t ∗⊗⊥ Σ ε t ∗⊗ (cid:15) (cid:15) ⊥ Σ α / / ⊥ Σ ⊥ Σ t ∗⊗⊥ Σ ε t ∗⊗ x x qqqqqqqqqq ⊥ Γ t ∗⊗ α / / ⊥ Σ t ∗⊗ ⊥ Σ t ∗⊗ . Here the left hand square commutes by naturality of ε ; the middle square is commutative byProposition 5.12; the right hand triangle commutes by applying ⊥ Σ to the commutative triangleof Proposition 5.14.For the second square, one proceeds similarly, by establishing the commutativity of the dia-gram:(5.4) ⊥ Γ ⊥ Γ t ∗⊗⊥ Γ ε t ∗⊗ & & ▼▼▼▼▼▼▼▼▼▼ ⊥ Γ α / / ⊥ Γ ⊥ Σ t ∗⊗⊥ Γ ε t ∗⊗ (cid:15) (cid:15) ∼ = / / ⊥ Σ ⊥ Γ t ∗⊗ ε ⊥ Γ t ∗⊗ (cid:15) (cid:15) ⊥ Σ α / / ⊥ Σ ⊥ Σ t ∗⊗ ε ⊥ Σ t ∗⊗ (cid:15) (cid:15) ⊥ Γ t ∗⊗ ⊥ Γ t ∗⊗ α / / ⊥ Σ t ∗⊗ . Assume that the compatibility in simplicial degrees i − i for i ≤ n − i ∈ { , . . . , n } :( ⊥ Γ ) n +1 t ∗⊗ ∼ = α n +1 / / δ Γ i =( ⊥ Γ ) i ε ( ⊥ Γ) n − it ∗⊗ (cid:15) (cid:15) ( ⊥ Σ ) n +1 t ∗⊗ e δ i Σ = δ Σ n − i =( ⊥ Σ ) n − i ε ( ⊥ Σ ) it ∗⊗ (cid:15) (cid:15) ( ⊥ Γ ) n t ∗⊗ ∼ = α n / / ( ⊥ Σ ) n t ∗⊗ . For i = n , the proof of the commutativity of the diagram is similar to that for the diagram (5.4).For i = n , using Lemma 5.19, this follows from the commutative diagram ( ⊥ Γ ) i ⊥ Γ ( ⊥ Γ ) n − i t ∗⊗ ( ⊥ Γ ) i ε Γ( ⊥ Γ) n − it ∗⊗ (cid:15) (cid:15) ( ⊥ Γ ) i ⊥ Γ α n − i / / ( ⊥ Γ ) i ⊥ Γ ( ⊥ Σ ) n − i t ∗⊗ ( ⊥ Γ ) i ε Γ( ⊥ Σ ) n − it ∗⊗ (cid:15) (cid:15) ∼ = / / ( ⊥ Γ ) i ( ⊥ Σ ) n − i ⊥ Γ t ∗⊗ ( ⊥ Γ ) i ( ⊥ Σ ) n − i ε t ∗⊗ (cid:15) (cid:15) ∼ = / / ( ⊥ Σ ) n − i ( ⊥ Γ ) i ⊥ Γ t ∗⊗ ( ⊥ Σ ) n − i ( ⊥ Γ ) i ε t ∗⊗ (cid:15) (cid:15) ( ⊥ Σ ) n − i α i +1 / / ( ⊥ Σ ) n − i ( ⊥ Σ ) i +1 t ∗⊗ ( ⊥ Σ ) n − i ε ( ⊥ Σ ) it ∗⊗ (cid:15) (cid:15) ( ⊥ Γ ) i ( ⊥ Γ ) n − i t ∗⊗ ( ⊥ Γ ) i α n − i / / ( ⊥ Γ ) i ( ⊥ Σ ) n − i t ∗⊗ ( ⊥ Γ ) i ( ⊥ Σ ) n − i t ∗⊗ ∼ = / / ( ⊥ Σ ) n − i ( ⊥ Γ ) i t ∗⊗ ( ⊥ Σ ) n − i α i / / ( ⊥ Σ ) n − i ( ⊥ Σ ) i t ∗⊗ , where the left hand square commutes by naturality of ε , the second square commutes by iteratingProposition 5.12, the third square commutes by naturality of the isomorphism ⊥ Γ ⊥ Σ F ∼ = → ⊥ Σ ⊥ Γ F and the last square is seen to commute by applying ( ⊥ Σ ) n − i to the commutative diagram obtainedby the inductive hypothesis in simplicial degrees i − i . PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 35
We now consider the degeneracies; these are induced by the respective coproducts ∆. For thecompatibility in simplicial degrees 0 and 1, we require to show that the diagram(5.5) ⊥ Γ t ∗⊗ ∆ (cid:15) (cid:15) ∼ = α / / ⊥ Σ t ∗⊗ ∆ (cid:15) (cid:15) ⊥ Γ ⊥ Γ t ∗⊗ ∼ = α / / ⊥ Σ ⊥ Σ t ∗⊗ commutes.Using the reindexation given in Remark 5.16, by Proposition 5.14, for b ∈ N , the map α b : ⊥ Γ t ∗⊗ ( b ) ∼ = M b = b (1)Γ ∐ b Γ0 t ∗⊗ ( b (1)Γ ) → ⊥ Σ t ∗⊗ ( b ) ∼ = M b = b (1)Σ ∐ b Σ0 t ∗⊗ ( b (1)Σ )sends the component indexed by b = b (1)Γ ∐ b Γ0 to the component with b (1)Σ = b (1)Γ and b Σ0 = b Γ0 .Proposition 3.31 identifies ∆ b : ⊥ Σ t ∗⊗ ( b ) → ⊥ Σ ⊥ Σ t ∗⊗ ( b ) as the morphism M b = b (1)Σ ∐ b Σ0 t ∗⊗ ( b (1)Σ ) → M b = b (2)Σ ∐ b Σ0 ∐ b Σ1 t ∗⊗ ( b (2)Σ ) , where the component indexed by the pair of decompositions ( b = b (1)Σ ∐ b Σ1 , b = b (2)Σ ∐ b Σ0 ∐ b Σ1 )is zero unless b (1)Σ = b (2)Σ , when it is the identity morphism t ∗⊗ ( b (1)Σ ) → t ∗⊗ ( b (2)Σ ).By Proposition 5.9 (4), the diagonal ∆ b : ⊥ Γ t ∗⊗ ( b ) → ⊥ Γ ⊥ Γ t ∗⊗ ( b ) identifies as the mor-phism M b = b (1)Γ ∐ b Γ0 t ∗⊗ ( b (1)Γ ) → M b = b (2)Γ ∐ b Γ0 ∐ b Γ1 t ∗⊗ ( b (2)Γ ) , where the component indexed by the pair of decompositions ( b = b (1)Γ ∐ b Γ0 , b = b (2)Γ ∐ b Γ0 ∐ b Γ1 )is zero unless b (1)Γ = b (2)Γ , when it is the identity morphism t ∗⊗ ( b (1)Γ ) → t ∗⊗ ( b (2)Γ ).Proposition 5.17 identifies α b : ⊥ Γ ⊥ Γ t ∗⊗ ( b ) → ⊥ Σ ⊥ Σ t ∗⊗ ( b ) as the morphism M b =( b (2)Γ ∐ b Γ0 ) ∐ b Γ1 t ∗⊗ ( b (2)Γ ) → M b =(( b (2)Σ ∐ b Σ0 ) ∐ b Σ1 t ∗⊗ ( b (2)Σ )sending the component indexed by b = ( b (2)Γ ∐ b Γ0 ) ∐ b Γ1 to the component with b (2)Σ = b (2)Γ and b Σ i = b Γ i for i ∈ { , } . The commutativity of the diagram (5.5) follows.The compatibility for degeneracy maps in higher simplicial degree is proved by induction,using the naturality of ∆, Proposition 5.12 and a similar commutative diagram as for the facemaps. (cid:3) Proof of Theorem 5.1.
We begin by constructing the cosimplicial object [ C • k Hom Ω ( − , a )] op ,appearing in the statement of Theorem 5.1, as a particular case of (4.1). For this we introducethe FI op -module k Hom Ω ( − , a ). Proposition 5.21.
For a ∈ N , the association b k Hom Ω ( b , a ) defines a functor in F ( FI op ; k ) ,where a generator f ∈ Hom Ω ( b , a ) is sent by a morphism i : b ′ ֒ → b of FI to the generator f ◦ i given by restriction along i , if this is a morphism of Ω , and to zero otherwise.Proof. The case a = 0 follows immediately from the fact that Hom Ω ( b , ) is empty unless b = .For a > f ◦ i belongs to Ω if and only if it is surjective. Since non-surjectivity is preservedunder restriction, it is clear that the above defines a functor on FI op . (cid:3) Remark . We have k Hom Ω ( − , ) = where is the FI op -module introduced in Notation4.14.The association ( a , b ) Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ), with the place permutation action on( t ∗ ) ⊗ a and ( t ∗ ) ⊗ b , defines a Σ × Σ op -module. Proposition 3.11 can be rephrased as follows: Proposition 5.23.
The maps DK induce an isomorphism of Σ × Σ op -modules: k DK : k Hom Ω ( − , − ) ∼ = −→ Hom F (Γ; k ) ( t ∗⊗ , t ∗⊗ ) . By Proposition B.6, the augmented simplicial object underlying C • k Hom Ω ( − , a ) identifies as( ⊥ Σ ) • +1 k Hom Ω ( − , a ). In the following Proposition, we deduce from Proposition 5.20 that theunderlying augmented simplicial objects of the cosimplicial objects appearing in Theorem 5.1are isomorphic. Proposition 5.24.
For a ∈ N , there is an isomorphism of augmented simplicial objects in F ( Σ op ; k ) : Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( ⊥ Γ ) • +1 t ∗⊗ ) ∼ = (cid:0) ( ⊥ Σ ) • +1 k Hom Ω ( − , a ) (cid:1) op . Proof.
By Proposition 5.20, there is an isomorphism of augmented simplicial objects in F ( Σ op ; k ):Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( ⊥ Γ ) • +1 t ∗⊗ ) ∼ = Hom F (Γ; k ) (( t ∗ ) ⊗ a , (cid:0) ( ⊥ Σ ) • +1 t ∗⊗ (cid:1) op ) . Now, for any G ∈ F (Γ × Σ op ; k ), since Hom F (Γ; k ) (( t ∗ ) ⊗ a , − ) is an additive functor, there is anatural isomorphism Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ G ) ∼ = ⊥ Σ Hom F (Γ; k ) (( t ∗ ) ⊗ a , G ) , since the Γ-module and Σ op -structures commute. This extends to give an isomorphism of aug-mented simplicial objects:Hom F (Γ; k ) (( t ∗ ) ⊗ a , (cid:0) ( ⊥ Σ ) • +1 t ∗⊗ (cid:1) op ) ∼ = (cid:0) ( ⊥ Σ ) • +1 Hom F (Γ; k ) (( t ∗ ) ⊗ a , t ∗⊗ ) (cid:1) op . The isomorphism of the statement follows from Proposition 5.23. (cid:3)
By Proposition 5.3, the augmented simplicial structure on Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • t ∗⊗ ) de-scribed in Proposition 5.24 extends to a cosimplicial structure, where the additional structurearises from the two morphisms d , d of F ( Σ op ; k ) given in (5.1). Composing with the isomor-phisms α ℓ +1 : ( ⊥ Γ ) ℓ +1 t ∗⊗ ∼ = → ( ⊥ Σ ) ℓ +1 t ∗⊗ , gives a cosimplicial structure on Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ • t ∗⊗ ).For example, in cosimplicial degrees 0 and 1 this cosimplicial structure gives the two morphismsof F ( Σ op ; k ): Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) d / / d / / Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ ( t ∗⊗ )( b ))where, for f ∈ Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ), d f := α ◦ ( ⊥ Γ f ) ◦ ψ ( t ∗ ) ⊗ a and d f := α ◦ ψ ( t ∗ ) ⊗ b ◦ f .We begin by proving that the cosimplicial objects Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ • t ∗⊗ ) and [ C • k Hom Ω ( − , a )] op are isomorphic in cosimplicial degrees 0 and 1 using the maps DK a , b introduced in Proposition3.11. Proposition 5.25.
For a ∈ N , we have the following commutative diagrams in F ( Σ op ; k ) : Hom F (Γ; k ) (( t ∗ ) ⊗ a , t ∗⊗ ) d / / Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ t ∗⊗ ) ∼ = ⊥ Σ Hom F (Γ; k ) (( t ∗ ) ⊗ a , t ∗⊗ ) k Hom Ω ( − , a ) k DK a , − ∼ = O O d = η k Hom Ω ( − , a ) / / ⊥ Σ k Hom Ω ( − , a ); ⊥ Σ k DK a , − ∼ = O O PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 37
Hom F (Γ; k ) (( t ∗ ) ⊗ a , t ∗⊗ ) d / / Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Σ t ∗⊗ ) ∼ = ⊥ Σ Hom F (Γ; k ) (( t ∗ ) ⊗ a , t ∗⊗ ) k Hom Ω ( − , a ) k DK a , − ∼ = O O d = ψ k Hom Ω ( − , a ) / / ⊥ Σ k Hom Ω ( − , a ) . ⊥ Σ k DK a , − ∼ = O O Proof.
It suffices to prove that the diagrams are commutative for b ∈ Ob Σ op .The case a = 0 is exceptional and can be treated directly; the details are left to the reader.Let us treat the case a = 1. In this case, for b = there is nothing to prove, so we mayassume that b >
0. By Definition 3.10, k DK , b sends the unique surjection b ։ to thestandard generator ξ b : t ∗ → ( t ∗ ) ⊗ b . Using Proposition 5.9 (2) and Proposition 5.14 d ( ξ b ) = α b ◦ ψ ( t ∗ ) ⊗ b ◦ ξ b : t ∗ → ⊥ Σ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ is equal to t ∗ ξ b −→ ( t ∗ ) ⊗ b i b −→ M b ′ ⊆ b ( t ∗ ) ⊗ b ′ where i b is the inclusion to the factor indexed by b ′ = b . By Proposition 3.35 and the functorialityof ⊥ Σ , this is equal to the other composition in the first commutative diagram.For the second commutative diagram, using Lemma 5.7 and Proposition 5.10 the map d ( ξ b ) = α ◦ ( ⊥ Γ ξ b ) ◦ ψ t ∗ : t ∗ → ⊥ Σ ( t ∗ ) ⊗ b ∼ = M b ′ ⊆ b ( t ∗ ) ⊗ b ′ has component t ∗ → ( t ∗ ) ⊗| b ′ | , the standard generator ξ | b ′ | for any ∅ 6 = b ′ ⊆ b . By Proposition3.33 and the functoriality of ⊥ Σ this is equal to the other composition in the second commutativediagram.The case a > a = 1 using the S op b -equivariant isomorphismgiven in Lemma 3.9 as follows.As in the Lemma, for a, b ∈ N ∗ there is a S op b -equivariant isomorphism M ( b i ) ∈ ( N ∗ ) a P ai =1 b i = b (cid:0) a O i =1 Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b i ) (cid:1) ↑ S b Q ai =1 S bi ∼ = → Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b ) . This is compatible with the isomorphism of Lemma 3.9 via the isomorphism of Proposition 3.11.Hence, for a >
1, it suffices to check the commutativity of the diagrams restricted to thefollowing submodules that are compatible via k DK a , b : a O i =1 k Hom Ω ( b i , ) ⊂ k Hom Ω ( b , a ) a O i =1 Hom F (Γ; k ) ( t ∗ , ( t ∗ ) ⊗ b i ) ⊂ Hom F (Γ; k ) (( t ∗ ) ⊗ a , ( t ∗ ) ⊗ b )(5.6)for each sequence ( b i ). Here, the inclusion in (5.6) is given by the tensor product of morphisms.The upper horizontal morphisms d , d in the statement of the Proposition are defined using ⊥ Γ , which is symmetric monoidal, by Corollary 3.22. Using this, their restriction to the submodulein (5.6) can be calculated directly from the case a = 1. The result follows by direct verification. (cid:3) Proof of Theorem 5.1.
We require to establish the isomorphism of cosimplicial Σ op -modules:Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • t ∗⊗ ) ∼ = C • k Hom Ω ( − , a ) op . Proposition 5.24 has already established that the underlying augmented simplicial objects areisomorphic. It remains to check that the first and last coface operators correspond.In cosimplicial degree 0 and 1 the isomorphism is given by Proposition 5.25. The highercosimplicial degrees are deduced from this case, proceeding as in the passage to higher simplicialdegrees in the proof of Proposition 5.20. (cid:3)
Remark . The result of Theorem 5.1 is stronger than stated, since it is also equivariant withrespect to automorphisms of a . Namely, there is an isomorphism of cosimplicial Σ × Σ op -modules:Hom F (Γ; k ) ( t ∗⊗ , ⊥ Γ • t ∗⊗ ) ∼ = C • k Hom Ω ( − , − ) op . Here, it is important to recall that t ∗⊗ is considered as an object of F (Γ × Σ op ; k ); this allowsthe two variances to be distinguished correctly.6. The cohomology of the Koszul complex of k Hom Ω ( − , a )The purpose of this section is to calculate, for a ∈ N , the cohomology of the Koszul complexof the FI op -module k Hom Ω ( − , a ) introduced in Proposition 5.21. Evaluated upon b , this cal-culation is achieved in Theorem 6.15, with information on the action of the symmetric groupsgiven in Theorem 6.20.By Corollary 5.2 and Theorem 5.4, the cohomology of this Koszul complex gives access tothe computation of Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) and Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) (see Theorem7.1) which are the main objects of interest of this paper.For typographical simplicity we introduce the following notation: Notation . For a ∈ N , let C ( − , a ) denote the complex Kz k Hom Ω ( − , a ) of Σ op -modules, sothat C ( b , a ) = Kz k Hom Ω ( − , a )( b ). Explicitly, for t ∈ N : C ( b , a ) t = M b ( t ) ⊂ b | b ( t ) | = b − t k Hom Ω ( b ( t ) , a ) ⊗ Or ( b \ b ( t ) ) . where Or is given in Notation 4.3 and the differential d : C ( b , a ) t → C ( b , a ) t +1 is the morphismof k -modules given on f ⊗ α , where f ∈ Hom Ω ( b ( t ) , a ) for b ( t ) ⊂ b such that | b ( t ) | = b − t and α ∈ Or ( b \ b ( t ) ), by d (cid:0) [ f ] ⊗ α (cid:1) = X y ∈ b ( t ) f ◦ ι y ∈ Hom Ω ( b ( t ) \{ y } , a ) [ f ◦ ι y ] ⊗ ( y ∧ α ) , where ι y : b ( t ) \{ y } ֒ → b ( t ) is the inclusion. Remark . By Proposition 4.8, the construction Kz is functorial. This implies that C ( − , − )is a complex of Σ × Σ op -modules, since the symmetric group S a acts on k Hom Ω ( − , a ) byautomorphisms of FI op -modules.First we record the following exceptional cases: Lemma 6.3.
For a ∈ N , (1) C ( b , a ) = 0 for b < a ; (2) C ( b , ) ∼ = Or ( b ) , concentrated in cohomological degree b , which thus coincides with thecohomology; PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 39 (3) C ( a , a ) ∼ = k [ S a ] , concentrated in cohomological degree zero, which thus coincides withthe cohomology. This allows us to concentrate on the case b > a >
The case a = 1 . This forms one of the initial steps of a double induction in the proof ofTheorem 6.15.
Proposition 6.4.
For a = 1 and b ∈ N ∗ , H ∗ ( C ( b , )) ∼ = (cid:26) Or ( b ) ∗ = b − otherwise.Proof. In the case b = 1, the result follows from Lemma 6.3, hence suppose that b > FI op -modules0 → → k → k Hom Ω ( − , ) → , where k denotes the constant FI op -module and is as in Notation 4.14. By Proposition 4.8 thisgives rise to a short exact sequence of cochain complexes in F ( Σ op ; k )0 → Kz → Kz k → Kz k Hom Ω ( − , ) → Kz = C ( − , ), by Lemma 6.3, for b ∈ N ∗ , ( Kz )( b ) has cohomology concentrated incohomological degree b , where it is isomorphic to Or ( b ).We prove that Kz k has zero cohomology. This is proved as for classical Koszul complexes byexhibiting a chain null-homotopy as follows. Firstly, ( Kz k ) t ( b ) identifies as M b ′′ ⊂ b | b ′′ | = t Or ( b ′′ ) , noting that this includes the module k in degree t = b .Define a chain null-homotopy h : ( Kz k ) t ( b ) → ( Kz k ) t − ( b ) using the generators introducedin Notation 4.3 by ω ( b ′′ ) (cid:26) b ′′ ω ( b ′′ \{ } ) 1 ∈ b ′′ , where b ′′ ⊂ b such that | b ′′ | = t .Note that, for b ′′ ⊂ b such that 1 ∈ b ′′ and x ∈ b \ b ′′ , we have h (1 ∧ x ∧ ω ( b ′′ \ { } )) = x ∧ ω ( b ′′ \ { } ) . The following calculations show that dh + hd = id, as required:(1) If 1 b ′′ , d ◦ h ( ω ( b ′′ )) = d (0) = 0 and h ◦ d ( ω ( b ′′ )) = h ( X x ∈ b \ b ′′ x ∧ ω ( b ′′ )) = X x ∈ b \ b ′′ h ( x ∧ ω ( b ′′ )) = h (1 ∧ ω ( b ′′ )) = h ( ω ( b ′′ ∪{ } )) = ω ( b ′′ ) . (2) If 1 ∈ b ′′ , d ◦ h ( ω ( b ′′ )) = d ( ω ( b ′′ \ { } )) = X x ∈ ( b \ b ′′ ) ∪{ } x ∧ ω ( b ′′ \ { } )= 1 ∧ ω ( b ′′ \ { } ) + X x ∈ b \ b ′′ x ∧ ω ( b ′′ \ { } ) = ω ( b ′′ ) + X x ∈ b \ b ′′ x ∧ ω ( b ′′ \ { } ) and h ◦ d ( ω ( b ′′ )) = h ( X x ∈ b \ b ′′ x ∧ ω ( b ′′ )) = X x ∈ b \ b ′′ h ( x ∧ ω ( b ′′ ))= X x ∈ b \ b ′′ h ( x ∧ ∧ ω ( b ′′ \ { } )) = X x ∈ b \ b ′′ − h (1 ∧ x ∧ ω ( b ′′ \ { } ))= X x ∈ b \ b ′′ − x ∧ ω ( b ′′ \ { } ) . (cid:3) The top cohomology of C ( b , a ) . The other ingredient to the double induction used inthe proof of Theorem 6.15 is the identification of the cohomology in the top degree b − a : Proposition 6.5. If b > a > , then there is a S a × S op b -equivariant surjection of chaincomplexes C ( b , a ) ։ Or ( a ) ⊗ Or ( b ) , (6.1) where Or ( a ) ⊗ Or ( b ) is placed in cohomological degree b − a .This induces an isomorphism of S a × S op b -modules: H b − a ( C ( b , a )) ∼ = Or ( a ) ⊗ Or ( b ) . In particular, for b ′ ⊂ b such that | b ′ | = a − , ( u, v ) ∈ ( b \ b ′ ) , f ∈ Hom Ω ( b ′ ∪ { u } , a ) , f ∈ Hom Ω ( b ′ ∪ { v } , a ) such that f | b ′ = f | b ′ and f ( u ) = f ( v ) then the cohomology classes ofthe generators [ f ] ⊗ ω and [ f ] ⊗ ω (where ω , ω denote the respective canonical orientationclasses) are equal, up to a possible sign ± .Proof. The result is proved by analysing the tail of the complex C ( b , a ): C ( b , a ) b − a − d / / C ( b , a ) b − a L ˜ b ⊂ b | ˜ b | = a +1 k Hom Ω (˜ b , a ) ⊗ Or ( b \ ˜ b ) / / L b ′ ⊂ b | b ′ | = a k Hom Ω ( b ′ , a ) ⊗ Or ( b \ b ′ ) , The surjection (6.1) is defined by the k -module morphism: ψ : k Hom Ω ( b ′ , a ) ⊗ Or ( b \ b ′ ) → Or ( a ) ⊗ Or ( b )that sends a generator [ f ] ⊗ ω ( b \ b ′ ), where f is a bijection f : b ′ ∼ = → a , to ω ( a ) ⊗ (cid:0) f − (1) ∧ . . . ∧ f − ( a ) ∧ ω ( b \ b ′ ) (cid:1) ∈ Or ( a ) ⊗ Or ( b ) . To see that this sends boundaries to zero, consider a generator [ g ] ⊗ ω ( b \ ˜ b ), where g : ˜ b ։ a with | ˜ b | = a + 1. For such a g , there exists a unique i ∈ a such that | g − ( i ) | = 2, with all otherfibres of cardinal one; write g − ( i ) = { u, v } . Then: d ([ g ] ⊗ ω ( b \ ˜ b )) = [ g | ˜ b \{ v } ] ⊗ ( v ∧ ω ( b \ ˜ b )) + [ g | ˜ b \{ u } ] ⊗ ( u ∧ ω ( b \ ˜ b )) . Now the images under ψ of the two terms of the right hand side differ only by the transpositionof the terms u , v in the wedge product in Or ( b ). The sum of these images is therefore zero in Or ( a ) ⊗ Or ( b ). PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 41
In particular, for b ′ ⊂ b such that | b ′ | = a − u, v, f , f as in the statement, taking˜ b = b ′ ∪ { u, v } and g ∈ Hom Ω (˜ b , a ) such that g ( u ) = g ( v ) and g | ˜ b \{ u,v } = f | b ′ = f | b ′ we have d ([ g ] ⊗ ω ( b \ ˜ b )) = [ f ] ⊗ ( v ∧ ω ( b ′ ∪ { u } )) + [ f ] ⊗ ( u ∧ ω ( b ′ ∪ { v } )) = 0 . Thus the cohomology classes of the generators [ f ] ⊗ ω and [ f ] ⊗ ω are equal, up to a possiblesign ± H b − a ( C ( b , a )) is generated by the class of the generator[Id a ] ⊗ ω ( b \ a ), where a is considered as a subset of b by the canonical inclusion. This element issent to the generator ω ( a ) ⊗ ω ( b ) of Or ( a ) ⊗ Or ( b ) by the surjection. It follows that the surjectioninduces an isomorphism in cohomology.Finally, by construction, the surjection of chain complexes is S a × S op b -equivariant, hence theinduced isomorphism in cohomology is one of S a × S op b -modules. (cid:3) The cohomology of C ( b , a ) . We begin by considering the case b = a + . Proposition 6.6.
For a ∈ N , H ∗ ( C ( a + , a )) ∼ = k ⊕ Rk( a +1 ,a ) ∗ = 0 k ∗ = 10 ∗ > , where Rk( a + 1 , a ) = (cid:26) a ∈ { , } ( a − ( a +1)!2 + 1 a > . In particular, the cohomology is k -free, concentrated in degrees and .Proof. The case a = 0 is given by Lemma 6.3.For a >
0, by Proposition 6.5, H ( C ( a + , a )) ∼ = k and the complex C ( a + , a ) is concen-trated in degrees 0 and 1, with terms that are free k modules with:rank k C ( a + , a ) = a ! (cid:18) a + 12 (cid:19) = a a + 1)!rank k C ( a + , a ) = a !( a + 1) = ( a + 1)! , by direct calculation. The result follows by observing that the image of the differential is a free k -module of rank rank k C ( a + , a ) − k -module of rank rank k C ( a + , a ) − rank k C ( a + , a ) + 1. (This can be viewed as anEuler-Poincar´e characteristic argument.) (cid:3) In order to compute the cohomology of C ( b , a ) for b ≥ a + we will consider several subcom-plexes of C ( b , a ). To construct these subcomplexes, we consider particular families of subsets ofHom Ω ( b ′ , a ), for b ′ ⊆ b , which we will refer to as families of subsets of Hom Ω ( − , a ), leaving therestriction to subsets of b implicit. Definition 6.7.
Let X ( b ′ ) ⊆ Hom Ω ( b ′ , a ), for b ′ ⊆ b , be a family of subsets of Hom Ω ( − , a ),denoted simply by X . We say that X is stable under restriction when, if f ∈ X ( b ′ ) and f | b ′′ issurjective for b ′′ ⊂ b ′ , then f | b ′′ ∈ X ( b ′′ ). Notation . For b >
0, denote the element b ∈ b by x ; this choice ensures that b \{ x } = b − . Example 6.9.
The following families of subsets of Hom Ω ( − , a ) are stable under restriction.(1) The family X such that X ( b ′ ) = (cid:26) Hom Ω ( b ′ , a ) if x b ′ , ∅ if x ∈ b ′ . (2) For y ∈ a , the family e X y such that e X y ( b ′ ) = (cid:26) Hom Ω ( b ′ , a ) if x b ′ , { f ∈ Hom Ω ( b ′ , a ) | f ( x ) = y } if x ∈ b ′ . (3) For y ∈ a , the family e X ,y ] such that e X ,y ] ( b ′ ) = (cid:26) Hom Ω ( b ′ , a ) if x b ′ , { f ∈ Hom Ω ( b ′ , a ) | ≤ f ( x ) ≤ y } if x ∈ b ′ . (4) For y ∈ a , the family X y such that X y ( b ′ ) = (cid:26) ∅ if x b ′ , { f ∈ Hom Ω ( b ′ , a ) | f − ( y ) = { x }} if x ∈ b ′ . By construction, there are inclusions of families of subsets X y ⊂ f X y ⊂ Hom Ω ( − , a ) ∪ X and e X = e X , ⊂ e X , ⊂ . . . ⊂ e X ,a − ⊂ e X ,a ] = Hom Ω ( − , a ) . A family of subsets of Hom Ω ( − , a ) stable under restriction gives rise to a subcomplex of C ( b , a ): Lemma 6.10. If X is a family of subsets of Hom Ω ( − , a ) stable under restriction, then the free k -submodules, for t ∈ N , M b ( t ) ⊂ b | b ( t ) | = b − t k X ( b ( t ) ) ⊗ Or ( b \ b ( t ) ) ⊂ C ( b , a ) t form a subcomplex of C ( b , a ) , called the subcomplex generated by X .Proof. This is clear from the definition of the differential of the complex C ( b , a ) that is recalledin Notation 6.1. (cid:3) Applying this lemma to the families introduced in Example 6.9 we obtain the following sub-complexes of C := C ( b , a ). Definition 6.11.
Suppose that b > a > x := b ∈ b .(1) Let D be the subcomplex of C generated by X .(2) For y ∈ a , let(a) f C y be the subcomplex of C generated by e X y ;(b) f y C be the subcomplex of C generated by e X ,y ] ;(c) C y be the subcomplex of C generated by X y .The inclusions of families of subsets give the following inclusions of complexes C y ⊂ f C y ⊂ C ∪ D and the increasing filtration of subcomplexes f C = f C ⊂ f C ⊂ . . . ⊂ f a − C ⊂ f a C = C . (6.2)These complexes are related by the following Lemma. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 43
Lemma 6.12.
For a > and y ∈ a , there is a short exact sequence of complexes: → D → f y C ⊕ ]C y +1 → f y +1 C → . (6.3) Proof.
The inclusions D ֒ → f y C and D ֒ → ]C y +1 induce short exact of complexes:0 → D → f y C → f y C / D →
0; 0 → D → ]C y +1 → ]C y +1 / D → . Considering the pushout of D ֒ → f y C and D ֒ → ]C y +1 in the category of chain complexes weobtain a short exact sequence of complexes0 → D → f y C ⊕ ]C y +1 → f y C ∪ D ]C y +1 → . For t ∈ N , consider the following k -modules A ty := M b ( t ) ⊂ b | b ( t ) | = b − tx ∈ b ( t ) k { f ∈ Hom Ω ( b ( t ) , a ) | ≤ f ( x ) ≤ y } ⊗ Or ( b \ b ( t ) ) B ty +1 := M b ( t ) ⊂ b | b ( t ) | = b − tx ∈ b ( t ) k { f ∈ Hom Ω ( b ( t ) , a ) | f ( x ) = y + 1 } ⊗ Or ( b \ b ( t ) ) . We have the following isomorphisms of k -modules( f y C / D ) t ∼ = A ty ; ( ]C y +1 / D ) t ∼ = B ty ; A ty +1 ∼ = A ty ⊕ B ty +1 ; ( f y C ) t ∼ = D t ⊕ A ty ; ( ]C y +1 ) t ∼ = D t ⊕ B ty +1 . We deduce that ( f y C ∪ D ]C y +1 ) t ∼ = D t ⊕ A ty ⊕ B ty +1 ∼ = D t ⊕ A ty +1 ∼ = ( f y +1 C ) t . Using the definition of the differential of C recalled in Notation 6.1 we see that this isomorphismof k -modules gives an isomorphism of complexes. (cid:3) The complexes D and C y , for y ∈ a , are identified in terms of complexes of the form C ( b ′ , a ′ )as follows, where [1] denotes the shift of cohomological degree (i.e., C [1] t := C t − by Notation1.4): Lemma 6.13.
For a, b ∈ N ∗ , we have the following isomorphisms of complexes (1) D ∼ = C ( b − , a )[1] ; (2) for y ∈ a , C y ∼ = C ( b − , a − ) .Proof. The isomorphism D t ∼ = C ( b − , a ) t − is induced by the obvious map noting that b ( t ) ⊂ b such that | b ( t ) | = b − t and x b ( t ) ⇐⇒ b ( t ) ⊂ b − such that | b ( t ) | = ( b − − ( t − . The isomorphism C ty ∼ = C ( b − , a − ) t is induced by the map C ty → C ( b − , a − ) t forget-ting the fibre above y together with the bijection of ordered sets a \ { y } ∼ = a − This proves that the underlying graded k -modules are isomorphic. The choice x = b ensuresthat the signs appearing in the differentials are compatible with these isomorphisms. (cid:3) Crucially, this also allows us to understand the cohomology of f C y , by the following result: Proposition 6.14. If b > a > , then j : C y ֒ → f C y is a quasi-isomorphism. Proof.
It suffices to check that the cokernel c C y of C y ֒ → f C y has zero cohomology.Using the notation introduced in the proof of Lemma 6.12, we have: f C yt ∼ = D t ⊕ B ty and C ty is a submodule of B ty so that c C yt ∼ = D t ⊕ B ty / C ty and D is a subcomplex of c C y . So D ֒ → c C y induces a short exact sequence of complexes0 → D → c C y → c C y / D → c C y / D ) t ∼ = B ty / C ty .We have the following isomorphisms( c C y / D ) t ∼ = B ty / C ty ∼ = M b ( t ) ⊂ b | b ( t ) | = b − tx ∈ b ( t ) k { f ∈ Hom Ω ( b ( t ) , a ) | f ( x ) = y } / k { f ∈ Hom Ω ( b ( t ) , a ) | f − ( y ) = { x }} ⊗ Or ( b \ b ( t ) ) ∼ = M b ( t ) ⊂ b | b ( t ) | = b − tx ∈ b ( t ) k { f ∈ Hom Ω ( b ( t ) , a ) | f ( x ) = y and | f − ( y ) | ≥ } ⊗ Or ( b \ b ( t ) ) ∼ = M b ( t +1) ⊂ b | b ( t +1) | = b − ( t +1) x b ( t +1) k { f ∈ Hom Ω ( b ( t +1) , a ) } ⊗ Or ( b \ b ( t +1) ) ∼ = D t +1 , where the fourth isomorphism is obtained taking the restriction given by removing the element x = b from b ( t ) . These isomorphisms of k -modules are compatible with the differential of thecomplexes, so that we have an isomorphism of complexes: c C y / D ∼ = D [ − x = b from b ′ . This operation corresponds to theisomorphism between c C y / D and D [ − c C y , as required. (cid:3) Theorem 6.15. If b > a > , then H ∗ ( C ( b , a )) is the free graded k -module H ∗ ( C ( b , a )) ∼ = k ∗ = b − a < ∗ < b − a or ∗ > b − a k ⊕ Rk( b,a ) ∗ = 0 where Rk( b, a ) = χ ( C ( b , a )) − ( − b − a , for χ the Euler-Poincar´e characteristic of the complex C ( b , a ) : χ ( C ( b , a )) = b − a X t =0 ( − t (cid:18) bt (cid:19) | Hom Ω ( b − t , a ) | . Proof.
We consider the proposition P ( b, a ): the cohomology of C = C ( b , a ) is free as a k -moduleconcentrated in degrees 0 and b − a , with H b − a ( C ) of rank one.We prove that P ( b, a ) is true for all b > a ≥ ≤ a and upon1 ≤ b − a . PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 45
The initial case P ( b,
1) is treated by Proposition 6.4 and the initial case P ( a + 1 , a ) is coveredby Proposition 6.6. The inductive step is to show that, given a pair ( b, a ) with a > b − a > P ( b − , a ) and P ( b − , a −
1) are true then P ( b, a ) holds.For this, we will prove, by induction on y , the following stronger result: if P ( b − , a ) and P ( b − , a −
1) are true then for all y ∈ { , . . . , a } the cohomology of f y C is free as a k -moduleconcentrated in degrees 0 and b − a , with H b − a ( f y C ) of rank one. The required result is thenobtained for y = a .For y = 1, by Lemma 6.13 and Proposition 6.14 we have H ∗ ( f C ) = H ∗ ( f C ) ∼ = H ∗ ( C ) ∼ = H ∗ ( C ( b − , a − ))so the result follows by the assumption on P ( b − , a − . . . → H t ( D ) → H t ( f y C ) ⊕ H t ( ]C y +1 ) → H t ( f y +1 C ) → H t +1 ( D ) → . . . (6.5)Lemma 6.13 together with the inductive hypothesis for P ( b − , a ) implies that H ∗ ( D ) isconcentrated in cohomological degrees 1 and b − a , with H b − a ( D ) ∼ = k and H ( D ) a free k -module.Moreover, as for the case y = 1, the cohomology of ]C y +1 is concentrated in cohomological degreeszero and b − a , with H b − a ( ]C y +1 ) ∼ = k and H ( ]C y +1 ) a free k -module. So using the inductivehypothesis on H ∗ ( f y C ), we obtain that the exact sequence (6.5) reduces to the following: • H t ( f y +1 C ) = 0 for 1 ≤ t ≤ b − a − t > b − a ; • the exact sequence:0 → H ( f y C ) ⊕ H ( ]C y +1 ) → H ( f y +1 C ) → H ( D ) → H ( D ) and H ( ]C y +1 ) are free k -modules; • the exact sequence, in which d is induced by the inclusions D ⊂ f y C and D ⊂ ]C y +1 (6.7)0 → H b − a − ( f y +1 C ) → H b − a ( D ) d −→ H b − a ( f y C ) ⊕ H b − a ( ]C y +1 ) → H b − a ( f y +1 C ) → . Using the exact sequence (6.6): by the inductive hypothesis H ( f y C ) is a free k -module; since H ( D ) is a free k -module, the short exact sequence splits as k -modules giving that H ( f y +1 C )is k -free, as required.Consider the exact sequence (6.7): recall that H b − a ( D ) ∼ = H b − a ( ]C y +1 ) ∼ = k . We prove thatthe inclusion i : D ֒ → ]C y +1 induces an isomorphism H b − a ( D ) ∼ = −→ H b − a ( ]C y +1 ).The cocycle [Id a ] ⊗ ω ( b \ a ) ∈ D b − a generates H b − a ( D ). We will see that the cohomology classof i ([Id a ] ⊗ ω ( b \ a )) ∈ ]C y +1 b − a generates H b − a ( ]C y +1 ).Consider b ′ = a \ { y + 1 } and f ∈ Hom Ω ( b ′ ∪ { x } , a ) given by f | b ′ = Id a | b ′ and f ( x ) = y + 1, the cocycle [ f ] ⊗ ω ( b \ ( { x } ∪ b ′ )) ∈ C b − ay +1 generates H b − a ( C y +1 ). By Proposition 6.14, j ([ f ] ⊗ ω ( b \ ( { x } ∪ b ′ ))) ∈ ]C y +1 b − a generates H b − a ( ]C y +1 ).Adapting the second part of Proposition 6.5 to the complex ]C y +1 we obtain that the coho-mology classes of i ([Id a ] ⊗ ω ( b \ a )) and j ([ f ] ⊗ ω ( b \ ( { x } ∪ b ′ ))) are equal (up to sign) (where j : C y +1 ֒ → ]C y +1 ). We deduce that i ([Id a ] ⊗ ω ( b \ a )) generates H b − a ( ]C y +1 ).It follows from the exact sequence that H b − a − ( f y +1 C ) = 0 and H b − a ( f y C ) ∼ = H b − a ( f y +1 C )which is of rank one by the recursive hypothesis.It remains to identify the rank of the free k -module H ( C ). By the first part of the proof, C ( b , a ) is a complex of free k -modules of finite rank, concentrated in degrees between 0 and b − a . So we have χ ( C ( b , a )) = b − a X t =0 ( − t rank k C ( b , a ) t = b − a X t =0 ( − t rank k H t ( C ( b , a )) . By the explicit form of the complex C ( b , a ) given in Notation 6.1 we have χ ( C ( b , a )) = b − a X t =0 ( − t (cid:18) bt (cid:19) | Hom Ω ( b − t , a ) | . Since H ∗ ( C ) is concentrated in degrees 0 and b − a , we obtainrank k H ( C ( b , a )) = χ ( C ( b , a )) − ( − b − a rank k H b − a ( C ( b , a ))and rank k H b − a ( C ( b , a )) = 1 by Proposition 6.5. (cid:3) Remark . By Lemma 6.3, for a, b ∈ N , H ∗ ( C ( b , a )) = 0 if b < a , H ∗ ( C ( b , )) is k , concen-trated in degree b and H ∗ ( C ( a , a )) is k [ S a ] concentrated in degree 0. Remark . The long exact sequence in cohomology given by [Gan16, Theorem 1] suggests analternative approach to proving Theorem 6.15, based upon a similar strategy. Namely, choosingthe element x in the above proof corresponds to using the shift functor of FI op -modules in[Gan16].To implement this strategy using [Gan16, Theorem 1] requires showing that the relevantconnecting morphism is surjective in cohomology.By Lemma 6.3 and Theorem 6.15, for ( b, a ) ∈ N × N , H ( C ( b , a )) is a free finitely-generated k -module. So we have a function Rk : N × N → N given byRk( b, a ) = rank k H ( C ( b , a )) . This function satisfies the following Proposition.
Proposition 6.18.
The function
Rk : N × N → N is determined by: Rk( b,
0) = (cid:26) b = 00 otherwise . (6.8) Rk( b, a ) = 0 for b < a (6.9) Rk( a, a ) = a !(6.10) Rk( b,
1) = 0 for b > a + 1 , a ) = ( a −
2) ( a + 1)!2 + 1 for a > b, a ) = a Rk( b − , a −
1) + ( a − b − , a ) for b − > a > . (6.13) Proof.
The equalities (6.8), (6.9) and (6.10) are consequences of Lemma 6.3; (6.11) follows fromProposition 6.4 and (6.12) from Proposition 6.6.To prove the recursive expression (6.13), observe that the short exact sequence (6.6) given inthe proof of Theorem 6.15 implies the following equality for all y ∈ { , . . . , a − } :rank k H ( f y +1 C ) = rank k H ( f y C ) + rank k H ( ]C y +1 ) + rank k H ( D ) . Taking the sum over y , for 1 ≤ y ≤ a −
1, of these equalities, we obtain:Rk( b, a ) = rank k H ( f a C ) = rank k H ( f C ) + a − X y =1 rank k H ( ]C y +1 ) + a − X y =1 rank k H ( D ) PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 47 = a X y =1 rank k H ( f C y ) + a − X y =1 rank k H ( D ) . By Lemma 6.13 we have: rank k H ( D ) = Rk( b − , a ), and by Lemma 6.13 and Proposition 6.14we have: rank k H ( f C y ) = rank k H ( C y ) = Rk( b − , a − (cid:3) Corollary 6.19.
The function
Rk : N × N → N satisfies the following equalities: Rk( b,
2) = 1 for b > b,
3) = 2 b − for b > . (6.15) Proof.
To prove (6.14), note that Rk(3 ,
2) = 1 by (6.12) and, for b >
3, we have b − > > b,
2) = 2Rk( b − ,
1) + Rk( b − ,
2) = Rk( b − , b,
2) = Rk(3 ,
2) = 1.To prove (6.15), note that Rk(4 ,
3) = 13 by (6.12) and for b >
4, we have b − > >
1; thus,by (6.13) and (6.14), we haveRk( b,
3) = 3Rk( b − ,
2) + 2Rk( b − ,
3) = 3 + 2Rk( b − , . Taking the linear combination of these equalities for b ≥ y ≥ b X y =5 b − y Rk( y,
3) = 3 b X y =5 b − y + 2 b X y =5 b − y Rk( y − ,
3) = 3 b − X y =0 y + b − X y =4 b − y Rk( y, b,
3) = 3(2 b − −
1) + 2 b − Rk(4 ,
3) = 2 b − (cid:3) The cohomology as a representation.
Recall that H ∗ ( C ( b , a )) takes values in graded S a × S op b -modules. Proposition 6.5 identifies this representation in the top degree, ∗ = b − a .Working in the Grothendieck group of finitely-generated S a × S op b -modules, Theorem 6.15(and its proof) give further information, beyond the dimension of H ( C ( b , a )). In the following,the class of a representation M in the Grothendieck group is denoted by [ M ]. Theorem 6.20.
Suppose that b > a ∈ N ∗ , then there is an equality in the Grothendieck groupof finitely-generated S a × S op b -modules: [ H ( C ( b , a ))] = ( − b − a +1 [ Or ( a ) ⊗ Or ( b )] + b − a X t =0 ( − t [ C ( b , a ) t ]= ( − b − a +1 [ Or ( a ) ⊗ Or ( b )]+ b − a X t =0 ( − t (cid:2)(cid:0) k Hom Ω ( b − t , a ) ⊗ Or ( b \ ( b − t )) (cid:1) ↑ S op b S op b − t × S op t (cid:3) . In particular, if k = Q , then this identifies the representation H ( C ( b , a )) up to isomorphism.Proof. Passage to the Grothendieck group allows the Euler-Poincar´e characteristic argumentused in the proof of Theorem 6.15 to be applied to give the first stated equality. The secondexpression then follows from Lemma 4.6.The category of Q [ S a × S op b ]-modules is semi-simple, so this suffices to identify the represen-tation up to isomorphism. (cid:3) Remark . For a < b ∈ N , 0 ≤ t ≤ b − a and k = Q , the permutation S a × S op b − t -modules Q Hom Ω ( b − t , a ) arising in the statement of Theorem 6.20 can be identified by standard methodsof representation theory, so H ( C ( b , a )) can be calculated effectively in this case. Example 6.22.
Consider the case a ∈ N ∗ and b = a + 1, so that: C ( a + , a ) = k Hom Ω ( a + , a ) C ( a + , a ) ∼ = k S a +1 , where the action in degree zero is the natural one and, in degree one, S a acts on the left byrestriction along S a ⊂ S a +1 .Theorem 6.20 gives the equality in the Grothendieck group of S a × S op a +1 -modules:[ H ( C ( a + , a ))] = [ Or ( a ) ⊗ Or ( a + )] + [ k Hom Ω ( a + , a )] − [ k S a +1 ] . (The case a = 0 is deduced from Lemma 6.3.) Consider the case a = 2 and restrict to the S -action (for the inclusion of the first factor in S × S op3 ). Both k S and k Hom Ω ( , ) are free k S -modules of rank three; this gives the equality in the Grothendieck group of S -modules:[ H ( C ( , ))] = [sgn S ] , by identifying Or ( ) as the signature representation.For k = Q , this shows that H ( C ( , )) is not a permutation representation as a S -moduleand hence cannot be a permutation representation as a S × S op3 -module.7. Computation of
Hom
Fin and
Ext Fin
In this section we combine our previous results to prove our main result, Theorem 7.1, cal-culating the groups that interest us. From this, we deduce properties of the objects k [ − ] ⊗ a ∈ Ob F ( Fin ; k ). Recall that the function Rk : N × N → N is defined before Proposition 6.18. Theorem 7.1.
For a, b ∈ N , there are isomorphisms Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = k ⊕ Rk( b,a ) Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = (cid:26) k b = a + 10 otherwise.Proof of Theorem 7.1. By Theorem 5.4, for a, b ∈ N there are isomorphisms:Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = H (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b ))Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = H (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • ( t ∗ ) ⊗ b )) . By Theorem 5.1 (or its Corollary 5.2), there is an isomorphism in graded Σ op -modules: H ∗ (Hom F (Γ; k ) (( t ∗ ) ⊗ a , ⊥ Γ • t ∗⊗ )) ∼ = H ∗ ( Kz k Hom Ω ( − , a ))By Lemma 6.3, for b < a , Kz k Hom Ω ( − , a )( b ) = 0, Kz k Hom Ω ( − , a )( a ) = k [ S a ], concentratedin cohomological degree zero and Kz k Hom Ω ( − , )( b ) = k concentrated in cohomological degree b . For b > a >
0, the calculation of these cohomology modules is given by Theorem 6.15. (cid:3)
Theorem 7.1 gives the following:
Corollary 7.2.
For a ∈ N , (1) k = k [ − ] ⊗ is injective if and only if k is an injective k -module; (2) k [ − ] ⊗ a is not injective if a > ; (3) k [ − ] ⊗ a is not projective. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 49
Proof.
The argument employed in the proof of Proposition 3.15 implies that k = k [ − ] ⊗ isinjective if and only if k is an injective k -module.The non-vanishing of Ext F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ a +1 ) ∼ = k for any a ∈ N , given by Theorem 7.1, implies the remaining non-projectivity and non-injectivitystatements. (cid:3) For the next Corollary we use the following Notation:
Notation . For a ∈ N ∗ and b ∈ N ∗ , let R b,a : k Hom Ω ( b , a ) → M b ′ ( b | b ′ | = | b |− k Hom Ω ( b ′ , a )denote the map induced by restriction along subsets b ′ ( b with | b ′ | = | b | − Corollary 7.4.
For a ∈ N ∗ and b ∈ N ∗ , we have Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) ∼ = H ( Kz k Hom Ω ( − , a )( b )) ∼ = ker( R b,a ) . In particular, this allows us to consider
Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b ) as a submodule of k Hom Ω ( b , a ) ,equipped with its natural S a × S op b -action.Proof. For the first isomorphism, combine Theorems 5.4 and 5.1 as in the beginning of the proofof Theorem 7.1. For the second, use Definition 4.5. (cid:3)
As in Section 6.4, the statement of Theorem 7.1 can be refined by taking into account thenatural S a × S op b -action. Proposition 7.5.
For a < b ∈ N ∗ , there is an equality in the Grothendieck group of S a × S op b -modules: [Hom F ( Fin ; k ) ( k [ − ] ⊗ a , k [ − ] ⊗ b )] = ( − b − a +1 [sgn S a ⊗ sgn S b ] + b − a X t =0 ( − t (cid:2)(cid:0) k Hom Ω ( b − t , a ) ⊗ sgn S t (cid:1) ↑ S op b S op b − t × S op t (cid:3) . Appendix A. Normalized (co)chain complexes
This short appendix fixes our conventions on the normalized cochain complex associated to acosimplicial object in an abelian category A . We start by reviewing the slightly more familiarsimplicial case.First recall (see [Wei94, Chapter 8], for example) that there is an unnormalized chain complex C ( A • ) associated to a simplicial object A • in A with C ( A • ) n = A n and differential given by thealternating sum of the face maps.The normalized chain complex N ( A • ) is defined (see [Wei94, Definition 8.3.6]) by N n ( A • ) := n − \ i =0 ker( ∂ i : A n → A n − ) , where ∂ i denote the face maps, and the differential is given by d := ( − n ∂ n . The inclusion N ( A • ) → C ( A • ) induces an isomorphism in homology. The chain complex C ( A • ) has an acyclic subcomplex D ( A • ) ⊂ C ( A • ), where D ( A • ) is thesubcomplex generated by the degenerate simplices. There is an isomorphism: N ( A • ) ∼ = → C ( A • ) /D ( A • ) , and hence the right hand side gives an alternative definition of the normalized complex.In the cosimplicial setting, for C • a cosimplicial object in A , the associated unnormalizedcochain complex ( C ∗ , d ) has differential d given by the alternating sum of the coface maps. Definition A.1.
For C • a cosimplicial object in A , the normalized cochain complex ( ˜ N ( C • ) , ˜ d ),is given by ˜ N ( C • ) n := Coker (cid:16) n − M i =0 C n − d i → C n (cid:17) with differential ˜ d := ( − n +1 d n +1 : ˜ N ( C • ) n → ˜ N ( C • ) n +1 .It can be more convenient to work with the isomorphic complex N ( C • ) that is given asa subcomplex of the cochain complex ( C ∗ , d ); this is analogous to working with the quotientcomplex C ( A • ) /D ( A • ) in the simplicial case. Definition A.2.
For C • a cosimplicial object in A , let ( N ( C • ) , d ) denote the subcomplex of( C ∗ , d ) given by: N ( C • ) n := ker (cid:16) C n s j → n − M j =0 C n − (cid:17) . As in the simplicial case, one has an isomorphism between the respective constructions of thenormalized cochain complex:
Proposition A.3.
The inclusion ( N ( C • ) , d ) ֒ → ( C ∗ , d ) is a cohomology equivalence and inducesan isomorphism of cochain complexes ( N ( C • ) , d ) ∼ = → ( ˜ N ( C • ) , ˜ d ) . Appendix B. Cosimplicial objects from comonads
In this appendix, we associate to a comonad two natural coaugmented cosimplicial objects ofa different nature. The first construction is a cobar-type cosimplicial resolution, extending thecanonical augmented simplicial object recalled in Proposition B.3, and the second one encodesthe notion of morphisms of comodules.
Notation
B.1 . For a category C , let ( ⊥ , ∆ , ε ) be a comonad, where ∆ : ⊥→⊥⊥ and ε : ⊥→ Idare the structure natural transformations.(1) The structure morphism of a ⊥ -comodule M is denoted ψ M : M →⊥ M and this ⊥ -comodule by ( M, ψ M ).(2) The category of ⊥ -comodules in C is denoted by C ⊥ and the morphisms in C ⊥ byHom ⊥ ( − , − ).(3) Morphisms in C are denoted simply by Hom( − , − ) to that there is a forgetful mapHom ⊥ ( − , − ) → Hom( − , − ).We record the standard fact: Proposition B.2.
Let C be an abelian category equipped with a comonad ( ⊥ , ∆ , ε ) such that ⊥ is exact. Then the category C ⊥ of ⊥ -comodules in C has a unique abelian category structure forwhich the forgetful functor C ⊥ → C is exact. Recall the natural augmented simplicial object ⊥ • +1 X associated to any X ∈ Ob C . PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 51
Proposition B.3. [Wei94, 8.6.4]
For a comonad ( ⊥ , ∆ , ε ) on the category C and X ∈ Ob C ,there is a natural augmented simplicial object ⊥ • +1 X , which is ⊥ n +1 X in simplicial degree n and has face operators induced by ε and degeneracies by ∆ : δ i := ⊥ i ε ⊥ n − i X : ⊥ n +1 X →⊥ n Xσ i := ⊥ i ∆ ⊥ n − i X : ⊥ n +1 X →⊥ n +2 X. The augmentation is given by ε X : ⊥ X → X . The augmented simplicial object ⊥ • +1 X of Proposition B.3 has the form : / / ⊥⊥⊥ X x x / / / / / / ⊥⊥ X t t t t ε ⊥ X / / ⊥ ε X / / ⊥ X ∆ X u u ε X / / X B.1.
A cobar-type cosimplicial construction.
In this section we show that, when X ∈ Ob C is a ⊥ -comodule and the comonad ⊥ is equipped with a coaugmentation η : Id →⊥ satisfyingHypothesis B.5, then the augmented simplicial object ⊥ • +1 X given in Proposition B.3 canbe extended naturally to a coaugmented cosimplicial object. This is a cobar-type cosimplicialconstruction for ⊥ -comodules.For this, we recall that there is an augmented simplicial object underlying any cosimplicialobject. (See Notation 1.2 for the notation used.) Proposition B.4.
There is an embedding of categories ∆ opaug ֒ → ∆ given by [ n ] [ n + 1] onobjects, for − ≤ n ∈ Z and δ i s i , σ j d j +1 . In particular, for any category C , restrictioninduces a functor ∆ C → ∆ opaug C from the category ∆ C of cosimplicial objects in C to thecategory ∆ opaug C of augmented simplicial objects in C .Proof. Analogous to the proof that Connes’ cyclic category is isomorphic to its opposite givenin [Lod98, Proposition 6.1.11]. (cid:3)
To define the coaugmented cosimplicial object of Proposition B.6 we require that the comonad ⊥ satisfies the following: Hypothesis
B.5 . Suppose that there exists a natural transformation η : Id →⊥ such that(1) ε ◦ η = 1 C ;(2) the following diagram commutes:Id η / / η (cid:15) (cid:15) ⊥ ∆ (cid:15) (cid:15) ⊥ η ⊥ / / ⊥⊥ . The augmented simplicial object of Proposition B.3 extends to give the following cosimplicialobject in C . Proposition B.6.
Let ( ⊥ , ∆ , ε ) be a comonad on the category C , equipped with a coaugmentation η satisfying Hypothesis B.5. For ( X, ψ X ) ∈ Ob C ⊥ , there is a natural cosimplicial object C • X in C with C n X = ⊥ n X in cosimplicial degree n and structure morphisms X d = η X / / d = ψ X / / ⊥ X s = ε X w w d = η ⊥ X / / d =∆ X / / d = ⊥ ψ X / / ⊥⊥ X s = ε ⊥ X u u s = ⊥ ε X u u / / . . . u u given explicitly as follows: d i ⊥ l X ∈ Hom C ( ⊥ l X, ⊥ l +1 X ) for ≤ i ≤ l + 1 and s j ⊥ l +1 X ∈ Hom C ( ⊥ l +1 X, ⊥ l X ) for ≤ j ≤ l , d i ⊥ l X = η ⊥ l X i = 0 ⊥ i − ∆ ⊥ l − i X < i < l + 1 ⊥ l ψ X i = l + 1; s j ⊥ l +1 X = ⊥ j ε ⊥ l − j X . The underlying augmented simplicial object given by Proposition B.4 is the augmented simpli-cial object ⊥ • +1 X of Proposition B.3.If C is abelian and ⊥ is exact, then C • is an exact functor C ⊥ → ∆ C .Proof. The construction in the statement is natural with respect to the ⊥ -comodule X . Moreover,by inspection, the underlying augmented simplicial object is ⊥ • +1 X . Hence, to verify thecosimplicial identities it suffices to check those involving η or ψ . Moreover, it is straightforwardto deduce these from those for the displayed part of the structure.The required identities follow from: ⊥ ψ X ◦ ψ X = ∆ X ◦ ψ X , by coassociativity of ψ X η ⊥ X ◦ ψ X = ⊥ ψ X ◦ η X , by naturality of η ∆ X ◦ η X = η ⊥ X ◦ η X , by Hypothesis B.5 (2) ε X ◦ ψ X = 1 X , by the counital property of ψ X ε X ◦ η X = 1 X , by Hypothesis B.5 (1) ε ⊥ X ◦ ⊥ ψ X = ψ X ◦ ε X , by naturality of ε ⊥ ε X ◦ η ⊥ X = η X ◦ ε X , by naturality of η . (cid:3) The above construction defines a cohomology theory for ⊥ -comodules in C : Definition B.7.
Let ( ⊥ , ∆ , ε ) be a comonad on the abelian category C such that ⊥ is exact,equipped with a coaugmentation η satisfying Hypothesis B.5. For ( X, ψ X ) ∈ Ob C ⊥ and n ∈ N ,let H n ⊥ ( X ) be the n -th cohomology of the complex associated to C • X . Example B.8.
Using the notation of Definition B.7, one has the identification H ⊥ ( X ) = equalizer (cid:0) X η X / / ψ X / / ⊥ X (cid:1) . This can be considered as the primitives of the ⊥ -comodule X (with respect to η ).In order to justify considering the above as a cohomology theory for ⊥ -comodules, it is impor-tant to understand its behaviour on cofree ⊥ -comodules; namely, for Y ∈ Ob C , one considersthe cofree comodule ( ⊥ Y, ∆ Y ) in C ⊥ . By Proposition B.6, we have the cosimplicial object C • ( ⊥ Y ). Proposition B.9.
Let ( ⊥ , ∆ , ε ) be a comonad on the category C , equipped with a coaugmentation η satisfying Hypothesis B.5. For Y ∈ Ob C , the cosimplicial object C • ( ⊥ Y ) associated to ( ⊥ Y, ∆ Y ) ∈ Ob C ⊥ is coaugmented via Y η Y →⊥ Y = C ( ⊥ Y ) . Moreover, the counit η Y : ⊥ Y → Y equips C • ( ⊥ Y ) with an extra codegeneracy.If C is abelian, the extra codegeneracy provides a chain nulhomotopy, so that the complexassociated to the coaugmented cosimplicial object Y → C • ( ⊥ Y ) is acyclic. PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 53
Proof.
The second condition of Hypothesis B.5 ensures that Y η Y →⊥ Y equalizes d and d , hencedefines a coaugmentation of the cosimplicial object C • ( ⊥ Y ).The counit ⊥ Y η Y → Y therefore provides an extra codegeneracy of the coaugmented cosimpli-cial object. Since ε Y ◦ η Y is the identity on Y by the first condition of Hypothesis B.5, the extracodegeneracy defines a chain nulhomotopy. (cid:3) The formal properties of H ∗⊥ are summarized by: Corollary B.10.
Let ( ⊥ , ∆ , ε ) be a comonad on the abelian category C such that ⊥ is exact,equipped with a coaugmentation η satisfying Hypothesis B.5.Then H ∗⊥ is a δ -functor on C such that, for Y ∈ Ob C : H n ⊥ ( ⊥ Y ) ∼ = (cid:26) Y n = 00 n > . Proof.
By Proposition B.6, C • is an exact functor from C ⊥ to ∆ C . In particular, it sends a shortexact sequence of ⊥ -comodules to a short exact sequence of ∆ C . Taking cohomotopy thereforegives a long exact sequence in cohomology, thus giving the δ -functor structure.Proposition B.9 then shows that this δ -functor is effacable (see [Wei94, Exercise 2.4.5], forexample, for this notion) by cofree ⊥ -comodules. (cid:3) B.2.
The cosimplicial object
Hom( M, ⊥ • N ) . In Proposition B.14 we show that, given acomonad ⊥ on a category C , the natural inclusion of morphisms of ⊥ -comodules extends to anatural coaugmented cosimplicial object.In the abelian setting, when ⊥ is an additive functor, this gives rise to an associated complex.Theorem B.23 gives an interpretation of the first two cohomology groups. Definition B.11.
For
M, N ∈ Ob C ⊥ , define the natural transformations:Hom( M, N ) d / / d / / Hom( M, ⊥ N ) s t t by d γ := ( ⊥ γ ) ψ M ; d γ := ψ N γ and s ζ := ε N ζ for γ ∈ Hom(
M, N ) and ζ ∈ Hom( M, ⊥ N )corresponding to composites in the (non-commutative) diagram: M ψ M / / γ (cid:15) (cid:15) ⊥ M ⊥ γ (cid:15) (cid:15) N ψ N / / ⊥ N ε N / / N. Proposition B.12.
Using the notation of Definition B.11, (1) Hom ⊥ ( M, N ) / / Hom(
M, N ) d / / d / / Hom( M, ⊥ N ) is an equalizer, where Hom ⊥ ( M, N ) → Hom(
M, N ) is the canonical inclusion; (2) there are equalities s d = id = s d . Proof.
The first statement is by definition of a morphism of ⊥ -comodules, corresponding to thecommutativity of the square in Definition B.11. The second is a straightforward verification,using the fact that ε N ψ N is the identity on N (likewise for M ) and the naturality of ε . (cid:3) The following is clear:
Lemma B.13.
For
M, N ∈ Ob C ⊥ , the natural augmented simplicial object ⊥ • +1 N in C givenby Proposition B.3 induces a natural augmented simplicial structure: Hom( M, ⊥ • +1 N ) . The following result shows that this extends (as in Proposition B.4) to a cosimplicial object:
Proposition B.14.
For
M, N ∈ Ob C ⊥ , there is a natural cosimplicial object Hom( M, ⊥ • N ) given by Hom( M, ⊥ t N ) in cosimplicial degree t , such that: (1) the underlying augmented simplicial object is that of Lemma B.13; (2) the remaining coface maps d , d t +1 : Hom( M, ⊥ t N ) ⇒ Hom( M, ⊥ t +1 N ) are given by d γ = ( ⊥ γ ) ψ M d t +1 γ = ( ⊥ t ψ N ) γ for γ ∈ Hom( M, ⊥ t N ) ; (3) the canonical inclusion Hom ⊥ ( M, N ) → Hom(
M, N ) defines a coaugmentation of thecosimplicial object Hom( M, ⊥ • N ) .If C is an abelian category and ⊥ is an additive functor, then this gives a coaugmented cosimplicialabelian group.Proof. Proposition B.12 establishes the cosimplicial structure up to cosimplicial degree one to-gether with the coaugmentation.Because of the nature of the construction, it is sufficient to verify the cosimplicial identitiesfor d i d j : Hom( M, N ) → Hom( M, ⊥⊥ N ) s i d j : Hom( M, ⊥ N ) → Hom( M, ⊥ N ) . These are consequences of the axioms for a comonad and comodules over this comonad. Forexample, d d γ is, by definition, the composite (cid:0) ⊥ (( ⊥ γ ) ψ M ) (cid:1) ψ M = ( ⊥⊥ γ )( ⊥ ψ M ) ψ M whereas d d γ is the composite ∆ N ( ⊥ γ ) ψ M . That these coincide follows from the commutativity of the following diagram: M ψ M / / ψ M (cid:15) (cid:15) ⊥ M ∆ M (cid:15) (cid:15) ⊥ γ / / ⊥ N ∆ N (cid:15) (cid:15) ⊥ M ⊥ ψ M / / ⊥⊥ M ⊥⊥ γ / / ⊥⊥ N, where the left hand square commutes by coassociativity for M and the right hand square bynaturality of ∆. The remaining verifications are similar and are left to the reader.Finally, if C is abelian and ⊥ is additive, then each Hom( M, ⊥ t N ) is an abelian group, thefunctor ⊥ induces a group morphism Hom( − , − ) → Hom( ⊥ − , ⊥ − ) so that, together withbiadditivity of composition, the above construction yields a coaugmented cosimplicial abeliangroup. (cid:3) PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 55
Example B.15.
For t = 1, the morphisms d i correspond to the three possible composites in the(non-commutative) diagram M ψ M / / ζ (cid:15) (cid:15) ⊥ M ⊥ ζ (cid:15) (cid:15) ⊥ N ∆ N / / ⊥ ψ N / / ⊥⊥ N and s , s are induced respectively by the composite with ε ⊥ N , ⊥ ε N : ⊥⊥ N ⇒ ⊥ N . Corollary B.16.
Suppose that C is abelian and ⊥ is additive, then the complex associatedto the coaugmented cosimplicial object of Proposition B.14 for M, N ∈ Ob C ⊥ yields a naturalcoaugmented cochain complex in abelian groups: → Hom ⊥ ( M, N ) → Hom(
M, N ) → Hom( M, ⊥ N ) → Hom( M, ⊥⊥ N ) → . . . , with Hom( M, ⊥ t N ) placed in cohomological degree t . In the rest of this section we assume:
Hypothesis
B.17 . The category C is abelian and ⊥ is exact. Remark
B.18 . Proposition B.2 implies that the category C ⊥ of ⊥ -comodules in C is an abeliancategory and the forgetful functor C ⊥ → C is exact.In order to give an interpretation of the first and second cohomology groups of the cochaincomplex obtained in Corollary B.16 we need the following notation. Notation
B.19 . Denote by(1) Ext ( − , − ) the group defined via classes of extensions in C ;(2) Ext ⊥ ( − , − ) the group of classes of extensions in C ⊥ .The following is clear: Proposition B.20.
The forgetful functor induces a natural transformation
Ext ⊥ ( − , − ) → Ext ( − , − ) . Notation
B.21 . Write Ext ⊥ ( − , − ) for the kernel of the transformation Ext ⊥ ( − , − ) → Ext ( − , − ). Remark
B.22 . If the underlying object of the ⊥ -comodule M is projective in C , then Ext ⊥ ( M, − ) ∼ =Ext ⊥ ( M, − ). Theorem B.23.
For
M, N ∈ Ob C ⊥ , the first two cohomology groups of the cochain complex (Hom( M, ⊥ • N ) , d ) associated to the cosimplicial object of Proposition B.14 identify as: H (Hom( M, ⊥ • N )) ∼ = Hom ⊥ ( M, N ) H (Hom( M, ⊥ • N )) ∼ = Ext ⊥ ( M, N ) . Proof.
By Proposition B.14, the beginning of the unnormalized cochain complex associated toHom( M, ⊥ • N ) is of the formHom( M, N ) d → Hom( M, ⊥ N ) d → Hom( M, ⊥⊥ N ) , (B.1)with Hom( M, N ) placed in cohomological degree zero, where d γ := ( ⊥ γ ) ψ M − ψ N γ ; d ζ := (cid:0) ( ⊥ ζ ) ψ M + ( ⊥ ψ N ) ζ (cid:1) − ∆ N ζ. The complex (Hom( M, ⊥ • N ) , d ) has the same cohomology as its normalized complex (as inDefinition A.2, this is defined as a subcomplex); in degree zero these coincide, whereas in degreeone Hom( M, ⊥ N ) is replaced by the kernelHom( M, ⊥ N ) := ker { Hom( M, ⊥ N ) Hom(
M,ε N ) −→ Hom(
M, N ) } . Thus, to calculate the cohomology in degrees 0 and 1, in the complex (B.1), one can replaceHom( M, ⊥ N ) by Hom( M, ⊥ N ).The identification of the kernel of d (and hence of H (Hom( M, ⊥ • N ))) follows from Propo-sition B.12. It remains to show that H (Hom( M, ⊥ • N )) is isomorphic to Ext ⊥ ( M, N ).An element of Ext ⊥ ( M, N ) represents an equivalence class of extensions of ⊥ -comodules ofthe form 0 → N → E → M → C , so that E ∼ = M ⊕ N . Fix a choice ofthis splitting.It follows that the structure morphism ψ E : E →⊥ E is determined by ψ N , ψ M and amorphism ζ ∈ Hom( M, ⊥ N ). Indeed, such a ζ gives a comodule structure on E (with respectto the given splitting in C as above) if and only if ε N ζ = 0 and d ζ = 0, since the first conditionis equivalent to the counit axiom and the second to the coassociativity axiom.Hence, the above construction defines a surjection:ker (cid:8) Hom( M, ⊥ N ) d → Hom( M, ⊥⊥ N ) } → Ext ⊥ ( M, N ) . It remains to show that the kernel of this surjection is the image of d .The morphism ζ maps to 0 in Ext ⊥ ( M, N ) if and only if there exists a retract of
N ֒ → E in ⊥ -comodules. With respect to the given isomorphism E ∼ = M ⊕ N in C , this is determined by amorphism γ : M → N in C .It is straightforward to check that γ induces a morphism of comodules E ζ → N (where E ζ denotes E equipped with the ⊥ -comodule structure corresponding to ζ ) if and only if ζ = − d γ . (cid:3) In the rest of this section we study the extra structure obtained when the category C is tensorabelian with respect to ⊗ and the functor ⊥ is symmetric monoidal.The following is standard: Lemma B.24.
The tensor product induces a symmetric monoidal structure on C ⊥ ; the comodulestructure morphism of M ⊗ N is M ⊗ N ψ M ⊗ ψ N → ( ⊥ M ) ⊗ ( ⊥ N ) ∼ = ⊥ ( M ⊗ N ) . The behaviour of (relative) extensions under tensor product with a comodule is described bythe following:
Proposition B.25.
Let
A, M, N be ⊥ -comodules. Then (1) the functor A ⊗ − : C → C induces a morphism of abelian groups Ext ⊥ ( M, N ) → Ext ⊥ ( A ⊗ M, A ⊗ N );(2) under this morphism, the extension represented by ζ : M →⊥ N is sent to the extensionrepresented by ψ A ⊗ ζ : A ⊗ M → ( ⊥ A ) ⊗ ( ⊥ N ) ∼ = ⊥ ( A ⊗ N ) . Proof.
The result follows from the definitions and by analysis of (the proof of) Theorem B.23. (cid:3)
PIRASHVILI-TYPE THEOREM FOR FUNCTORS ON NON-EMPTY FINITE SETS 57
References [CE17] Thomas Church and Jordan S. Ellenberg,
Homology of FI-modules , Geom. Topol. (2017), no. 4,2373–2418. MR 3654111[Gan16] Wee Liang Gan, A long exact sequence for homology of FI-modules , New York J. Math. (2016),1487–1502. MR 3603074[KS06] Masaki Kashiwara and Pierre Schapira, Categories and sheaves , Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006.MR 2182076[Lod98] Jean-Louis Loday,
Cyclic homology , second ed., Grundlehren der Mathematischen Wissenschaften [Fun-damental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E byMar´ıa O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili. MR 1600246[Pir00a] Teimuraz Pirashvili,
Dold-Kan type theorem for Γ -groups , Math. Ann. (2000), no. 2, 277–298.MR 1795563[Pir00b] , Hodge decomposition for higher order Hochschild homology , Ann. Sci. ´Ecole Norm. Sup. (4) (2000), no. 2, 151–179. MR 1755114[PV18] Geoffrey Powell and Christine Vespa, Higher Hochschild homology and exponential functors , ArXiv e-prints (2018), arXiv:1802.07574.[PV20] ,
Extensions of outer functors , In preparation.[TW19] Victor Turchin and Thomas Willwacher,
Hochschild-Pirashvili homology on suspensions and represen-tations of
Out( F n ), Ann. Sci. ´Ec. Norm. Sup´er. (4) (2019), no. 3, 761–795. MR 3982870[Ves18] Christine Vespa, Extensions between functors from free groups , Bulletin of the London MathematicalSociety (2018), no. 3, 401–419.[Wei94] Charles A. Weibel, An introduction to homological algebra , Cambridge Studies in Advanced Mathematics,vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324[Zie95] G¨unter M. Ziegler,
Lectures on polytopes , Graduate Texts in Mathematics, vol. 152, Springer-Verlag,New York, 1995. MR 1311028
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