3.1 The functor category $ \mathcal{F} (\Gamma;\, \unicode{x1D55C}) $

The functor category $ \mathcal{F} (\Gamma;\, \unicode{x1D55C}) $ has been studied by Pirashvili and his co-authors (see, e.g., [Reference Pirashvili5,Reference Pirashvili6]). We begin this section by some basic facts concerning this category.

Since $\Gamma$ is a pointed category, one has the splitting:

The projectives given in Notation 3.1 have the following property:

Proof. Use that the object $\textbf{m+n}_+$ of $\Gamma$ is isomorphic to the coproduct $\textbf{m}_+ \bigvee \textbf{n}_+$ in $\Gamma$ .

One of the most important results concerning the category $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ is Pirashvili’s Dold–Kan type theorem for $\Gamma$ -modules (see Theorem 3.7). To recall this, we introduce the following:

One has the following, in which the image of the generator $[x] \in \unicode{x1D55C}[\textbf{n}_+]$ (corresponding to $x \in \textbf{n}_+$ ) in $t^* (\textbf{n}_+)$ is written [[x]], which is zero if and only if $x=+$ .

Proof. The canonical splitting is provided by Lemma 3.2 and the remaining statements follow by explicit identification of the functors.

Lemma 3.5 together with Proposition 3.3 lead directly to the following proposition, in which $(t^* )^{\otimes 0}=\unicode{x1D55C}$ :

The category $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ is symmetric monoidal for the structure induced by $\otimes$ on $\unicode{x1D55C}$ -modules and ${\boldsymbol{\Omega}}$ has a symmetric monoidal structure (see Remark 2.2).

In the following result, $\unicode{x1D55C} {\boldsymbol{\Omega}} $ denotes the $\unicode{x1D55C}$ -linearisation of the category ${\boldsymbol{\Omega}}$ .

Theorem 3.7 [Reference Pirashvili6]. There is a $\unicode{x1D55C}$ -linear, symmetric monoidal embedding

\begin{equation*}(\unicode{x1D55C} {\boldsymbol{\Omega}})^\textrm{op}\rightarrow \mathcal{F} (\Gamma;\, \unicode{x1D55C})\end{equation*}

that is induced by $\textbf{1} \mapsto t^*$ . This is fully faithful and induces an equivalence of categories between ${\mathcal{F} (\Gamma;\, \unicode{x1D55C})}$ and $\mathcal{F} ({\boldsymbol{\Omega}};\, \unicode{x1D55C})$ .

In particular, for $a, b\in \mathbb{N}$ , there is a natural isomorphism

\begin{equation*}\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})}\left(\left(t^*\right)^{\otimes a}, \left(t^*\right)^{\otimes b} \right)\cong \unicode{x1D55C} \textrm{Hom}_{\boldsymbol{\Omega}} (\textbf{b}, \textbf{a}).\end{equation*}

In Section 5, it is necessary to have an explicit description of the natural isomorphism given in the previous theorem. So, the rest of this section is devoted to make explicit the isomorphism

\begin{equation*}\unicode{x1D55C} \textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}, \textbf{a})\stackrel{\cong }{\rightarrow }\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})}\left(\left(t^*\right)^{\otimes a}, \left(t^*\right)^{\otimes b} \right).\end{equation*}

For $a=0$ , we have

\begin{equation*}\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} (\unicode{x1D55C} , (t^*)^{\otimes b} )\cong \left\{\begin{array}{l@{\quad}l}\unicode{x1D55C} & b=0\\[4pt]0 & b>0,\end{array}\right.\end{equation*}

where, for $b=0$ , a generator is given by the identity $\textrm{Id}_{\unicode{x1D55C}}\,:\, \unicode{x1D55C} \to \unicode{x1D55C}$ .

For $a>0$ , understanding the morphisms $ \textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} (t^* , (t^*)^{\otimes b} ) $ is a key ingredient. By Lemma 3.5, for (Z, z) a finite pointed set, $ \{[y] - [z]\ |\ y \in Z \backslash \{z \} \}$ is a basis $t^*(Z)$ . We use this basis in the following:

Lemma 3.8. For $b \in \mathbb{N}$ ,

\begin{equation*}\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} (t^* , (t^*)^{\otimes b} )\cong \left\{\begin{array}{l@{\quad}l}\unicode{x1D55C} & b>0\\[4pt]0 & b=0,\end{array}\right.\end{equation*}

where, for $b>0$ , a generator is given by the morphism $\xi_b\,:\, t^* \rightarrow (t^*)^{\otimes b}$ given upon evaluation on (Z,z) by $ [y]-[z] \mapsto ([y]-[z])^{\otimes b}$ , where $y \in Z \backslash \{z \}$ .

In particular, the action of $\mathfrak{S}_b$ on $\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} (t^* , (t^*)^{\otimes b} )$ induced by the place permutation action on $(t^*)^{\otimes b}$ is trivial.

Proof. If $b=0$ , since $t^*$ is a reduced $\Gamma$ -module, the statement follows from Lemma 3.2. For $b>0$ , the inclusion $t^* \subset P^{\Gamma}_{\textbf{1}_+} \cong t^* \oplus \unicode{x1D55C}$ (where the isomorphism is given by Lemma 3.5) induces an isomorphism

\begin{equation*}\textrm{Hom}_{\mathcal{F}\!\left(\Gamma;\, \unicode{x1D55C}\right)} \left( P^{\Gamma}_{\textbf{1}_+}, \left(t^*\right)^{\otimes b} \right) \cong \textrm{Hom}_{\mathcal{F}\!\left(\Gamma;\, \unicode{x1D55C}\right)} \left(t^* , \left(t^*\right)^{\otimes b} \right) \end{equation*}

by Lemma 3.2. Yoneda’s lemma gives that the left-hand side is $\unicode{x1D55C}$ , with trivial $\mathfrak{S}_b$ -action, where the given generator corresponds to the element $([1]-[+])^{\otimes b} \in t^*(\textbf{1}_+)^{\otimes b}$ .

The symmetric monoidal structure of ${\boldsymbol{\Omega}}$ (see Remark 2.2) gives the following:

Lemma 3.9. For $a \in \mathbb{N}^*$ and $ b \in \mathbb{N}$ , the symmetric monoidal structure of ${\boldsymbol{\Omega}}$ induces a $\mathfrak{S}_b^\textrm{op}$ -equivariant isomorphism

\begin{equation*}\coprod_{\substack { \left(b_i\right)\in (\mathbb{N}^*)^{a} \\ \sum_{i=1}^a b_i = b }}\left( \prod_{i=1}^a \textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}_{\textbf{i}}, \textbf{1})\right) \uparrow_{\prod_{i=1}^a \mathfrak{S}_{b_i}}^{\mathfrak{S}_b}\stackrel{\cong }{\rightarrow }\textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}, \textbf{a} ),\end{equation*}

in which $\textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}_{\textbf{i}}, \textbf{1}) \cong \{ * \}$ .

Proof. For $b=0$ , the two sides of the bijection are the empty set. For $b \not = 0$ , the component of the morphism indexed by $(b_i) \in (\mathbb{N}^*)^a$ is induced by the set map

\begin{eqnarray*}\prod_{i=1}^a\textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}_{\textbf{i}}, \textbf{1})&\rightarrow &\textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}, \textbf{a})\\(\xi_i) &\mapsto &\textbf{b}\cong \amalg_{i=1}^a \textbf{b}_{\textbf{i}}\stackrel{\amalg_{i=1} ^a \xi_i}{\longrightarrow}\amalg_{i=1}^a \textbf{1}\cong \textbf{a}.\end{eqnarray*}

This induces the given isomorphism, since a surjection $f \,:\, \textbf{b} \twoheadrightarrow \textbf{a}$ is uniquely determined by the partition of $\textbf{b}$ given by the fibres $\textbf{b}_i\,:\!=\, f^{-1}(i)$ (cf. Lemma 2.6).

This allows the following definition to be given.

Proposition 3.11. For $a, b\in \mathbb{N}$ , the $\unicode{x1D55C}$ -linear extension of $\mathfrak{DK}_{\textbf{a}, \textbf{b}}$ induces a $\mathfrak{S}_a \times \mathfrak{S}_b^\textrm{op}$ -equivariant isomorphism

\begin{equation*}\unicode{x1D55C} \mathfrak{DK}_{\textbf{a}, \textbf{b}} \,:\,\unicode{x1D55C} \textrm{Hom}_{{\boldsymbol{\Omega}}} (\textbf{b}, \textbf{a})\stackrel{\cong }{\rightarrow }\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})}\left(\left(t^*\right)^{\otimes a}, \left(t^*\right)^{\otimes b} \right).\end{equation*}

Proof. The construction of the map $\mathfrak{DK}_{\textbf{a}, \textbf{b}}$ ensures that the $\unicode{x1D55C}$ -linearisation is $\mathfrak{S}_a \times \mathfrak{S}_b^\textrm{op}$ -equivariant. The fact that it is an isomorphism follows from Pirashvili’s result, Theorem 3.7.

3.2 The functor category $ \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$

The aim of this section is to give an analysis of the category $ \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ parallel to that of $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ given in the previous section.

While $\emptyset$ is the initial object of $\textbf{Fin}$ , it is not final, so that $\textbf{Fin}$ is not pointed and there is no splitting analogous to that of Lemma 3.2. Instead one has:

Proof. Let $I_{\textbf{1},\unicode{x1D55C}}^{\textbf{Fin}}\in \mathsf{Ob}\, \mathcal{F} (\textbf{Fin};\, \unicode{x1D55C})$ be the functor given by the maps from $\textrm{Hom}_\textbf{Fin} ({-}, \textbf{1})$ to $\unicode{x1D55C}$ , denoted by $\unicode{x1D55C}^{\textrm{Hom}_\textbf{Fin} ({-}, \textbf{1})} $ ; similarly $I_{\textbf{1},\unicode{x1D55C}}^{{\overline{\textbf{Fin}}}}\,:\!=\, \unicode{x1D55C}^{\textrm{Hom}_{\overline{\textbf{Fin}}} ({-}, \textbf{1})}$ in $\mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ . Since $\textbf{1}$ is the terminal object of both $\textbf{Fin}$ and ${\overline{\textbf{Fin}}}$ , there are isomorphisms $I_{\textbf{1},\unicode{x1D55C}}^{\textbf{Fin}} \simeq \unicode{x1D55C}$ and $I_{\textbf{1},\unicode{x1D55C}}^{{\overline{\textbf{Fin}}}} \simeq \overline{\unicode{x1D55C}}$ .

By Yoneda’s lemma, there are natural isomorphisms for $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{Fin};\, \unicode{x1D55C})$ and $G \in \mathsf{Ob}\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ :

\begin{equation*}\textrm{Hom}_{\mathcal{F}\!\left(\textbf{Fin};\, \unicode{x1D55C}\right)}\left(F,I_{\textbf{1},\unicode{x1D55C}}^{\textbf{Fin}}\right) \simeq \textrm{Hom}_{\unicode{x1D55C}} (F(\textbf{1}) ,\unicode{x1D55C});\, \qquad \textrm{Hom}_{\mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)}\left(G,I_{\textbf{1},\unicode{x1D55C}}^{{\overline{\textbf{Fin}}}}\right) \simeq \textrm{Hom}_{\unicode{x1D55C}} (G(\textbf{1}) ,\unicode{x1D55C}).\end{equation*}

It follows that $\textrm{Hom}_{\mathcal{F} (\textbf{Fin};\, \unicode{x1D55C})}\left({-},I_{\textbf{1},\unicode{x1D55C}}^{\textbf{Fin}}\right)$ (respectively $\textrm{Hom}_{\mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)}({-},I_{\textbf{1},\unicode{x1D55C}}^{{\overline{\textbf{Fin}}}})$ ) is exact iff $\unicode{x1D55C}$ is injective as a $\unicode{x1D55C}$ -module. This gives the equivalence of the conditions of the statement.

Proposition 3.3 has the following analogue for $ \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ ; the proof is similar.

We introduce the functor $\overline{\unicode{x1D55C}[{-}]} \in \mathsf{Ob}\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ , which is the analogue of the functor $t^* \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ introduced in Definition 3.4.

This gives the defining short exact sequence:

(3.1) \begin{eqnarray}0\rightarrow \overline{\unicode{x1D55C}[{-}]}\rightarrow P^{\overline{\textbf{Fin}}}_{\textbf{1}}\rightarrow \overline{\unicode{x1D55C}}\rightarrow 0.\end{eqnarray}

The functor $\overline{\unicode{x1D55C}[{-}]}$ plays an essential rôle in analysing the category of ${\overline{\textbf{Fin}}}$ -modules. The principal difference between $t^* \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ and $\overline{\unicode{x1D55C}[{-}]} \in \mathsf{Ob}\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ is that the functor $\overline{\unicode{x1D55C}[{-}]}$ is not projective (see Corollary 7.2). We record immediately the following important observation:

Proof. The morphism $P^{\overline{\textbf{Fin}}}_{\textbf{1}} \twoheadrightarrow \overline{\unicode{x1D55C}}$ is an isomorphism when evaluated on $\textbf{1}$ hence, were a section to exist, it would send $1 \in \overline{\unicode{x1D55C}}$ to the generator $[\textbf{1}]$ . However, this morphism cannot extend to a natural transformation from the constant functor $\overline{\unicode{x1D55C}}$ to $P^{\overline{\textbf{Fin}}}_{\textbf{1}}$ , since $[\textbf{1}]$ is not invariant under morphisms of ${\overline{\textbf{Fin}}}$ . The remaining statements follow immediately.

3.3 Comparison of the functor categories $ \mathcal{F} (\Gamma;\, \unicode{x1D55C}) $ and $ \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$

In this section, we relate the categories $ \mathcal{F} (\Gamma;\, \unicode{x1D55C}) $ and $ \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ using the functors $\vartheta$ and $({-})_+$ introduced in Section 2.

Recall that a functor $\psi$ is conservative if a morphism f is an isomorphism if and only if $\psi (f)$ is.

Proposition 3.19. The adjunction $({-})_+ \,:\, {\overline{\textbf{Fin}}} \rightleftarrows \Gamma \,:\, \vartheta$ induces an adjunction

\begin{equation*} \vartheta^* \,:\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right) \rightleftarrows \mathcal{F} (\Gamma;\, \unicode{x1D55C}) \,:\, ({-})_+^* \end{equation*}

of exact functors that are symmetric monoidal. Moreover:

1. (1) $\vartheta^* \,:\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)\rightarrow \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ is faithful and conservative;

2. (2) $\vartheta^*$ sends projectives to projectives. (Explicitly, for $n \in \mathbb{N}^*$ , $\vartheta^* P^{\overline{\textbf{Fin}}}_{\textbf{n}} \cong P^\Gamma_{\textbf{n}_+}$ .)

Proof. The first statement is general: precomposition with a functor induces an exact, symmetric monoidal functor; precomposition with an adjunction yields an adjunction, with the rôle of the adjoints reversed.

The fact that $\vartheta^*$ is faithful and conservative follows from the fact that $\vartheta$ is essentially surjective and faithful. (This is the reason for restricting attention to ${\overline{\textbf{Fin}}}$ .)

Finally, it follows formally from the adjunction that $\vartheta^*$ sends projectives to projectives. The explicit identification follows from the adjunction isomorphism $\textrm{Hom}_{{\overline{\textbf{Fin}}}} (\textbf{n}, \vartheta(X)) \cong \textrm{Hom}_{\Gamma} (\textbf{n}_+, X)$ , for X a finite pointed set.

Proof. It is clear that, with respect to the identification of the action of $\vartheta^*$ on projectives given in Proposition 3.19, $\vartheta^*\left(P^{\overline{\textbf{Fin}}} _{\textbf{1}} \twoheadrightarrow \overline{\unicode{x1D55C}}\right)$ is the projection $P^\Gamma _{\textbf{1}_+} \twoheadrightarrow \unicode{x1D55C}$ . Since $\vartheta^*$ is exact, this gives the first statement, together with the isomorphism $\vartheta^* \left(\overline{\unicode{x1D55C}[{-}]}\right) \cong t^*$ .

The second statement then follows from the fact that $\vartheta^*$ is symmetric monoidal.

In the rest of the section, we establish the comonadic description of $\mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ using the Barr–Beck theorem (see Theorem 3.24).

Explicitly, for $F \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ , ${\perp^{ \scriptscriptstyle \Gamma}} F \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ is defined by ${\perp^{ \scriptscriptstyle \Gamma}} F (Y,y)=F(Y_+,+)$ for $(Y,y) \in \mathsf{Ob}\, \Gamma$ and, for $\alpha\,:\, (Y,y) \to (Z,z)$ a morphism in $\Gamma$ , ${\perp^{ \scriptscriptstyle \Gamma}} F(\alpha)= F (\alpha_+)\,:\, F(Y_+,+) \to F(Z_+, +)$ .

One has the following immediate consequence of Proposition 3.19:

The structure maps of the comonad ${\perp^{ \scriptscriptstyle \Gamma}}$ are described in the following proposition.

Proof. By Proposition 3.19, the adjunction giving rise to ${\perp^{ \scriptscriptstyle \Gamma}}$ is induced by the adjunction $({-})_+ \,:\, {\overline{\textbf{Fin}}} \rightleftarrows \Gamma \,:\, \vartheta$ ; hence, the structure morphisms of the comonad ${\perp^{ \scriptscriptstyle \Gamma}}$ can be deduced from Lemma 2.4.

Note that, by definition of ${\perp^{ \scriptscriptstyle \Gamma}}$ , we have

\begin{equation*}\qquad\qquad\qquad{\perp^{ \scriptscriptstyle \Gamma}}_1 {\perp^{ \scriptscriptstyle \Gamma}}_2 F (Y) =F \circ ({-})_{+_2} \circ \vartheta \circ ({-})_{+_1} \circ \vartheta(Y)=F \left(\left(Y_{+_1}\right) _{+_2}\right).\qquad\qquad\qquad\end{equation*}

Recall that a ${\perp^{ \scriptscriptstyle \Gamma}}$ -comodule in $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ is an object $F \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ equipped with a structure morphism $\psi_F\,:\, F \to {\perp^{ \scriptscriptstyle \Gamma}} F$ such that the following diagrams commute:

A morphism $(F, \psi_F) \rightarrow (G, \psi_G)$ of ${\perp^{ \scriptscriptstyle \Gamma}}$ -comodules is a morphism $F \rightarrow G$ of $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ that is compatible with the structure morphisms.

The Barr–Beck theorem (see, e.g., [Reference Kashiwara and Schapira3, Theorem 4.3.8] for the dual monadic statement) then gives the following:

Remark 3.25. The Barr–Beck correspondence can be understood explicitly as sketched below.

Suppose that $G \in \mathsf{Ob}\, \mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right)$ . Let $\psi_{\vartheta^*G}\,:\!=\,\vartheta^*(\eta_G)\,:\, \vartheta^*G \to {\perp^{ \scriptscriptstyle \Gamma}} \vartheta^*G$ where $\eta\,:\, Id \to ({-})_+^* \circ \vartheta^*$ is the unit of the adjunction given in Proposition 3.19. Explicitly, for $X \in {\overline{\textbf{Fin}}}$ , $(\eta_G)_X=G(i_X)\,:\, G(X) \to G(X \amalg \{ + \})$ where $i_X\,:\, X \to X \amalg \{ + \}$ is the canonical inclusion. Then $(\vartheta^*G, \psi_{\vartheta^*G})$ is a ${\perp^{ \scriptscriptstyle \Gamma}}$ -comodule and the correspondence $G \mapsto (\vartheta^*G, \psi_{\vartheta^*G})$ defines a functor $\mathcal{F}\!\left({\overline{\textbf{Fin}}};\, \unicode{x1D55C}\right) \to \mathcal{F} (\Gamma;\, \unicode{x1D55C})_{\perp^{ \scriptscriptstyle \Gamma}}$ .

Suppose that $F \in \mathsf{Ob}\, \mathcal{F} (\Gamma;\, \unicode{x1D55C})_{\perp^{ \scriptscriptstyle \Gamma}}$ ; the associated functor on ${\overline{\textbf{Fin}}}$ is given for $n \in \mathbb{N}^*$ by $\textbf{n}\mapsto F ((\textbf{n-1})_+)$ . This corresponds to choosing a basepoint for each object of the skeleton of ${\overline{\textbf{Fin}}}$ .

The ${\perp^{ \scriptscriptstyle \Gamma}}$ -comodule structure allows the full functoriality to be recovered. For $X \in \mathsf{Ob}\, \Gamma$ a finite pointed set, one has the morphisms

\begin{equation*}F(X)\rightarrow {\perp^{ \scriptscriptstyle \Gamma}} F(X) \cong F(X_+)\rightarrow F(X),\end{equation*}

where the first is the comodule structure map and the second is induced by sending the added basepoint $+$ 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 any basepoints $x, y \in X$ .

3.4 Comparison of the functor categories $ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) $ and $ \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$

In this section, we relate the categories $ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) $ and $ \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

We begin by exhibiting the right adjoint of the restriction functor $\downarrow \,:\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) \rightarrow \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ induced by ${\boldsymbol{\Sigma}} ^\textrm{op} \subset \textbf{FI}^\textrm{op}$ . Consider the functor in $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op} \times \textbf{FI};\, \unicode{x1D55C})$ given by $(\textbf{b}, \textbf{n}) \mapsto \unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n})$ . (Observe that $\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n})= 0$ for $b >n$ .) Hence, for G a ${\boldsymbol{\Sigma}}^\textrm{op}$ -module,

\begin{equation*}\textbf{n} \mapsto \textrm{Hom} _{\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})} (\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \cdot , \textbf{n}) , G ({\cdot}))\end{equation*}

is an $\textbf{FI}^\textrm{op}$ -module and the right-hand side identifies as

(3.2) \begin{eqnarray}\bigoplus _{b\leq n}\textrm{Hom} _{\mathfrak{S}_b^\textrm{op}} (\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \textbf{b} , \textbf{n}) , G (\textbf{b}))\subset\bigoplus _{b\leq n}\textrm{Hom} _{\unicode{x1D55C}} (\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \textbf{b} , \textbf{n}) , G (\textbf{b})).\end{eqnarray}

Definition 3.26. Let $\uparrow\,:\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}) \to \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ be the functor given by

\begin{equation*}\uparrow G \,:\!=\, \left(\textbf{n} \mapsto \textrm{Hom}_{\mathcal{F} \left({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}\right)} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \cdot , \textbf{n}) , G ({\cdot})\right)\right)\end{equation*}

for $G \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

Proposition 3.27. The functor $\uparrow\,:\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}) \to \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ is the right adjoint of the restriction functor $\downarrow \,:\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) \rightarrow \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ . In other words, for $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ and $G \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ there is a natural isomorphism:

\begin{equation*}\textrm{Hom} _{\mathcal{F} \left({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}\right)} \left(\downarrow F, G\right)\cong \textrm{Hom}_{ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})} \left(F,\uparrow G\right).\end{equation*}

Proof. We analyse $\textrm{Hom}_{ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) } (F ({-}),\textrm{Hom} _{\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})} (\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \cdot , -) , G ({\cdot})))$ , based upon the identification in equation (3.2), which gives:

\begin{equation*}\textrm{Hom}_{ \mathcal{F} \left(\textbf{FI}^\textrm{op};\, \unicode{x1D55C}\right) } \left(F ({-}),\textrm{Hom} _{\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \cdot , -) , G ({\cdot})\right)\right)\end{equation*}

\begin{equation*}\cong \bigoplus _{b\leq n}\textrm{Hom}_{ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) } \left(F ({-}),\textrm{Hom} _{\mathfrak{S}_b^\textrm{op}} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \textbf{b} , -) , G (\textbf{b})\right)\right).\end{equation*}

Fixing $b\in \mathbb{N}$ and neglecting the $\mathfrak{S}_b$ -action, we consider

\begin{equation*}\textrm{Hom}_{ \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) } \left(F ({-}),\textrm{Hom} _{\unicode{x1D55C}} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} (\textbf{b}, - ) , G (\textbf{b})\right)\right).\end{equation*}

Fixing $n \in \mathbb{N}$ and neglecting the functoriality with respect to $\textbf{FI}$ , one has the standard natural adjunction isomorphisms for $\unicode{x1D55C}$ -modules:

\begin{eqnarray*} \textrm{Hom}_{\unicode{x1D55C}} \left(F \left(\textbf{n}\right),\textrm{Hom} _{\unicode{x1D55C}} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} \left(\textbf{b}, \textbf{n} \right) , G \left(\textbf{b}\right)\right)\right)&\cong &\textrm{Hom}_{\unicode{x1D55C}} \left(F \left(\textbf{n}\right) \otimes \unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} \left(\textbf{b}, \textbf{n} \right) , G \left(\textbf{b}\right)\right)\\[4pt]& \cong &\textrm{Hom}_{\unicode{x1D55C}} \left( \unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} \left(\textbf{b}, \textbf{n}\right) ,\textrm{Hom}_{\unicode{x1D55C}} \left(F \left(\textbf{n}\right), G \left(\textbf{b}\right)\right)\right).\end{eqnarray*}

Then, taking into account the functoriality with respect to $\textbf{FI}$ gives the first isomorphism below:

\begin{eqnarray*}\textrm{Hom}_{\mathcal{F}\!\left(\textbf{FI}^\textrm{op};\, \unicode{x1D55C}\right)} \left(F \left({-}\right),\textrm{Hom} _{\unicode{x1D55C}} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} \left(\textbf{b}, - \right) , G \left(\textbf{b}\right)\right)\right)&\cong &\textrm{Hom}_{\mathcal{F}\!\left(\textbf{FI};\, \unicode{x1D55C}\right)} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} \left(\textbf{b}, - \right), \right. \\&& \left.\textrm{Hom} _{\unicode{x1D55C}} \left( F\left({-}\right) , G \left(\textbf{b}\right)\right)\right)\\[4pt]&\cong &\textrm{Hom} _{\unicode{x1D55C}} \left( F\left(\textbf{b}\right) , G \left(\textbf{b}\right)\right),\end{eqnarray*}

the second isomorphism being given by Yoneda’s lemma.

Finally, taking into account the naturality with respect to ${\boldsymbol{\Sigma}}^\textrm{op}$ , one arrives at the required isomorphism.

One can identify the values taken by the functor $\uparrow\,:\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}) \to \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ by the following:

Proof. Consider the first statement. We have

\begin{equation*}\uparrow G(\textbf{n}) =\textrm{Hom}_{\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( - , \textbf{n}) , G ({-}))\right)\cong \bigoplus _{b\leq n}\textrm{Hom} _{\mathfrak{S}_b^\textrm{op}} \left(\unicode{x1D55C} \textrm{Hom}_{\textbf{FI}} ( \textbf{b} , \textbf{n}) , G (\textbf{b})\right).\end{equation*}

For $b, n\in \mathbb{N}$ , $\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n})$ is a free $\mathfrak{S}_b^\textrm{op}$ -set on the set of subsets $\textbf{n}^{\prime}\subseteq \textbf{n}$ of cardinal b. More precisely,

\begin{equation*}\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n})\cong \coprod _{\substack{\textbf{n}^{\prime} \subseteq \textbf{n}\\|\textbf{n}^{\prime}|=b}}\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n}^{\prime})\end{equation*}

where $\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n}^{\prime}) \cong \mathfrak{S}_b$ as a right $\mathfrak{S}_b^\textrm{op}$ -set. Hence:

\begin{equation*}\uparrow G(\textbf{n})\cong \bigoplus _{b\leq n} \bigoplus_{\substack{\textbf{n}^{\prime} \subseteq \textbf{n}\\ |\textbf{n}^{\prime}|=b}} G(\textbf{b})\cong \bigoplus _{b\leq n} \bigoplus_{\substack{\textbf{n}^{\prime} \subseteq \textbf{n}\\ |\textbf{n}^{\prime}|=b}} G(\textbf{n}^{\prime})\cong \bigoplus_{\textbf{n}^{\prime} \subseteq \textbf{n}} G(\textbf{n}^{\prime}).\end{equation*}

The second statement follows by considering the inclusion $\textrm{Hom}_{\textbf{FI}} ({-}, \textbf{l}) \hookrightarrow \textrm{Hom}_{\textbf{FI}} ({-}, \textbf{n})$ induced by f. Evaluated on $\textbf{b}$ , this corresponds to the inclusion

\begin{equation*}\coprod _{\substack{\textbf{l}^{\prime} \subseteq \textbf{l}\\ |\textbf{l}^{\prime}|=b}}\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{l}^{\prime})\hookrightarrow\coprod _{\substack{\textbf{n}^{\prime} \subseteq f(\textbf{l})\\ |\textbf{n}^{\prime}|=b}}\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n}^{\prime})\subseteq\coprod_{\substack{\textbf{n}^{\prime} \subseteq \textbf{n}\\ |\textbf{n}^{\prime}|=b}}\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{n}^{\prime}),\end{equation*}

where the first map sends $\textrm{Hom}_{\textbf{FI}} (\textbf{b}, \textbf{l}^{\prime})$ to $\textrm{Hom}_{\textbf{FI}} (\textbf{b}, f(\textbf{l}^{\prime}))$ by postcomposition with $f|_{\textbf{l}^{\prime}}$ .

The result follows.

Proof. The first two statements follow from Proposition 3.28.

Since ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ , we can apply (3.3) to this functor to obtain

\begin{equation*}{\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} \left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G\right) (\textbf{b}) \cong \bigoplus _{\textbf{b}^{\prime} \subseteq \textbf{b}} \left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G\right)(\textbf{b}^{\prime}).\end{equation*}

Applying (3.3) to each term ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G (\textbf{b}^{\prime}) $ gives the expression for ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G (\textbf{b}) $ .

The natural morphism $\Delta \,:\, {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G$ is induced by the $\textbf{FI}^\textrm{op}$ -module structure of $\uparrow G$ , which is given by Proposition 3.28 (2). This leads to the stated identification.

Remark 3.32. The decomposition of $ {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_0 {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 G$ (we label the functors for clarity) given in Proposition 3.31 (3) can be reindexed as follows:

\begin{equation*} {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_0 \left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 G\right) (\textbf{b})\cong \bigoplus_{\textbf{b}= \textbf{b}^{(1)}_\Sigma \amalg \textbf{b}^{\Sigma}_0 }{\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 G (\textbf{b}^{(1)}_\Sigma)\cong \bigoplus_{\textbf{b}= \left(\textbf{b}^{(2)} _\Sigma\amalg \textbf{b}^{\Sigma}_1\right) \amalg \textbf{b}^{\Sigma}_0 }G \left(\textbf{b}^{(2)}_\Sigma\right).\end{equation*}

The sum is indexed by ordered decompositions of $\textbf{b}$ into three subsets (possibly empty). With respect to the decomposition given in Proposition 3.31 (3), we have $\textbf{b}^{(2)}_\Sigma \,:\!=\, \textbf{b}^{\prime\prime}$ , $\textbf{b}^{\Sigma}_1\,:\!=\,\textbf{b}^{\prime}\backslash \textbf{b}^{\prime\prime}$ and $\textbf{b}^{\Sigma}_0\,:\!=\,\textbf{b}\backslash \textbf{b}^{\prime}$ . The indices in $\textbf{b}^{\Sigma}_0, \textbf{b}^{\Sigma}_1$ record from which application of ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ the subset $\textbf{b}^{\Sigma}_i$ arises. More generally, we have

(3.4) \begin{equation} {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_0 {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 \ldots {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_\ell G (\textbf{b})\cong \bigoplus_{\textbf{b}=\left.\left(\ldots \left(\left(\textbf{b}^{(\ell+1)}_\Sigma \amalg \textbf{b}^{\Sigma}_\ell\right) \amalg \textbf{b}^{\Sigma}_{\ell-1}\right)\right) \ldots \amalg \textbf{b}^{\Sigma}_1\right) \amalg \textbf{b}^{\Sigma}_0 }G \left(\textbf{b}^{(\ell+1)}_\Sigma\right).\end{equation}

The general theory of comonads associated with an adjunction implies that, if $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , the underlying object $F \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ has a canonical ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule structure.

Proposition 3.33. For $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , the ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule structure of the underlying object $\downarrow F \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ (denoted below simply by F) identifies with respect to the isomorphism (3.3) of Proposition 3.31 as follows. For $b \in \mathbb{N}$ , the structure morphism $\psi_F \,:\, F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F $ is given by

\begin{equation*}\psi _F \,:\, F (\textbf{b})\rightarrow \bigoplus_{\textbf{b}^{\prime} \subseteq \textbf{b}}F (\textbf{b}^{\prime})\end{equation*}

with component $F (\textbf{b}) \rightarrow F(\textbf{b}^{\prime}) $ given by the $\textbf{FI}^\textrm{op}$ structure of F for $\textbf{b}^{\prime} \subset \textbf{b}$ .

Proposition 3.33 gives rise to a functor:

\begin{equation*}\mathcal{F}\!\left(\textbf{FI}^\textrm{op};\, \unicode{x1D55C}\right) \to \mathcal{F}\!\left({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}\right)_{\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}.\end{equation*}

Proof. The functor $\mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) \to \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})_{\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ is fully faithful; to see that it is an equivalence, it suffices to show that it is essentially surjective.

By definition, a ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule is an object G of $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ equipped with a structure morphism $\psi _G\,:\, G \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G$ satisfying the comodule axioms. We claim that this provides G with the structure of an $\textbf{FI}^\textrm{op}$ -module exhibiting the ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule $(G, \psi_G)$ as lying in the essential image of $\mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) \to \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})_{\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ .

The underlying ${\boldsymbol{\Sigma}}^\textrm{op}$ -module determines the action of the isomorphisms of $\textbf{FI}^\textrm{op}$ . To define the morphism $G (\textbf{b}) \rightarrow G(\textbf{b}^{\prime})$ associated with a proper inclusion $\textbf{b}^{\prime} \subset \textbf{b}$ , we apply the structure morphism $\psi _G$ to $\textbf{b}$ :

\begin{equation*}(\psi _G)_\textbf{b}\,:\, G(\textbf{b}) \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G(\textbf{b}) \cong \bigoplus_{\textbf{b}^{\prime} \subseteq \textbf{b} } G (\textbf{b}^{\prime})\end{equation*}

where the last isomorphism is obtained using Proposition 3.31 (1). The morphism $G (\textbf{b}) \rightarrow G(\textbf{b}^{\prime})$ is the corresponding component of the morphism $(\psi _G)_\textbf{b}$ .

To verify that this induces a $\textbf{FI}^\textrm{op}$ -module structure, consider morphisms in $\textbf{FI}$ : $\textbf{b}^{\prime\prime} \xrightarrow{i^{\prime}} \textbf{b}^{\prime} \xrightarrow{i} \textbf{b}$ ; the relation $G(i \circ i^{\prime})=G(i^{\prime}) \circ G(i)$ follows from the coassociativity of the ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule structure if i and i ^′ are proper inclusions and the fact that $G \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G$ is a morphism of ${\boldsymbol{\Sigma}}^\textrm{op}$ -modules if i or i ^′ is an isomorphism.

Proposition 3.36 below shows that the comonad ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ is equipped with a natural transformation $\eta \,:\, \textrm{Id} \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} $ , which can be considered as a form of coaugmentation. This form of additional structure is considered in Appendix B.1; it is important that this satisfies Hypothesis B.5.

Proposition 3.36. There is a natural transformation $\eta \,:\, \textrm{Id} \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} $ of endofunctors of $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ , where the morphism $\eta_G \,:\, G(\textbf{b}) \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} G(\textbf{b}) $ , for $G \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ and $b \in \mathbb{N}$ , is given by

\begin{equation*}G(\textbf{b})\rightarrow \bigoplus_{\textbf{b}^{\prime} \subseteq \textbf{b}}G(\textbf{b}^{\prime}),\end{equation*}

the inclusion of the factor indexed by $\textbf{b}^{\prime}=\textbf{b}$ .

Hence, the following diagrams commute:

Thus, the structure $(\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C});\, {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}, \Delta, \varepsilon, \eta)$ satisfies Hypothesis B.5.

Proof. This is proved by direct verification, using Proposition 3.31.

4.1 The Koszul complex $\mathsf{Kz}\ F$

In this section, we construct, for an $\textbf{FI} ^\textrm{op}$ -module F, the complex $\mathsf{Kz}\ F$ in ${\boldsymbol{\Sigma}}^\textrm{op}$ -modules. To keep track of the signs in the construction, we introduce the following notation:

This construction is compatible with the action of the group $\mathfrak{S}_b$ :

Proof. The inclusion $F (\textbf{b} - \textbf{t})\otimes\mathsf{Or} (\textbf{t}^{\prime})\hookrightarrow(\mathsf{Kz}\ F) ^t(\textbf{b})$ corresponding to the summand indexed by $\textbf{b}-\textbf{t}$ is $(\mathfrak{S}_{b-t} \times \mathfrak{S}_t)^\textrm{op}$ -equivariant, thus induces up to a morphism of $\mathfrak{S}_b^\textrm{op}$ -modules. It is straightforward to check that this is an isomorphism, as required.

It remains to check that the morphism d is equivariant with respect to the $\mathfrak{S}_b^\textrm{op}$ action. This follows from the explicit formula for d given in Definition 4.5.

Proof. It suffices to show that d is a differential. Using the notation of Definition 4.5,

\begin{equation*}d^2 (Y \otimes \alpha)=\sum_{\{x, y \} \subset \textbf{b}^{(t)}}\iota^*_{x,y} Y\otimes ( (y \wedge x \wedge \alpha) + (x \wedge y \wedge \alpha)) ,\end{equation*}

where $\iota_{x,y}$ denotes the inclusion $\textbf{b}^{(t)} \backslash \{x, y \} \subset \textbf{b}^{(t)} $ . This is zero, as required.

The following proposition will be used in Section 6.

Proposition 4.8. The Koszul construction defines an exact functor

\begin{equation*}\mathsf{Kz}\ \,:\, \mathcal{F} (\textbf{FI} ^\textrm{op};\, \unicode{x1D55C}) \to \textrm{coCh}(\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})),\end{equation*}

where $\textrm{coCh}(\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C}))$ is the category of cochain complexes in $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

Proof. The functoriality is clear from the construction. Moreover, the functor $F \mapsto (\mathsf{Kz}\ F)^t (\textbf{b})$ as in Definition 4.5 is exact, since the underlying $\unicode{x1D55C}$ -module of each $\mathsf{Or} (\textbf{b} \backslash \textbf{b}^{(t)})$ is flat (more precisely, free of rank one). This implies that $\mathsf{Kz}\ $ is exact considered as a functor to cochain complexes.

4.2 The cosimplicial ${\boldsymbol{\Sigma}}^\textrm{op}$ -module $(\mathfrak{C}^\bullet F)^\textrm{op}$ and its normalised cochain complex

In this section, we identify the cosimplicial ${\boldsymbol{\Sigma}}^\textrm{op}$ -module $(\mathfrak{C}^\bullet F)^\textrm{op}$ and its associated normalised cochain complex $N (\mathfrak{C}^\bullet F)^\textrm{op}$ .

In order to describe $(\mathfrak{C}^\bullet F)^\textrm{op}$ , we will use the following decompositions, given by Remark 3.32 (3.4), distinguishing the indices of $({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} F (\textbf{b}) $ by a $\tilde{\ }$ :

\begin{equation*} \left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}\right)^{\ell+1} F (\textbf{b}) = {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_0 {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 \ldots {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_\ell F (\textbf{b})\cong \bigoplus_{\textbf{b}=\left.\left(\ldots \left(\left(\textbf{b}^{(\ell+1)}_\Sigma \amalg \textbf{b}^{\Sigma}_\ell\right) \amalg \textbf{b}^{\Sigma}_{\ell-1}\right)\right) \ldots \amalg \textbf{b}^{\Sigma}_1\right) \amalg \textbf{b}^{\Sigma}_0 }F \left(\textbf{b}^{(\ell+1)}_\Sigma\right)\end{equation*}

\begin{equation*} \left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}\right)^{\ell} F \left(\textbf{b}\right) = {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_0 {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_1 \ldots {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}_{\ell-1} F \left(\textbf{b}\right)\cong \bigoplus_{\textbf{b}=\left.\left(\ldots \left(\left(\tilde{\textbf{b}}^{\left(\ell\right)}_\Sigma \amalg \tilde{\textbf{b}}^{\Sigma}_{\ell-1}\right) \amalg \tilde{\textbf{b}}^{\Sigma}_{\ell-2}\right)\right) \ldots \amalg \tilde{\textbf{b}}^{\Sigma}_1\right) \amalg \tilde{\textbf{b}}^{\Sigma}_0 }F \left(\tilde{\textbf{b}}^{\left(\ell\right)}_\Sigma\right).\end{equation*}

The structure of $\mathfrak{C}^\bullet F$ is described in (4.1) and, by Notation 1.3, $(\mathfrak{C}^\bullet F)^\textrm{op}$ is the cosimplicial object with coface operators $\tilde{d}^i\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} F \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} F$ and codegeneracy operators $\tilde{s}^j\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} F \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} F$ satisfying $\tilde{s}^j=s^{\ell-j}$ and $\tilde{d}^j=d^{\ell+1-j}$ .

Proof. Using Notation 1.3 (recalled before the statement) we have $\tilde{s}^j=s^{\ell-j}$ and, by Proposition B.6, ${s}^{\ell-j}= ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell-j} \varepsilon^\Sigma_{({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{j}F} $ .

Using the decomposition of $({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{j+1} F (\textbf{b})$ given by Remark 3.32 (3.4), we have

\begin{equation*}\varepsilon^\Sigma_{\left({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}\right)^{j}F} \,:\, \bigoplus_{\textbf{b}=\left.\left(\ldots \left(\left(\textbf{b}^{\left(j+1\right)}_\Sigma \amalg \textbf{b}^{\Sigma}_j\right) \amalg \textbf{b}^{\Sigma}_{j-1}\right)\right) \ldots \amalg \textbf{b}^{\Sigma}_1\right) \amalg \textbf{b}^{\Sigma}_0 }F \left(\textbf{b}^{\left(j+1\right)}_\Sigma\right) \to \bigoplus_{\textbf{b}=\left.\left(\ldots \left(\left(\tilde{\textbf{b}}^{\left(j\right)}_\Sigma \amalg \tilde{\textbf{b}}^{\Sigma}_{j-1}\right) \amalg \tilde{\textbf{b}}^{\Sigma}_{j-2}\right)\right) \ldots \amalg \tilde{\textbf{b}}^{\Sigma}_1\right) \amalg \tilde{\textbf{b}}^{\Sigma}_0 }F \left(\tilde{\textbf{b}}^{\left(j\right)}_\Sigma\right).\end{equation*}

Proposition 3.31 (2) gives that, if $\textbf{b}^\Sigma_{j} \neq \emptyset$ , the restriction of $\varepsilon^\Sigma_{({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{j}F} $ to the factor indexed by $\textbf{b}= \textbf{b}_\Sigma^{(j+1)} \amalg \coprod_{i=0}^{j} \textbf{b}^{\Sigma}_{i}$ is zero; if $\textbf{b}^\Sigma_{j} = \emptyset$ , it is the projection onto the direct summand of $({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{j} F$ such that:

\begin{equation*}\tilde{\textbf{b}}^{(j)}_\Sigma= \textbf{b}^{(j +1)}_\Sigma \quad \text{and} \quad \tilde{\textbf{b}}^{\Sigma}_i= \textbf{b}^{\Sigma}_i \quad \text{for}\ i < j.\end{equation*}

The result for $\tilde{s}^j$ then follows by applying $({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell-j}$ .

For $\tilde{d}^i\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} F \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} F$ the computation is similar, using the general form of $d^i\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} F \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} F$ given in Proposition B.6, together with Propositions 3.31 (4), 3.33 and 3.36.

For the version of the normalised cochain complex used in the following, see Definition A.2.

Proposition 4.10. For $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , the associated normalised cochain complex $N (\mathfrak{C}^\bullet F)^\textrm{op}$ evaluated on $\textbf{b}$ , for $b \in \mathbb{N}$ , has terms:

\begin{equation*}(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b}) =\bigoplus_{\substack{\textbf{b} = \textbf{b}^{(t)} \amalg \coprod_{i=0}^{t -1} \textbf{b}_i^{(t)}\\ {\textbf{b}^{(t)}_i \neq \emptyset}}}F \left(\textbf{b}^{(t)}\right).\end{equation*}

In particular, $(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b}) =0$ if $t <0$ or if $t >|\textbf{b}|$ .

The differential $d\,:\, ( N (\mathfrak{C}^\bullet F)^\textrm{op})^t \rightarrow (N(\mathfrak{C}^\bullet F)^\textrm{op}) ^{t +1}$ is given by $\sum_{i=0}^{t }({-}1)^i \tilde{d}^i=({-}1)^{t+1}\sum_{i=1}^{t+1}({-}1)^i {d}^i$ .

Proof. The identification of the normalised subcomplex follows from the form of the codegeneracies $\tilde{s}^j$ given in Proposition 4.9. The vanishing statement follows immediately.

By Definition A.2, the differential on $N (\mathfrak{C}^\bullet F)^\textrm{op}$ is the restriction of that on the cochain complex associated with $(\mathfrak{C}^\bullet F)^\textrm{op}$ , that is, $\sum_{i=0}^{t + 1}({-}1)^i \tilde{d}^i$ . By Proposition 4.9 (2c) the restriction of $\tilde{d}^{t+1}$ to $N (\mathfrak{C}^\bullet F)^\textrm{op}$ is zero.

4.3 Relating the Koszul complex and the normalised cochain complex

In this section, we define a natural surjection from $N (\mathfrak{C}^\bullet F)^\textrm{op}$ to the Koszul complex $\mathsf{Kz}\ F$ . The proof that this surjection is a quasi-isomorphism is given in the subsequent sections.

To define this natural surjection, we use the following direct summand of the underlying $\mathbb{N}$ -graded object of the normalised complex $N(\mathfrak{C}^\bullet F)^\textrm{op}$ ; we stress that it is not in general a direct summand as a complex.

The $\mathfrak{S}_b^\textrm{op}$ -action on $\overline{(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b})}$ is described as follows, in a form suitable for comparison with the Koszul complex:

Proof. The first statement is clear.

The inclusion $\textbf{b}-\textbf{t} \subset \textbf{b}$ induces a $\mathfrak{S}_{b-t}^\textrm{op}$ -equivariant inclusion $F (\textbf{b}-\textbf{t} ) \hookrightarrow \overline{(N(\mathfrak{C}^\bullet F)^\textrm{op})^t (\textbf{b})}$ corresponding to the summand indexed by taking $\textbf{b}_i ^{(t)} = \{b-t+1+i\}$ for $0 \leq i \leq t-1$ . This induces a morphism of $\mathfrak{S}_b^\textrm{op}$ -modules, as in the statement. By inspection, this is an isomorphism.

Proposition 4.13. There is a surjection $N (\mathfrak{C}^\bullet F)^\textrm{op}\twoheadrightarrow\mathsf{Kz}\ F $ of cochain complexes in $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ given, for $t,b \in \mathbb{N}$ , as the composite

\begin{equation*}(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b})\twoheadrightarrow\overline{(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b})}\twoheadrightarrow(\mathsf{Kz}\ F)^t(\textbf{b}) ,\end{equation*}

where the first morphism is the projection of Proposition 4.12 and the second is

\begin{equation*}\overline{(N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b})}\cong \big(F (\textbf{b} - \textbf{t})\otimes\unicode{x1D55C} \textrm{Aut} (\textbf{t}^{\prime})\big)\uparrow _{\mathfrak{S}_{b-t}^\textrm{op} \times \mathfrak{S}_t ^\textrm{op}}^{\mathfrak{S}_b^\textrm{op}}\twoheadrightarrow\big(F(\textbf{b} - \textbf{t})\otimes\mathsf{Or} (\textbf{t}^{\prime})\big)\uparrow _{\mathfrak{S}_{b-t}^\textrm{op} \times \mathfrak{S}_t ^\textrm{op}}^{\mathfrak{S}_b^\textrm{op}}\cong (\mathsf{Kz}\ F)^t (\textbf{b}),\end{equation*}

induced by the surjection $\unicode{x1D55C} \textrm{Aut} (\textbf{t}^{\prime})\twoheadrightarrow\mathsf{Or} (\textbf{t}^{\prime})$ of $\mathfrak{S}_t^\textrm{op}$ -modules that sends the generator $[\textrm{id}_{\textbf{t}^{\prime}}]$ to $\omega (\textbf{t}^{\prime})$ .

Explicitly, an element $Y \in F (\textbf{b}^{(t)})$ that represents a class of $(N (\mathfrak{C}^\bullet F)^\textrm{op})^t (\textbf{b})$ in the summand indexed by $\left(\textbf{b}^{(t)};\, \textbf{b}_0^{(t)} , \ldots , \textbf{b}_{t-1} ^{(t)}\right)$ in the decomposition given in Proposition 4.10 is sent to zero, unless $|\textbf{b}_i ^{(t)}|=1$ for all i, $0 \leq i \leq t-1$ , when it is sent to

\begin{equation*}Y \otimes \left(\textbf{b}_0^{(t)} \wedge \ldots \wedge\textbf{b}_{t-1}^{(t)} \right)\in F\left(\textbf{b}^{(t)}\right) \otimes \mathsf{Or} \left(\textbf{b}\backslash \textbf{b}^{(t)}\right) \subset(\mathsf{Kz}\ F)^t (\textbf{b}).\end{equation*}

Proof. Lemma 4.6 together with Proposition 4.12 implies that the morphism $N (\mathfrak{C}^\bullet F)^\textrm{op} \twoheadrightarrow \mathsf{Kz}\ F$ is a surjection of $\mathbb{N}$ -graded ${\boldsymbol{\Sigma}}^\textrm{op}$ -modules.

It remains to prove that this surjection is compatible with the respective differentials. First observe that, in general, the kernel of the projection

\begin{equation*}(N (\mathfrak{C}^\bullet F)^\textrm{op}) ^* (\textbf{b}) \twoheadrightarrow\overline{(N (\mathfrak{C}^\bullet F)^\textrm{op}) ^* (\textbf{b})}\end{equation*}

is not stable under the differential of $N (\mathfrak{C}^\bullet F)^\textrm{op}$ . This is due to the fact that a coface map of $(\mathfrak{C}^\bullet F)^\textrm{op} (\textbf{b}) $ can split a set $\textbf{b}_i^{(t)}$ with two elements $\{u, v\}$ into the ordered decompositions as singletons $(\{u\}, \{v \})$ and $(\{v\}, \{u\})$ . However, these contributions will vanish on passing to $\mathsf{Kz}\ F$ , due to the relation $u \wedge v = - v \wedge u$ . Using the previous observation, one verifies that the kernel (denoted here simply by $\ker$ ) of the morphism $(N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) \twoheadrightarrow (\mathsf{Kz}\ F)^* (\textbf{b})$ is stable under the differential.

The quotient differential on $ (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) /\ker $ is induced by the coface operator $\tilde{d}^0$ on $(\mathfrak{C}^\bullet F)^\textrm{op}(\textbf{b})$ . The operator $\tilde{d}^0$ is described in Proposition 4.9 (2a).

We claim that this quotient complex is isomorphic to $(\mathsf{Kz}\ F)^* (\textbf{b})$ via the given morphism. For instance, consider the differential from cohomological degree zero to degree one. This is induced by the structure morphism $F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F$ and, in the quotient complex, identifies with

\begin{equation*}F( \textbf{b})\rightarrow \bigoplus_{\substack{\textbf{b}^{\prime}\subset \textbf{b} \\ \left|\textbf{b}\backslash \textbf{b}^{\prime} \right|=1}}F(\textbf{b}^{\prime})\end{equation*}

given by the restriction morphisms $F (\textbf{b} ) \rightarrow F(\textbf{b}^{\prime})$ associated with the inclusions. This identifies with the differential of the Koszul complex.

The general case follows similarly.

4.4 The complex of ordered partitions

The key to proving Theorem 4.1, that is, that $N(\mathfrak{C}^\bullet F)^\textrm{op} \twoheadrightarrow \mathsf{Kz}\ F$ is a quasi-isomorphism, is the case $F= \unicode{x1D7D9} \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , where $\unicode{x1D7D9}$ is the following $\textbf{FI}^\textrm{op}$ -module:

The aim of this section is to give an explicit description of the complex $N (\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op}$ in terms of the complex given by ordered partitions (see Proposition 4.23) described below.

Henceforth in this section, X denotes a non-empty finite set.

Consider the refinement of ordered partitions as follows:

Definition 4.16. Let $\mathfrak{p}= (\mathfrak{p}_1, \ldots , \mathfrak{p}_t)\in \mathfrak{Part}_t (X)$ and $\mathfrak{q} = (\mathfrak{q}_1, \ldots , \mathfrak{q}_s) \in \mathfrak{Part}_s (X)$ , then $\mathfrak{q}$ is an ordered refinement of $\mathfrak{p}$ , denoted $\mathfrak{q} \leq \mathfrak{p}$ , if there exists an order-preserving surjection $\alpha \,:\, \textbf{s} \twoheadrightarrow \textbf{t}$ such that

\begin{equation*}\mathfrak{p}_i = \bigcup_{j \in \alpha^{-1} (i)} \mathfrak{q}_j.\end{equation*}

(In particular, this requires that $s \geq t$ , with equality if and only if $\mathfrak{p} = \mathfrak{q}$ .)

The following is clear:

The following lemma highlights properties of the poset $(\mathfrak{Part}_* (X), \leq )$ that are exploited in the proof of Proposition 4.27.

Proof. For the forward implication of the first point, suppose that there exists $\mathfrak{q} \in \mathfrak{Part}_{|X|-1}(X)$ such that $\mathfrak{p}\leq \mathfrak{q}$ and $\mathfrak{p}^{\prime}\leq \mathfrak{q}$ . By Definition 4.16, we have order-preserving surjections $\alpha\,:\, X\to |X|-1$ and $\alpha^{\prime}\,:\, X\to |X|-1$ such that

(4.2) \begin{equation}\mathfrak{q}_k = \coprod _{j \in \alpha^{-1} (k)} \mathfrak{p}_j \qquad \textrm{and} \qquad \mathfrak{q}_k = \coprod_{j \in \alpha^{\prime-1} (k)} \mathfrak{p}^{\prime}_j.\end{equation}

Since $\alpha$ and $\alpha^{\prime}$ are order-preserving surjections there is $i, j\in \{1, \ldots, |X|-1\}$ such that $\alpha^{-1}(i) = \{i , i+1 \}$ and $\alpha^{\prime-1}(j) = \{j , j+1 \}$ with all other fibres of cardinal one. We deduce from (4.2) that $i=j$ and $\mathfrak{q}_i = \mathfrak{p}_i \amalg \mathfrak{p}_{i+1}= \mathfrak{p}^{\prime}_i \amalg \mathfrak{p}^{\prime}_{i+1}$ so $\mathfrak{p}_i=\mathfrak{p}^{\prime}_{i+1}$ , $\mathfrak{p}_{i+1}=\mathfrak{p}^{\prime}_i$ and $\mathfrak{p}_j=\mathfrak{p}^{\prime}_j$ for $j \not\in \{i,i+1\}$ .

For the converse, pick $\mathfrak{p}$ and $\mathfrak{p}^{\prime}$ as in the statement. The following partition $\mathfrak{q} \in \mathfrak{Part}_{|X|-1}(X)$ satisfies the inequalities of the statement:

\begin{equation*}\mathfrak{q}_j=\mathfrak{p}_j\ \textrm{ for }\ 1\leq j \leq i-1,\ \mathfrak{q}_i=\mathfrak{p}_i \amalg \mathfrak{p}_{i+1},\ \mathfrak{q}_j=\mathfrak{p}_{j-1}\ \textrm{ for } \ i+1\leq j \leq |X|-1.\end{equation*}

For the second point, by hypothesis, $\mathfrak{r} \leq \mathfrak{p}$ . By Definition 4.16, this corresponds to giving $\mathfrak{r}$ together with an order-preserving surjection $\alpha \,:\, \textbf{t}+\textbf{2} \twoheadrightarrow \textbf{t}$ . To prove the result, it suffices to show that there are precisely two possible factorisations of $\alpha$ via order-preserving surjections:

\begin{equation*}\textbf{t}+\textbf{2} \stackrel{\alpha^{\prime}}{\twoheadrightarrow} \textbf{t}+\textbf{1} \stackrel{\alpha^{\prime\prime}}{\twoheadrightarrow}\textbf{t}.\end{equation*}

The factorisation condition implies that $\alpha^{\prime\prime}$ is uniquely determined by $\alpha^{\prime}$ .

There are two possibilities:

1. (1) there exists $i \in \textbf{t}$ such that $|\alpha^{-1}(i)|=3$ , all other fibres having cardinal one;

2. (2) there exists $i<j \in \textbf{t}$ such that $|\alpha^{-1}(i)|=|\alpha^{-1}(j)|=2$ , with all other fibres of cardinal one.

In the first case, one has either $\alpha^{\prime-1}(i) = \{i , i+1 \}$ or $\alpha^{\prime-1}(i+1) = \{i+1 , i+2 \}$ , with all other fibres of cardinal one. In the second case, either $\alpha^{\prime-1}(i) = \{i , i+1 \}$ or $\alpha^{\prime-1}(j) = \{j , j+1 \}$ , again with all other fibres of cardinal one.

Definition 4.20. Let $C_{\mathsf{perm}} (X)$ denote the cochain complex with

\begin{equation*}C_{\mathsf{perm}} (X)^t \,:\!=\,\left\{\begin{array}{l@{\quad}l}\unicode{x1D55C} [\mathfrak{Part}_{t} X] & 1 \leq t \leq |X|\\[5pt]0 & \mbox{otherwise}.\end{array}\right.\end{equation*}

The differential $d \,:\, C_{\mathsf{perm}} (X)^t \rightarrow C_{\mathsf{perm}} (X)^{t+1}$ is the alternating sum $\sum_{i=1}^t ({-}1)^i \delta^i$ , where $\delta^i$ sends a generator $[\mathfrak{p}]$ corresponding to the ordered partition $\mathfrak{p}= (\mathfrak{p}_1, \ldots, \mathfrak{p}_i, \ldots , \mathfrak{p}_{t} ) $ to the sum of ordered partitions of the form $(\mathfrak{p}_1, \ldots, \mathfrak{p}^{\prime}_i, \mathfrak{p}^{\prime\prime}_i, \ldots , \mathfrak{p}_{t} )$ where $\mathfrak{p}_i = \mathfrak{p}^{\prime}_i \amalg \mathfrak{p}^{\prime\prime}_i$ .

Proposition 4.23. For X a non-empty finite set, there is an isomorphism of complexes

\begin{equation*}(N (\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op}) (X)\cong C_{\mathsf{perm}} (X).\end{equation*}

Proof. By Proposition 4.10 and Notation 4.14, since $\unicode{x1D7D9}$ vanishes on non-empty sets, we have

\begin{equation*}\left(N \left(\mathfrak{C}^\bullet \unicode{x1D7D9}\right)^\textrm{op}\right)^t (X)=\bigoplus_{\substack{X= \textbf{b}^{(t)} \amalg \coprod_{i=0}^{t -1} \textbf{b}_i^{(t)}\\{\textbf{b}^{(t)}_i \neq \emptyset}}}\unicode{x1D7D9} (\textbf{b}^{(t)})=\bigoplus_{\substack{X= \coprod_{i=0}^{t -1} \textbf{b}_i^{(t)}\\{\textbf{b}^{(t)}_i \neq \emptyset}}}\unicode{x1D55C}\cong \unicode{x1D55C} [\mathfrak{Part}_{t} X]\end{equation*}

and the coface $\tilde{d}^0\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} \unicode{x1D7D9} \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} \unicode{x1D7D9} $ is zero. So, using Proposition 4.10, the differential of $(N (\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op})$ is given by $\sum_{i=1}^{t }({-}1)^i \tilde{d}^i$ .

For $1\leq i\leq t$ , comparing the definition of $\delta^i$ given in Definition 4.20 and the explicit description of $\tilde{d}^i\,:\, ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell} \unicode{x1D7D9} \rightarrow ({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}})^{\ell+1} \unicode{x1D7D9}$ given in Proposition 4.9, we obtain that $\tilde{d}^i$ corresponds, via the above isomorphism, to $\delta^i$ .

4.5 Cohomology of the complex of ordered partitions

Let X be a non-empty finite set. To calculate the cohomology of $C_{\mathsf{perm}} (X)$ , we relate it to a cellular complex as follows. The permutohedron $\Pi_X$ is an (abstract) $(|X|-1)$ -dimensional polytope, with k-faces in bijection with $\mathfrak{Part}_{|X|-k} X$ (see [Reference Ziegler12, Example 0.10]). In particular, the vertices (i.e., 0-faces) are indexed by elements of $\mathfrak{Part}_{|X|} (X)$ (i.e., by permutations of the set X).

We record the necessary information on the face inclusions of $\Pi_X$ :

Lemma 4.19 can be rephrased as follows:

The polytope $\Pi_X$ has a geometric realisation $|\Pi_X|$ as the convex hull in $\mathbb{R}^{|X|}$ of the vectors with pairwise distinct coordinates from $\{1, 2, \ldots, |X| \}$ ; in particular, $|\Pi_X|$ is contractible. Moreover, by the above, the geometric polytope $|\Pi_X|$ is equipped with a cellular structure with the cells of dimension k indexed by the elements of $\mathfrak{Part}_{|X|-k} (X)$ , for $0\leq k \leq |X|-1$ . The cellular structure is understood using Lemma 4.24.

Example 4.26. The geometric realisation of the permutohedron $\Pi_{\{1, 2, 3\}}$ is a hexagon with vertices indexed by the elements of $\mathfrak{S}_3$ :

where, for notational clarity, the braces have been omitted from singletons.

In this case, the 0-faces are the vertices, the 1-faces are the edges and the 2-face is the interior of the hexagon. Note that a 0-face lies in precisely two 1-faces.

In the following proposition, $C_{\mathsf{perm}} (X)$ is considered with homological degree and $[-|X|]$ denotes the shift of homological degree (i.e., $C_{\mathsf{perm}} (X)[-|X|]_t=C_{\mathsf{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 chain complexes

\begin{equation*}\gamma_X \,:\,C_{\mathsf{perm}} (X)[-|X|]_*\stackrel{\cong }{\rightarrow }C^{\textrm{cell}}_*\left(|\Pi_X|;\, \unicode{x1D55C}\right),\end{equation*}

where $C^{\textrm{cell}}_*(|\Pi_X|;\, \unicode{x1D55C})$ is the cellular complex of $|\Pi_X|$ with coefficients in $\unicode{x1D55C}$ .

In particular, $C_{\mathsf{perm}}(X)$ has cohomology $\unicode{x1D55C}$ concentrated in degree $|X|$ .

Proof. The case $|X|=1$ is clear, hence we suppose that $|X| \geq 2$ .

The cellular complex $C^{\textrm{cell}}_*(|\Pi_X|;\, \unicode{x1D55C})$ has underlying graded $\unicode{x1D55C}$ -module which is free on $\mathfrak{Part}_* (X)$ , so that there is an isomorphism of $\unicode{x1D55C}$ -modules

(4.3) \begin{equation}C_{\mathsf{perm}} (X)^{|X|-t}\cong C^{\textrm{cell}}_t(|\Pi_X|;\, \unicode{x1D55C}).\end{equation}

Moreover, for $[\mathfrak{p}]_{\textrm{cell}}$ the generator corresponding to $\mathfrak{p} \in \mathfrak{Part}_t (X)$ , $d [\mathfrak{p}]_{\textrm{cell}}$ is a signed sum of the generators $[\mathfrak{q}]_{\textrm{cell}}$ , where $\mathfrak{q} \in \mathfrak{Part}_{t+1} (X)$ and $\mathfrak{q} \leq \mathfrak{p}$ . (Here and in the following, the subscripts indicate in which complex the generators live.) With the regrading used here, this is analogous to the 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

\begin{eqnarray*}C_{\mathsf{perm}} (X) ^{|X|} &\rightarrow & C^{\textrm{cell}}_0(|\Pi_X|;\, \unicode{x1D55C})\\[4pt]\ [\sigma]_{\textrm{perm}} & \mapsto & \mathsf{sign} (\sigma)[\sigma]_{\textrm{cell}},\end{eqnarray*}

using the identification of ordered partitions of X into singletons with the symmetric group $\textrm{Aut} (X)$ to define $\mathsf{sign}(\sigma)$ .

To prove the statement, we consider the cellular filtration of $|\Pi_X|$ . More precisely, we prove, by induction on n, that we have an isomorphism of chain complexes

such that, at each degree, $\gamma_X [\mathfrak{p}]_{\textrm{perm}} = \pm [\mathfrak{p}]_{\textrm{cell}}$ and where $\tau_{\leq n}C_*$ is the truncation of the complex $C_*$ defined by $(\tau_{\leq n}C_*)_i=0 $ if $i>n$ and $(\tau_{\leq n}C_*)_i=C_i $ if $i\leq n$ .

To prove the initial step, note that the 1-skeleton of $|\Pi_X|$ can be given the structure of an oriented graph, where each edge is oriented from the vertex with negative signature to that of 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 the associated chain complex.

It follows that the identity on $\mathfrak{Part}_{|X|-1} (X)$ induces an isomorphism of chain complexes $\gamma_X\,:\, \tau_{\leq 1}C_{\mathsf{perm}} (X)[-|X|]_*\to \tau_{\leq 1}C^{\textrm{cell}}_*(|\Pi_X|;\, \unicode{x1D55C})$ :

For $n\geq 2$ assume that $\gamma_X\,:\, \tau_{\leq n-1}C_{\mathsf{perm}} (X)[-|X|]_*\to \tau_{\leq n-1}C^{\textrm{cell}}_*(|\Pi_X|;\, \unicode{x1D55C})$ is defined and is an isomorphism. To prove the inductive step, choose $\mathfrak{p} \in \mathfrak{Part}_{|X|-n} (X)$ ; as observed above, the differential of the generator $[\mathfrak{p}]_{\textrm{cell}} \in C^{\textrm{cell}}_n(|\Pi_X|;\, \unicode{x1D55C})$ is a signed sum of the generators $[\mathfrak{q}]_{\textrm{cell}}$ corresponding to its $(n-1)$ -faces:

\begin{equation*}d [\mathfrak{p}]_{\textrm{cell}} = \sum_{\substack{\mathfrak{q} \in \mathfrak{Part}_{|X|-n+1} (X) \\ \mathfrak{q}\leq \mathfrak{p}}} \eta_{\mathfrak{q}} [\mathfrak{q}]_{\textrm{cell}},\end{equation*}

with $\eta_{\mathfrak{q}}\in \{\pm 1 \}.$

Now, by the inductive hypothesis, we have an isomorphism $\gamma_X \,:\, C_{\mathsf{perm}} (X)^{|X|-n+1} \stackrel{\cong }{\rightarrow } C_{n-1}^{\textrm{cell}}(|\Pi_X|;\, \unicode{x1D55C}) $ that is compatible with the differential and which has the form $[\mathfrak{q}]_{\textrm{perm}} \mapsto \pm [\mathfrak{q}] _{\textrm{cell}}$ . Hence, by Remark 4.21:

\begin{equation*}\gamma_Xd[\mathfrak{p}]_{\textrm{perm}}= \sum_{\substack{\mathfrak{q} \in \mathfrak{Part}_{|X|-n+1} (X) \\ \mathfrak{q}\leq \mathfrak{p}}} \alpha_{\mathfrak{q}} [\mathfrak{q}]_{\textrm{cell}},\end{equation*}

with $\alpha_{\mathfrak{q}}\in \{\pm 1 \}.$ ; moreover, $\gamma_Xd[\mathfrak{p}]_{\textrm{perm}}$ is a cycle.

We claim that $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} =\varepsilon _{\mathfrak{p}} d[\mathfrak{p}]_{\textrm{cell}}$ , for some $\varepsilon_{\mathfrak{p}} \in \{ \pm 1 \}$ . If $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} = d[\mathfrak{p}]_{\textrm{cell}}$ , take $\varepsilon _{\mathfrak{p}}=1$ . Otherwise, consider

\begin{equation*} \gamma_X d[\mathfrak{p}]_{\textrm{perm}} + d[\mathfrak{p}]_{\textrm{cell}}=\sum_{\substack{\mathfrak{q} \in \mathfrak{Part}_{|X|-n+1} (X) \\ \mathfrak{q}\leq \mathfrak{p}}} \kappa_{\mathfrak{q}}[\mathfrak{q}]_{\textrm{cell}}, \end{equation*}

where $\kappa_{\mathfrak{q}} = \alpha_{\mathfrak{q}}+\eta_ {\mathfrak{q}}$ .

By the hypothesis, there is some $\mathfrak{q} \in \mathfrak{Part}_{|X|-n+1} (X)$ such that $\kappa_{\mathfrak{q}} =0$ .

If $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} + d[\mathfrak{p}]_{\textrm{cell}} \not =0$ , there is some $\mathfrak{q}^{\prime}$ such that $\kappa_{\mathfrak{q}^{\prime}}\not =0$ . Using Lemma 4.25 (2), we can find such a pair $(\mathfrak{q}, \mathfrak{q}^{\prime})$ such that $\mathfrak{q}$ and $\mathfrak{q}^{\prime}$ have a $(n-2)$ -face in common, denoted by $\mathfrak{q} \cap \mathfrak{q}^{\prime}$ ; moreover, $\mathfrak{q}, \mathfrak{q}^{\prime}$ are the only $(n-1)$ -faces of which $\mathfrak{q} \cap \mathfrak{q}^{\prime}$ is a face. We deduce that $d(\mathfrak{q} \cap \mathfrak{q}^{\prime}) \not = 0$ : this contradicts the fact that $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} + d[\mathfrak{p}]_{\textrm{cell}}$ is a cycle, therefore $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} + d[\mathfrak{p}]_{\textrm{cell}}=0$ . Taking $\varepsilon _{\mathfrak{p}}=-1$ in this case establishes the claim.

By the uniqueness property of the cycle $d[\mathfrak{p}]_{\textrm{cell}}$ explained above, it follows that $ \gamma_X d[\mathfrak{p}]_{\textrm{perm}} =\varepsilon _{\mathfrak{p}} d[\mathfrak{p}]_{\textrm{cell}}$ , where $\varepsilon_{\mathfrak{p}} \in \{ \pm 1 \}$ . Thus, an extension of $\gamma_X$ compatible with the differential and of the required form is given by defining $\gamma_X [\mathfrak{p}]_{\textrm{perm}}\,:\!=\, \varepsilon _{\mathfrak{p}} [\mathfrak{p}]_{\textrm{cell}}$ . Applying this argument for all cells $\mathfrak{p}$ of dimension n completes the inductive step.

The isomorphism of chain complexes $\gamma_X$ induces an isomorphism in homology. Now $|\Pi_X|$ is contractible, hence has homology $\unicode{x1D55C}$ concentrated in degree zero. Returning to the cohomological, unshifted grading, this implies that $C_{\mathsf{perm}} (X)$ has cohomology isomorphic to $\unicode{x1D55C}$ concentrated in cohomological degree $|X|$ .

Corollary 4.28. There is an isomorphism of cohomologically graded ${\boldsymbol{\Sigma}}^\textrm{op}$ -modules

\begin{equation*}H^* (N (\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op})\cong \mathsf{Or}.\end{equation*}

Proof. This follows from Propositions 4.23 and 4.27, noting that the result holds evaluated on $\textbf{0}$ by inspection. For X a non-empty finite set, the isomorphism in cohomology is induced by the morphism of $\unicode{x1D55C}$ -modules $C_{\mathsf{perm}} (X)^{|X|}\rightarrow \unicode{x1D55C} $ given by $[\sigma] \mapsto \mathsf{sign} (\sigma)$ .

4.6 Proof of Theorem 4.1

In this section, we pass from the case $\unicode{x1D7D9}$ treated in Corollary 4.28 to that of an arbitrary $\textbf{FI}^\textrm{op}$ -module F.

The complex $(N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) $ has a finite filtration given as follows:

Lemma 4.29. Let F be an $\textbf{FI}^\textrm{op}$ -module and $b \in\mathbb{N}$ . For $s \in \mathbb{N}$ , let $\mathfrak{f}^s (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) \subset(N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) $ be the graded $\unicode{x1D55C}$ -submodule:

\begin{equation*}\mathfrak{f}^s (N(\mathfrak{C}^\bullet F)^\textrm{op}) ^t (\textbf{b}) =\bigoplus_{\substack{\textbf{b} = \textbf{b}^{(t)} \amalg \coprod_{i=0}^{t -1} \textbf{b}_i^{(t)}\\ {\textbf{b}^{(t)}_i \neq \emptyset}\\\left|\textbf{b}^{(t)} \right|\leq b-s}}F \left(\textbf{b}^{(t)}\right).\end{equation*}

Then $\mathfrak{f}^s (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) $ is a subcomplex and there is a finite filtration

\begin{equation*}0\subset\mathfrak{f}^b (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})\subset\mathfrak{f}^{b-1} (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})\subset \ldots \subset\mathfrak{f}^0 (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})=(N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}).\end{equation*}

Moreover, the quotient complex $\mathfrak{f}^s (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})/\mathfrak{f}^{s+1} (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})$ decomposes as the direct sum of complexes

\begin{equation*}\bigoplus_{\substack{\textbf{b}^{\prime} \subset \textbf{b} \\ \left|\textbf{b}^{\prime}\right|=b-s}}F(\textbf{b}^{\prime})\otimes(N(\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op})^* (\textbf{b}\backslash \textbf{b}^{\prime}).\end{equation*}

Proof. From the construction of $\mathfrak{C}^\bullet F$ , it is clear that $\mathfrak{f}^s (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b}) $ is stable under the differential and that one has a finite filtration as given.

The subquotient $\mathfrak{f}^s /\mathfrak{f}^{s+1}$ only has contributions from terms with $|\textbf{b}^{\prime} | = b-s$ . Moreover, by construction, the differential behaves as though all non-isomorphisms of $\textbf{FI}^\textrm{op}$ act as zero. By inspection, the subquotient identifies as stated.

Proof of Theorem 4.1. The natural surjection $N (\mathfrak{C}^\bullet F)^\textrm{op}\twoheadrightarrow\mathsf{Kz}\ F $ of cochain complexes in $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ is given by Proposition 4.13. It suffices to show that $(N (\mathfrak{C} ^\bullet F)^\textrm{op})^* (\textbf{b})\twoheadrightarrow(\mathsf{Kz}\ F)^*(\textbf{b})$ is a quasi-isomorphism, for each $b \in \mathbb{N}$ . This follows from the spectral sequence associated with the filtration $\mathfrak{f}^\bullet (N (\mathfrak{C}^\bullet F)^\textrm{op})^* (\textbf{b})$ that calculates the cohomology of $(N(\mathfrak{C}^\bullet F)^\textrm{op})^*(\textbf{b})$ . By Lemma 4.29, the $E_0$ -page is given by

\begin{equation*}E_0^{s,t}=\mathfrak{f}^s (N (\mathfrak{C}^\bullet F)^\textrm{op})^{s+t} (\textbf{b})/\mathfrak{f}^{s+1} (N (\mathfrak{C}^\bullet F)^\textrm{op})^{s+t} (\textbf{b})\cong \bigoplus_{\substack{\textbf{b}^{\prime} \subset \textbf{b} \\ \left|\textbf{b}^{\prime}\right|=b-s}}F(\textbf{b}^{\prime})\otimes\left(N(\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op}\right)^{s+t} (\textbf{b}\backslash \textbf{b}^{\prime}).\end{equation*}

The $E_1$ -page is calculated using Corollary 4.28 to identify the cohomology of each complex $(N(\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op})^* (\textbf{b}\backslash \textbf{b}^{\prime})$ and by applying universal coefficients. In particular, for $|\textbf{b}^{\prime}|=b-s$ the cohomology of $(N(\mathfrak{C}^\bullet \unicode{x1D7D9})^\textrm{op})^* (\textbf{b}\backslash \textbf{b}^{\prime})$ is concentrated in degree s.

It follows that the $E_1$ -page is concentrated in the line $t=0$ . This line, equipped with the differential $d_1$ , identifies with the complex $(\mathsf{Kz}\ F)^*(\textbf{b})$ . The spectral sequence degenerates at the $E^2$ -page, since there is no space for differentials. The result follows.

4.7 Relating to $\textbf{FI}^\textrm{op}$ -cohomology

In [Reference Church and Ellenberg1,Reference Gan2], the authors define the $\textbf{FI}$ -homology of a $\textbf{FI}$ -module F as the left derived functors of a right exact functor $H_0^{\textbf{FI}}$ . The functor $H_0^{\textbf{FI}}$ can be defined as the left adjoint of the functor $\mathcal{F} ({\boldsymbol{\Sigma}};\, \unicode{x1D55C}) \rightarrow \mathcal{F} (\textbf{FI};\, \unicode{x1D55C})$ given by extension by zero on morphisms (see Definition 4.30 for a precise definition in the dual case).

In this section, we consider the dual situation to define the $\textbf{FI}^\textrm{op}$ -cohomology of a $\textbf{FI}^\textrm{op}$ -module F. This allows us to rephrase Theorem 4.1 in Corollary 4.36, using $\textbf{FI}^\textrm{op}$ -cohomology. Theorem 6.15 can also be reinterpreted as the computation of the $\textbf{FI}^\textrm{op}$ -cohomology of the $\textbf{FI}^\textrm{op}$ -module $\unicode{x1D55C} \textrm{Hom}_{\boldsymbol{\Omega}} ({-}, \textbf{a})$ considered in Proposition 5.21 (see Theorem 4).

By Theorem 3.34, a $\textbf{FI}^\textrm{op}$ -module F can be considered equivalently as a ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ -comodule in $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ , with structure morphism $\psi_F \,:\, F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F$ . Moreover, by Proposition 3.36, one has the natural coaugmentation $\eta_F \,:\, F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F$ .

The following constructions should be compared with the general framework that is given in Appendix B.2, working with the category of $\textbf{FI}^\textrm{op}$ -modules, which is abelian and has enough injectives (see below for the latter).

Definition 4.30. (Cf. Example B.8.) For $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , let $H^0_{\textbf{FI}^\textrm{op}}(F) \in \mathsf{Ob}\, \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ be the equaliser of the diagram

which defines a functor $H^0_{\textbf{FI}^\textrm{op}} \,:\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C}) \rightarrow \mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

Remark 4.31. Using the description of $\psi_F \,:\, F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F$ given in Proposition 3.33 and of $\eta_F \,:\, F \rightarrow {\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}} F$ given in Proposition 3.36, for $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ and $b \in \mathbb{N}$ , one has the explicit identification:

\begin{equation*}H^0_{\textbf{FI}^\textrm{op}}(F)(\textbf{b})=\left\{x \in F(\textbf{b})\ |\ \forall i\,:\, \textbf{b}^{\prime} \subsetneq \textbf{b}, \quad F(i)(x)=0 \right\}.\end{equation*}

Proof. One checks that, for F an $\textbf{FI}^\textrm{op}$ -module, $H^0_{\textbf{FI}^\textrm{op}}F$ is the largest sub $\textbf{FI}^\textrm{op}$ -module that lies in the image of $\textbf{Z}$ , hence $H^0_{\textbf{FI}^\textrm{op}}$ is the right adjoint to $\textbf{Z}$ . This implies, in particular, that $H^0_{\textbf{FI}^\textrm{op}}$ is left exact.

It is a standard fact that the category $\mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ has enough injectives. Explicitly, for M an injective $\unicode{x1D55C}$ -module and $\textbf{n} \in \mathsf{Ob}\, \textbf{FI}^\textrm{op}$ , the functor $I_{\textbf{n},M}^{\textbf{FI}^\textrm{op}} \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ given by the maps from ${\textrm{Hom}_{\textbf{FI} ^\textrm{op}} ({-}, \textbf{n})}={\textrm{Hom}_{\textbf{FI}} (\textbf{n}, -)}$ to M, denoted by $M^{\textrm{Hom}_{\textbf{FI}} (\textbf{n}, -)}$ , is an injective $\textbf{FI}^\textrm{op}$ -module. This follows from the natural isomorphism for $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ given by Yoneda’s lemma:

\begin{equation*}\textrm{Hom}_{\mathcal{F}\!\left(\textbf{FI}^\textrm{op};\, \unicode{x1D55C}\right)}\left(F,I_{\textbf{n},M}^{\textbf{FI}^\textrm{op}}\right) \simeq \textrm{Hom}_{\unicode{x1D55C}} (F(\textbf{n}) ,M),\end{equation*}

which shows that $\textrm{Hom}_{\mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})}\left({-},I_{\textbf{n},M}^{\textbf{FI}^\textrm{op}}\right)$ is exact, since M is injective. Then, since the category of $\unicode{x1D55C}$ -modules has enough injectives, one deduces that $\mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ does also.

Thus, one can define $\textbf{FI}^\textrm{op}$ -cohomology as follows:

Proposition 3.36 shows that the coaugmentation $\eta$ of ${\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}$ satisfies Hypothesis B.5, hence one can form the cosimplicial object $\mathfrak{C}^\bullet F$ , as in Proposition B.6, with respect to the structure $({\perp^{ {\scriptscriptstyle {\boldsymbol{\Sigma}}}}}, \eta)$ on $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

Proposition 4.35. For $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , there is a natural isomorphism

\begin{equation*} H^*_{\textbf{FI}^\textrm{op}} (F)\cong \pi^* (\mathfrak{C}^\bullet F),\end{equation*}

where the right-hand side is the cohomotopy of the cosimplicial object $\mathfrak{C}^\bullet F $ of $\mathcal{F} ({\boldsymbol{\Sigma}}^\textrm{op};\, \unicode{x1D55C})$ .

Proof. By definition, $H^0 _{\textbf{FI}^\textrm{op}}(F) = \pi^0 (\mathfrak{C}^\bullet F)$ . Moreover, Proposition B.6 implies that $F \mapsto \mathfrak{C}^\bullet F$ defines an exact functor to cosimplicial objects, so $F \mapsto \pi^* (\mathfrak{C}^\bullet F)$ forms a cohomological $\delta$ -functor. The usual argument of homological algebra (cf. [Reference Weibel11, Exercise 2.4.5], for example), using the effacability property established in Corollary B.10, then provides the isomorphism.

This allows Theorem 4.1 to be restated using $\textbf{FI}^\textrm{op}$ -cohomology:

Corollary 4.36. For $F \in \mathsf{Ob}\, \mathcal{F} (\textbf{FI}^\textrm{op};\, \unicode{x1D55C})$ , there is a natural isomorphism

\begin{equation*}H^*_{\textbf{FI}^\textrm{op}} (F) \cong H^* (\mathsf{Kz}\ F).\end{equation*}

Proof. The normalised complex $N (\mathfrak{C}^\bullet F)^\textrm{op} $ is canonically isomorphic to $N \mathfrak{C} ^\bullet F$ , in particular has canonically isomorphic cohomology. Thus, the statement follows immediately from Theorem 4.1.

5.1 The cosimplicial abelian group $\textbf{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} ((t^*)^{\otimes a}, {\perp^{ \scriptscriptstyle \Gamma}}^\bullet (t^*)^{\otimes b} )$

In this section, we apply the general constructions given in Appendix B.2 for $\mathcal{C}= \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ and ${\perp^{ \scriptscriptstyle \Gamma}} \,:\, \mathcal{F} (\Gamma;\, \unicode{x1D55C}) \rightarrow \mathcal{F} (\Gamma;\, \unicode{x1D55C})$ the comonad introduced in Notation 3.21.

By Proposition B.12, for $a, b \in \mathbb{N}$ , there is an equaliser:

(5.1)

where, for $f \in \textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} \left((t^*)^{\otimes a},(t^*)^{\otimes b}\right) $ , $d^0 f \,:\!=\, \left({\perp}^\Gamma f\right) \circ \psi_{(t^*)^{\otimes a}} $ , $d^1 f \,:\!=\, \psi_{ (t^*)^{\otimes b}} \circ f$ and $\mathcal{F} (\Gamma;\, \unicode{x1D55C})_{{\perp^{ \scriptscriptstyle \Gamma}}}$ is the category of ${\perp^{ \scriptscriptstyle \Gamma}}$ -comodules in $\mathcal{F} (\Gamma;\, \unicode{x1D55C})$ .

The comonad ${\perp^{ \scriptscriptstyle \Gamma}}$ provides the usual augmented simplicial object ${{\perp^{ \scriptscriptstyle \Gamma}}}^{\bullet +1} (t^*)^{\otimes b}$ (cf. Proposition B.3) and hence an augmented simplicial object $\textrm{Hom}_{\mathcal{F} (\Gamma;\, \unicode{x1D55C})} \left((t^*)^{\otimes a}, {\perp^{ \scriptscriptstyle \Gamma}}^{\bullet +1} (t^*)^{\otimes b} \right)$ (cf. Lemma B.13).