The category FI and FI-modules over a commutative ring

Let $FI$ be the category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) whose objects are the finite sets and whose morphisms are the injections; composition is composition of functions and identities are identity functions. Every finite set $S$ is in bijection with exactly one of the sets $[n]:=\{0,1,\dots ,n-1\}$, $n\ge 0$, and injections between these representatives form a full skeletal subcategory: its inclusion is fully faithful, and every object of $FI$ is isomorphic to one of its objects. This skeletal description requires no simultaneous selection of bijections or quasi-inverse on the class of all finite sets; under the realization $S_n=\mathrm{Sym}(\{0,\dots ,n-1\})$ of The finite symmetric group $S_n$, one-line notation, and cycle notation this gives ${\mathrm{End}}_{FI}([n])=S_n$ for every $n\ge 0$, and between different objects there are no endomorphisms.

An FI-module over a commutative ring $k$ (Commutative ring) is a covariant functor $V\text{\hspace{0.17em}⁣}:FI\to k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ (Covariant functor, identity functor, composite functor, and contravariant functor, Unital left and right modules over a ring; unqualified module means left module). Write $V_S:=V(S)$, $V_n:=V([n])$ and $f_{\ast }:=V(f)\text{\hspace{0.17em}⁣}:V_S\to V_T$ for a morphism $f\text{\hspace{0.17em}⁣}:S\to T$; functoriality means $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{i}\mathrm{d}}_S{)}_{\ast }={\mathrm{i}\mathrm{d}}_{V_S}$. Since ${\mathrm{End}}_{FI}([n])=S_n$, the maps $V_n\to V_n$ induced by permutations make $V_n$ a left $k{S}_n$-module via the group ring (The group ring $R[G]$ of finitely supported formal $R$-linear combinations of group elements, For a commutative ring $R$, $R$-linear $G$-actions are exactly the compatible left $R[G]$-module structures): functoriality gives ${\sigma }_{\ast }{\tau }_{\ast }=(\sigma \tau {)}_{\ast }$ and ${(\mathrm{i}\mathrm{d})}_{\ast }=\mathrm{i}\mathrm{d}$, and the group-ring axioms follow.

For an injection $f\text{\hspace{0.17em}⁣}:[m]\to [n]$ define ${\mathrm{\Phi }}_f\text{\hspace{0.17em}⁣}:S_m\to S_n$ by ${\mathrm{\Phi }}_f(\sigma )(y):=f(\sigma (f^{-1}(y)))$ for $y\in f([m])$ and ${\mathrm{\Phi }}_f(\sigma )(y):=y$ for $y\mathrm{\notin }f([m])$; then ${\mathrm{\Phi }}_f$ is a group homomorphism, because $f^{-1}(y)$ is defined on $f([m])$, the two factors of a product commute through $f$, and both sides fix the complement pointwise. Since $f\circ \sigma ={\mathrm{\Phi }}_f(\sigma )\circ f$, functoriality gives $f_{\ast }\circ {\sigma }_{\ast }=({\mathrm{\Phi }}_f(\sigma ){)}_{\ast }\circ f_{\ast }$ for all $\sigma \in S_m$: each $f_{\ast }\text{\hspace{0.17em}⁣}:V_m\to V_n$ is equivariant for the conjugation homomorphism ${\mathrm{\Phi }}_f$ determined by $f$.

A sub-FI-module of $V$ is a subfunctor $W\subseteq V$ (Submodule of a module): submodules $W_S\le V_S$ with $f_{\ast }(W_S)\subseteq W_T$ for all injections $f\text{\hspace{0.17em}⁣}:S\to T$. A map of FI-modules $V\to W$ is a natural transformation, and kernels, images and cokernels of such maps are computed pointwise, so that e.g. $\ker (F{)}_S=\ker (F_S\text{\hspace{0.17em}⁣}:V_S\to W_S)$ with the restrictions of the transition maps (Quotient module $M\mathrm{/}N$ with scalar multiplication on additive cosets, Abelian category). With these pointwise operations the category of FI-modules over $k$ inherits its additive structure and its abelian-category structure from $k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ (Modules over a ring form an abelian category): finite direct sums, kernels, cokernels and the compatibility axioms are checked objectwise, and a pointwise surjective map is an epimorphism and a pointwise injective map is a monomorphism.

An FB-module over $k$ is a functor from the groupoid $FB$ of finite sets and bijections to $k\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$, with natural transformations as maps. Since all morphisms of $FB$ are isomorphisms and every finite set is isomorphic to exactly one $[n]$, an FB-module is determined up to isomorphism by the $k{S}_n$-modules $W_n$ alone and carries no transition maps between different $n$. No choice principle is used beyond the definable-class conventions of Category, object, morphism, domain, codomain, identity, composition, and hom-collection.