The n-Category Café

Representation stable sequences

Let $k$ be a commutative ring and let $S_n$ denote the symmetric group on $n$ letters. Write $V = (V_n)_{n \geq 0}$ for a collection of $S_n$-representations $V_n$. Equivalently, $V$ defines a functor $\oplus S_a \to \mathbf{Mod}_k$ whose domain is the disjoint union of the symmetric groups $S_n$, regarded as one-object groupoids (or, if you prefer, the category of finite sets and isomorphisms). This data is sometimes called a symmetric sequence or a species. The symmetric groups come with injective group homomorphisms

$$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \hookrightarrow \cdots$$

where the image of $S_n \hookrightarrow S_{n+1}$ consists of those permutations that fix the element $n+1$. The “mapping telescope” $\mathbf{Tel}$ (more precisely, the lax colimit) of this sequence is the category defined by gluing $\oplus S_n$ to the poset $\omega$ along the discrete category $\mathbb{N}$.

Definition. A consistent sequence of representations is a functor $V\colon \mathbf{Tel} \to \mathbf{Mod}_k$.

Explicitly, a consistent sequence is a symmetric sequence $V$ together with $S_n$-equivariant maps (natural transformations) $V_n \to \text{res}(V_{n+1})$ from $V_n$ to the restriction of the $S_{n+1}$-representation $V_{n+1}$ along $S_n\hookrightarrow S_{n+1}$. Here’s another way to think about this. Define a category $\mathbf{Rep}$ over $\omega$ whose objects in the fiber over $n$ are $S_n$-representations $V_n$. Hom-sets in Rep are defined by

$$\text{Hom}_{\mathbf{Rep}}(V_m, V_n) = \text{Hom}_{S_m}(V_m, \text{res} V_{n}) = \text{Hom}_{S_{n}}(\text{ind}V_m, V_{n})$$

if $m \leq n$ and empty otherwise. Here $\text{ind} \colon \mathbf{Mod}_k^{S_n} \to \mathbf{Mod}_k^{S_{m}}$ is left adjoint to the restriction along the composite inclusion $S_n\hookrightarrow S_m$. It’s precisely the left Kan extension. In representation theory, this functor is called induction, which explains the notation I’m using here. Now a consistent sequence is exactly a section $V$ to the projection functor $\mathbf{Rep} \to \omega$, i.e., a countable sequence of composable morphism in $\mathbf{Rep}$, with exactly one object in the fiber over each $n \in \omega$.

Definition. A consistent sequence of representations $V$ is a representation stable sequence if

(i) All but finitely many of the maps in $V \colon \omega \to \mathbf{Rep}$ are monomorphisms.

(ii) All but finitely many of the maps in $V \colon \omega \to \mathbf{Rep}$ are epimorphisms.

(iii) For each partition $\lambda$, the coefficients $c_{n,\lambda}$ of the irreducible representation $V(\lambda)$ in the decomposition $V_n \cong \oplus_{\lambda} V(\lambda)^{c_{n,\lambda}}$ is independent of $n$ for $n$ sufficiently large.

By definition, a morphism $V_m \to V_{n}$ in $\mathbf{Rep}$ is represented by either of the two adjunct arrows $V_m \to \text{res}V_{n}$ in $\mathbf{Mod}_k^{S_m}$ or $\text{ind}V_m \to V_{n}$ in $\mathbf{Mod}_k^{S_{n}}$. The map $V_m \to V_{n}$ is a monomorphism if and only if $V_n \to \text{res}V_{n}$ is injective and an epimorphism if and only if $\text{ind}V_m \to V_{n}$ is surjective.

The sequence is uniformly representation stable if the stable range in condition (iii) can be chosen independently of $\lambda$.

For example, consider the space $\text{Conf}_n(\mathbb{C})$ of configurations of $n$ labeled points in the plane. “Forgetting the last point” defines a continuous map $\text{Conf}_{n+1}(\mathbb{C}) \to \text{Conf}_n(\mathbb{C})$, inducing a map $H^i(\text{Conf}_n(\mathbb{C});\mathbb{Q}) \to H^i(\text{Conf}_{n+1}(\mathbb{C}),\mathbb{Q})$ that is $S_n$-equivariant in the sense described above. So $i$-th cohomology of configuration spaces defines a consistent sequence of representations. The theorem mentioned at the end of Part I now takes the following form:

Theorem. The consistent sequence of representations $H^i(\text{Conf}_n(\mathbb{C});\mathbb{Q})$ is uniformly representation stable with stable range $n \geq 4i$.

Later, this theorem was generalized to configurations of ordered points in any connected, orientable, manifold of finite-type (meaning that the rational cohomology consists of finite dimensional vector spaces).

FI-modules

The second-generation account of representation stability begins with the following observation. The functor $\text{Conf}_{(-)}(\mathbb{C}) \colon \mathbf{Tel}^{op} \to \mathbf{Top}$ factors through a quotient of the category $\mathbf{Tel}$ that turns out to be much easier to describe. The consistent sequence of configuration spaces was defined using maps that “forget the last point.” But of course, we could have forgotten any of the points or any number of points. Moreover we could have chosen new labels for the points that we do not forget. These “forgetful” maps assemble into a functor $\mathbf{FI}^{op} \to \mathbf{Top}$, where FI is the category of finite sets and injections. (When notationally convenient, we’ll consider $\mathbf{FI}$ to be skeletal.)

Definition. An FI-module is a functor $\mathbf{FI} \to \mathbf{Mod}_k$. The category FI-Mod is the category of FI-modules and natural transformations.

A consistent sequence is an FI-module if and only if $V \colon \mathbf{Tel}(S) \to \mathbf{Mod}_k$ factors through the functor $\mathbf{Tel} \to \mathbf{FI}$ induced by the obvious cone under the pushout diagram defining the category $\mathbf{Tel}$. This is the case just when, for each $v$ in the image of the canonical map $V_m \to V_n$ ($m \lt n$), and for any $\sigma, \tau \in S_n = hom(n,n)$ with common restriction along $m \hookrightarrow n$, we have $\sigma \cdot v= \tau \cdot v$.

A consistent sequence that is not an FI-module is the sequence of regular representations

$$k \to k[S^1] \to k[S^2] \to k[S^3] \to \cdots$$

in which $k[S^n]$ is the free $k$-module module with basis $S_n$.

Representable FI-modules and free FI-modules

For each $n$, there is a free FI-module that Church, Ellenberg, and Farb denote by $M(a)$ defined to be the composite of the functor represented by $a$ and the free $k$-module functor $k[-] \colon \mathbf{Set}\to \mathbf{Mod}_k$. So $M(a)_m = k[hom(a,m)]$. By the Yoneda lemma and the free-forgetful adjunction, a map of $FI$-modules $M(a) \to V$ corresponds to a vector $v \in V_a$.

There are more general notions of free FI-modules defined by forming the left Kan extension of an $S_a$-representation or of a symmetric sequence. (We might regard an $S_a$-representation as a symmetric sequence in which all of the other representations are 0.)

Write $\oplus S_a$-Rep for the functor category $\mathbf{Mod}_k^{\oplus S_a}$. Note that $\oplus S_a$ is the maximal subgroupoid of $\mathbf{FI}$. Left Kan extension defines the free-FI-module functor

$$M(-)\colon \oplus S_a\text{-}\mathbf{Rep} \to \mathbf{FI}\text{-}\mathbf{Mod}$$

which is left adjoint to restriction: the functor that sends an FI-module to its underlying symmetric sequence. Because the category $\mathbf{Mod}_k$ is cocomplete, we obtain an explicit formula for $M(-)$ as a special case of the formula for pointwise left Kan extensions:

$$M(W)_n = \oplus_a \left( M(a)_n \otimes_{k[S_a]} W_a \right).$$

If $k$ is a field (so that $W_a$ is free), this formula tells us that

$$\text{dim} M(W)_n = \sum_a \left(\begin{array}{c} n \\ a \end{array}\right) \text{dim} W_a.$$

Categorical properties of FI-$\mathbf{Mod}$

FI-$\mathbf{Mod}$ is a category of functors taking values in a complete and cocomplete abelian category. Hence FI-$\mathbf{Mod}$ is also abelian and has these limits and colimits (and kernels and cokernels), defined pointwise. The tensor product over the commutative ring $k$ also induces a number of monoidal structures that we decline to discuss here. Another monoidal structure is induced by Day convolution from the disjoint union on FI.

The abelian category structure of FI-modules will have computational implications.

Definition. An FI-module $V$ is finitely generated if there exists a finite sequence of integers $m_i$ and an epimorphism $\oplus_i M(m_i) \to V$ in FI-Mod.

The map $\oplus_i M(m_i) \to V$ identifies a sequence of elements $v_i \in V_{m_i}$ that span $V$ in the sense that $V$ is the largest submodule containing each of the $v_i$. We have the elementary result:

Proposition. Consider a short exact sequence $U \to V \to W$ of FI-modules. If $V$ is finitely generated, then so is $W$. If both $U$ and $W$ are finitely generated, then so is $V$.

If $k$ is a Noetherian ring containing $\mathbb{Q}$, then FI-$\mathbf{Mod}$ is Noetherian, meaning that any submodule of a finitely generated FI-module is finitely generated. This is harder to prove.

FI-modules and representation stability

The reason we care about finite generation is that in good cases it is equivalent to uniform representation stability.

Theorem. An FI-module over a field of characteristic zero is finitely generated if and only if its underlying consistent sequence is uniformly representation stable and each vector space is finite dimensional.

Moreover, it is computationally advantageous to show that a sequence of FI-modules is finitely generated. For instance, we have the following theorem, which is a special case of a more general result about character polynomials. (These are polynomials whose variables $X_i$ are the class functions $X_i \colon S_n \to \mathbb{N}$ that carry a permutation to the number of $i$-cycles in its cycle decomposition.)

Theorem. If $V$ is a finitely generated FI-module over a field of characteristic zero, then there is a polynomial $P$ with rational coefficients so that $dim V_n = P(n)$ for all $n$ sufficiently large.

FI$\sharp$-modules

In some examples, FI-modules come with even more structure. Let us return our attention to the configuration spaces $\text{Conf}_n(M)$ but now assume that the manifold $M$ has non-empty boundary. This allows us to define maps $\text{Conf}_m(M) \to \text{Conf}_{n}M$ for $m$ less than $n$, which introduce new points in a small neighborhood of the boundary. Composition of maps of this form isn’t associative on the nose, but it is well-defined up to homotopy, which means that the cohomology $H^i(\text{Conf}_n(M);k)$ for $i$ fixed has both the structure of an FI-module and of a co-FI-module (a contravariant functor $\mathbf{FI}^{op} \to \mathbf{Mod}_k$).

Church, Ellenberg, and Farb call symmetric sequences of this form FI$\sharp$-modules, which are $\mathbf{Mod}_k$-valued functors whose domain is the category $\mathbf{FI}\sharp$ of finite sets are partially-defined injections. It’s convenient to think of $\mathbf{FI}\sharp$ as the category of (isomorphism classes of) spans in $\mathbf{FI}$. The span

$$m \leftarrowtail s \rightarrowtail n$$

specifies a subset of $m$ (the domain of the partial function) and a bijection (mediated by $s$) with its image, a subset of $n$. We call $|s|$ the rank of this map. There are natural bijective-on-objects inclusions $\mathbf{FI} \to \mathbf{FI}\sharp$ and $\mathbf{FI}^\op \to \mathbf{FI}\sharp$ whose images consists of those spans with one component the identity. In this way we see that an FI$\sharp$-module has both an underlying FI-module and co-FI-module structure.

The representable FI-module $M(a)$ can be extended to an FI$\sharp$-module, which we also denote by $M(a)$, in a unique way. Recall that $M(a)_s = 0$ for all $s \lt a$. Note that in $\mathbf{FI}\sharp$, the span $m \leftarrowtail s \rightarrowtail n$ factors through $s$ in the evident way (as a map in $\mathbf{FI}^\op$ followed by a map in $\mathbf{FI}$). So an immediate consequence of functoriality is that any map in $\mathbf{FI}\sharp$ of rank less than $a$ must define the zero map on the FI$\sharp$-module $M(a)$.

To define the linear map

$$M(a)_m = k[\mathrm{hom}_{\mathbf{FI}}(a,m)] \to k[\mathrm{hom}_{\mathbf{FI}}(a,n)]$$

corresponding to $m \leftarrowtail s \rightarrowtail n$, it of course suffices to specify the image for each basis element $a \rightarrowtail m$. To do so, form the composite span $a \leftarrowtail t \rightarrowtail n$. Either $a =t$, in which case this map lies in the subcategory $\mathbf{FI}\subset \mathbf{FI}\sharp$, and we define the image of $a \rightarrowtail m$ to be the basis vector $a \rightarrowtail n$. If not, this composite has rank less than $a$ and we must define its image to be zero.

Note the FI$\sharp$-module $M(a)$ is not the left Kan extension of the FI-module $M(a)$ along $\mathbf{FI}\hookrightarrow \mathbf{FI}\sharp$: the left Kan extension would be the representable FI$\sharp$-module. (Recall that Kan extensions are only guaranteed to be extensions in the case where the map being extended along is full and faithful.) This confused me for a while. A big difference between $\mathbf{FI}$ and $\mathbf{FI}\sharp$ is that objects in the latter category have non-invertible endomorphisms. Indeed, each subset $s \subset a$ defines an idempotent

$$a \leftarrowtail s \rightarrowtail a$$

whose splitting is the object $s$; note the category $\mathbf{FI}\sharp$ is idempotent complete.

Now allowing $a$ to vary, the $M(a)$ define a symmetric sequence of FI$\sharp$-modules (with $S_a$ acting on $M(a)$ by precomposition). So we can define a functor $M(-)$ from symmetric sequences to FI$\sharp$-modules by functor tensor product:

$$M(W) = \oplus_a \left( M(a) \otimes_{k[S_a]} W_a \right).$$

This functor induces a classification of FI$\sharp$-modules.

Theorem. $M(-) \colon \oplus S_a\text{-}\mathbf{Rep} \to \mathbf{FI}\sharp\text{-}\mathbf{Mod}$ defines an equivalence of categories.

A stronger version of this theorem was proven by Teimuraz Pirashvili in Dold-Kan type theorem for Gamma-groups.

If $k$ is a field, a corollary of this result and the dimension calculation above is that the dimension of any finitely generated FI$\sharp$-module $V$ is given by a single polynomial in $n$ for all $n$. In fact, over a field finite generation of an FI$\sharp$-module $V$ is equivalent to the dimensions dim $V_n$ being bounded. This, together with the Noetherianness mentioned above, provides useful tools for proving that FI-modules are finitely generated.