## February 2, 2014

### An Emerging Pattern in Algebra and Topology II

#### Posted by Emily Riehl Last time, I described an emerging pattern in algebra and topology, exemplified by the cohomology of the space of configurations of ordered points in the plane. In language I’ll introduce below, cohomology with rational coefficients defines a uniformly representation stable sequence, the fundamental data of which is an $S_n$-representation for each $n$.

In this post, I want to tell you in more detail about the representation stable sequences introduced in a paper by Tom Church and Benson Farb. I’ll also discuss the second generation approach to representation theory, which centers of the functor category of what they call FI-modules. Jordan Ellenberg, the third author of this paper, has blogged about this also. The first post in his series can be found here. In contrast with Part I, my exposition here will be more explicitly directed at the categorically-minded reader.

## 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.

Posted at February 2, 2014 3:24 AM UTC

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### Re: An Emerging Pattern in Algebra and Topology II

Thanks for writing these two articles. Tom Church gave a series of lectures at a workshop in Copenhagen in August, Christine Vespa gave a parallel series on a similar topic. It was a great opportunity to learn about Representation Stability and to begin to figure out how it fits into the larger picture. We also got a sneak peak at future results and perhaps how a future attempt at writing down Representation Stability might look.

Just to let you know there’s a typo, non-invertible automorphism’, perhaps endomorphism.

I would also like to take issue with the idea that H(Conf(C)) is an FI#-module. It is! But this is evil’ since the chain complex C(Conf(C)) is not a homotopy FI#-module, at least not over Z and it requires the formality of the little discs to be true over Q. I don’t think it’s true over fields of finite characteristic. It is true that adding a point onto the boundary is homotopy associative, but it is not homotopy commutative.

The non-evil version would be some statement along the lines of: the family Conf(C) is a (homotopy) braided monoidal category. It can also be written in the language of operads; the little discs operad E2 is a left module over itself.

I have some further thoughts on all of this, but wanted to check first: is there going to be a part III?

Posted by: James Griffin on February 2, 2014 4:21 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Indeed, I meant “non-invertible endomorphism”! I’ll fix it now. Thanks!

Could you elaborate on this point:

I would also like to take issue with the idea that H(Conf(C)) is an FI#-module. It is! But this is `evil’ since the chain complex C(Conf(C)) is not a homotopy FI#-module, at least not over Z and it requires the formality of the little discs to be true over Q. I don’t think it’s true over fields of finite characteristic. It is true that adding a point onto the boundary is homotopy associative, but it is not homotopy commutative.

When $M$ is the interior of a compact, connected, oriented manifold with non-empty boundary, $\text{Conf}_n(M)$ is a “homotopy FI$\sharp$-space” (a functor to the homotopy category of spaces, so $C^*(\text{Conf}_n(M))$ is a “homotopy FI$\sharp$-chain complex” (taking values in the category of chain complexes and chain homotopy classes of maps), and this works with any coefficients.

But you’re making a more interesting and more subtle point, I think. What exactly fails to be homotopy commutative?

Also, there’s no part III, so please comment away!

Posted by: Emily Riehl on February 2, 2014 6:12 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Tomorrow I will more carefully read the definition of FI#, I am more familiar with the categories used by Pirashvili. But if it’s the category I think it is then I do not believe that Conf(M) is a homotopy FI#-module (taking values in the homotopy category of spaces).

Counter-example, let M be the unit interval I = [0,1], with chosen boundary component 0.

We can factor the inclusion {1} —> {1,2,3} in two ways, either {1} —> {1,3} —> {1,2,3} or {1} —> {1,2} —> {1,2,3}. Both of these act contravariantly, providing maps from Conf1(M) to Conf3(M), but Conf1(M) is contractible and with the given action of adding in points on the boundary these maps land in different connected components.

For manifolds of higher dimension Conf(M) is closer to a homotopy FI#-module, but one can only fill in all of the homotopies when M is infinite dimensional (say the colimit of manifolds of increasing dimension).

Is this right?

Posted by: James Griffin on February 2, 2014 10:21 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Sorry, of course I meant dim$(M) \geq 2$ so that the configuration spaces are connected. But now I think the claim that $\text{Conf}_n(M)$ is a homotopy FI$\sharp$-space is correct.

Posted by: Emily Riehl on February 3, 2014 2:56 AM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Ah, I was worried that you might say that. Disputing the higher dimensional case is harder. I did try to write out an explicit argument taking M as the 2-disc, it got quite fiddly. This is my second go and uses a little technology.

Let X be any FIop-module, and let Y be any space then I claim that the space

(1)$X(Y) = \coprod_n X(n) \times_{S_n} Y^n$

is equipped with maps $p_y$ for $y\in Y$. For example for $\text{Conf}_n(M)$ this construction gives the disjoint union of all $Y$-labelled points of M and $p_y$ adds a point $y$ onto the boundary. Now the important part of the claim is that all the maps $p_y$ commute.

For a homotopy module this means that all the maps $p_y$ should homotopy commute. But this does not happen with configurations in a finite dimensional n-disc.

One of the ways to see this is to take Y to be a finite set and then to group complete with respect to the actions of the $p_y$. The resulting space turns out to be homotopy equivalent to the n-fold loop space of a bouquet of Y n-spheres. This is an n-fold loop space and the actions of the $p_y$ are the actions of the generators for $\pi_n$ of the bouquet. But the homotopy commutativity of the $p_y$ would then imply that the n-fold loop space of a bouquet of spheres was an infinite loop space, in particular so would be a bouquet of n-spheres.

This homotopy commutativity was I suppose what I was talking about in my first reply. Apologies if this isn’t very clear, but it’s already my second try and I’d quite like to write about the really interesting bit!

Posted by: James Griffin on February 3, 2014 8:39 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Dear James,

The restriction that $\text{dim} \,M\geq 2$ is necessary — otherwise the configuration spaces are not connected, and the claim is definitely not true. But as long as $\text{dim} \,M\geq 2$ it should be true that $\text{Conf}(M)$ forms a homotopy $\text{FI}\sharp$-space. The dispute seems to be about what it means to commute up to homotopy — perhaps it is resolved by the fact that (if I’m using these terms correctly) an $E_2$-algebra really is homotopy commutative, even if it’s not as “coherently commutative” as an $E_\infty$-algebra?

By the way, if you’re trying to understand this claim about $\text{Conf}(M)$ (or disprove it!), it is probably easier to focus on the claim that $\text{Conf}(M)$ forms a homotopy $\text{FI}$-space. The remaining structure of an $\text{FI}\sharp$-module comes from the fact that $\text{Conf}(M)$ forms a (strict) $\text{FI}^{\text{op}}$-space, about which I don’t think there’s any disagreement. (Plus this way you don’t have to worry about the definition of $\text{FI}\sharp$. It is indeed a subcategory of the category that Pirashvili works with, though.)

You can find our proof that $\text{Conf}(M)$ is a homotopy $\text{FI}$-space in Proposition 4.6 of arXiv:1204.4533, at the bottom of page 45. However we don’t say so much more than you already have, except to note that $\text{dim} \,M\geq 2$ is necessary so that the space of configurations to add is connected.

Best regards, Tom

Posted by: Tom Church on February 4, 2014 4:06 AM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

By the way, please feel free to set this aside if you like — I’m eager to hear about the other thoughts you mentioned as well!

Posted by: Tom Church on February 4, 2014 3:05 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Apologies, I’ve been busy with job/fellowship applications (yuck!) and only just checked the thread late last night.

I am happy now that $\text{Conf}_n(M)$ is a functor $\text{FI}\sharp \rightarrow \text{hTop}$. I suppose that my initial objection, which I still hold, is calling it a homotopy FI$\sharp$-space. And that’s because it isn’t in any way an FI$\sharp$-space and adding an adjective should make things stronger, not weaker. This is coupled with the fact that there is a good notion of homotopy FI$\sharp$-space interpreted as an FI$\sharp$-space up to homotopy. Ok, one could make the objection that this isn’t strictly an FI$\sharp$-space either, but there are homotopy equivalent, it’s certainly morally fine.

In line with these comments I personally would avoid thinking of an $E_2$-algebra as homotopy commutative. In the algebra of operads at least this means an $E_\infty$-algebra, see the ncatlab entry for homotopy algebra.

So, what is $\text{Conf}_n(M^d)$? Well it’s a right-module for a unital $E_d$-operad. The unit takes care of the right FI-space structure, the more subtle part by the operad action. To this operad there is an associated (topological) prop $P(E_d)$, which replaces FI$\sharp$: $\text{Conf}_n(M^d)$ is a $P(E_d)$-space. One now recovers your Theorem by noting that when $d\geq 2$

(1)$\pi_0(P(E_d)) \cong \text{FI}\sharp.$

This richer structure is the start of factorisation homology, although many of the ideas are older.

Posted by: James Griffin on February 8, 2014 12:08 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Much of this stuff is stirring up memories from 40 years ago about the classifying topos for the theory of infinite separated objects. This is the theory with a single separation relation, say #, such that x#x is false, x#z implies the disjunction of x#y and y#z, and x#y implies y#x together with the axiom scheme that says if n elements are mutually separate then there exists another element separate from them all. Its classifying topos can be identified with the category of combinatorial functors, or equivalently with continuous G-sets for G the direct limit of S_n.

There is a nice argument that G is acyclic. Let E be the topos, and X in E be the generic model, and N the natural number object in E. Then X is classified by the identity map E -> E and N by the unique point of E. The canonical injections X -> X + N <- N provide a homotopy from the identity map of E to its point.

Posted by: Gavin Wraith on February 6, 2014 6:32 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Let be write $S_\infty$ for your $G$. The topos you mention, of continuous $S_\infty$-sets, has come up in the context of FI-modules before.

This topos is a reflective subcategory of $\mathbf{Set}^{\mathbf{FI}}$. The inclusion takes a continuous $S_\infty$-set to the FI-set defined using fixed points for the subgroup of $S_\infty$ that fixes $\{1,\ldots, n\}$. The left adjoint takes the colimit of the composite diagram $\omega \hookrightarrow \mathbf{FI} \to \mathbf{Set}$ and endows it with a continuous $S_\infty$ action.

The reason I mention this is there is another characterization of the essential image: it’s the category of sheaves for the atomic topology on $\mathbf{FI}^{op}$ (each singleton arrow is covering). More explicitly, it’s the category of pullback preserving functors $\mathbf{FI} \to \mathbf{Set}$.

It’s this last characterization that leads to the connection with $\mathbf{FI}$-modules. Here I may have a few of the details slightly wrong, as I’m trying to remember a conversation I had with Jordan and Tom about a year ago.

For any finitely generated FI-module $V$, there is a surjection from a free FI-module $M$ with kernel $W$:

$0 \to W \hookrightarrow M \twoheadrightarrow V \to 0.$

You can check that $W$ is “torsion free”: given any $w \in W_m$ and $f \colon m \rightarrowtail n$ if $f_*(w)=0$ then $w=0$.

Now if $W$ is finitely generated (e.g., in the Noetherian contexts mentioned above), there is another surjection from a free module $M'$ with kernel $W'$:

$0 \to W' \hookrightarrow M' \twoheadrightarrow W \to 0$

and I think the FI-modules $W'$ that arise in this way preserve pullbacks.

Posted by: Emily Riehl on February 6, 2014 8:29 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Posted by: Emily Riehl on February 6, 2014 8:29 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

By a combinatorial functor on the category of finite sets and injections I mean one that preserves intersections. I am trying to remember from whom I first heard this usage: Anders Kock, Stephen Schanuel, Andre Joyal? An erratum: 40 years ago may be slightly too many, but at least 30. The other interesting atomic topos that I remember from the same context replaced “finite sets and injections” by “linearly ordered finite sets and monotone injection”. Stephen pointed out that two finite combinatorial functors F1, F2 for which #F1(n) = #F2(n) for all n have isomorphic restrictions to the linearly ordered case. But I guess the rigidity of linear ordering is a very familiar observation to programmers.

Posted by: Gavin Wraith on February 7, 2014 10:50 AM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

I mentioned above that I have some thoughts on where the study of FI-modules is going. I suspect these aren’t new to Tom.

The jump from considering sequences of $S_n$-reps with some added maps, or even sequences of tableau with some structure, to functors out of the category FI, is empowering. It grants us all of the tools of functor homology. But we know that FI-modules are more special than arbitrary $\mathcal{C}$-modules. So what’s special about the category FI? Well we have already learnt that it’s Noetherian in the sense that finitely generated modules are also finitely presented. It is this that implies the homological stability results.

So is this it, have we boiled down representation stability to a little applied category theory? Well of course not, there’s more going on here (much more).

Actually the study of FI-modules and representation stability is commutative algebra. This should get us excited.

I’ll try to explain. There is a symmetric monoidal category $\mathbb{S}-mod$, in which there is a commutative monoid $C$ for which the category of left FI-modules (covariant functors into abelian groups) is the category of $C$-modules. The symmetric monoidal category is familiar to operad theorists, it is the category of $\mathbb{S}$-modules, where $\mathbb{S}$ is the category of finite sets and permutations. The monoidal structure is defined by:

(1)$(F \square G)(A) = \bigoplus_{A=S\amalg T} F(S) \otimes G(T).$

The algebra $C$ is the constant functor sending every set to $\mathbb{Z}$. I’ll leave the remaining details as an exercise.

So why is this exciting? Well for starters module categories for commutative rings have interesting homological algebra. In particular the algebra $C$ is Koszul, leading to Koszul resolutions for FI-modules. Actually something stronger is true, there is a Koszul duality between the category of left FI-modules and another category, the category of comodules for the Koszul dual of $C$. I haven’t checked but this could be equivalent to the category of right FI-modules.

In Tom’s workshop series in Copenhagen he talked about torsion, saturated and k-saturated FI-modules and used them to great effect. I’m afraid any explanation I gave wouldn’t do this justice. My intuition is that this theory can be understood in the language of local rings and Ext groups.

What other theories can be carried over from commutative algebra? What about other functor categories?

I think there are many reasons to be excited about FI-modules; in the addition to the algebraic aspects studied by Tom we should expect the homotopy theory to be rich.

Posted by: James Griffin on February 8, 2014 1:15 PM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II James, without saying too much, I think you’ll like the paper Tom and I are just finishing up now! It includes some of the results Tom talked about in Copenhagen – but I think the way we set it up now is even a little bit better than the “k-saturated” formulation we were using last summer. And the flavor of the paper is very much “find in FI-Mod analogues of familiar notions from commutative algebra over finitely generated rings.”

When talking about this connection it’s also really important to mention the work of Snowden and Sam, who work with a category equivalent to that of FI-mod/C , and in fact with a whole bouquet of related categories which are really interesting and in many ways substantially harder – in many cases, the Noetherianness is not known, and seems to be a very interesting question.

Among other things, they work out a very remarkable duality, which they call a “Fourier transform,” between (in our language) the category of torsion FI-modules and the Serre quotient of FI-Mod by that category.

Snowden-Sam paper

Posted by: JSE on February 12, 2014 3:14 AM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

Jordan, that’s great. I’m sure I will like this next paper, I’ll look out for it. Will I find Ext groups? Also, thanks for the reference to Snowden and Sam.

If you’re looking at analogues for results from commutative algebra over finitely generated rings, then there is no better way than to state that FI is a commutative finitely generated ring and that modules in the sense of functors precisely are equivalent to modules in the familiar sense.

As for generalisations, I have worked with a category where morphisms consist of an injection but where the complement of the image is equipped with a free action of $S_2$. It is also a free commutative algebra; and it’s modules have a nice Koszul duality theory. I found them while thinking about Kontsevich’s graph complexes.

Posted by: James Griffin on February 18, 2014 10:03 AM | Permalink | Reply to this

### Re: An Emerging Pattern in Algebra and Topology II

James: your generalization is something that Snowden and I have considered. A basic question which we have not been able to answer is whether or not its module category is Noetherian. Let me give some references briefly.

All of the papers that we have publically shared are over characteristic 0 since we were mostly interested in exploiting semisimple representation theory to prove our results (some of the results that Jordan mentions from our papers probably fail in positive characteristic actually).

The category you mention (free S2 action on complement of an injection) is a commutative algebra in the “downwards Brauer category” which we introduce in (4.2.5) of http://arxiv.org/abs/1302.5859

Our interest there is the connection the stable polynomial representation theory of the orthogonal group. For a closer connection to commutative algebra, see Theorem 4.3.1.

It would be interesting to know how it’s related to graph complexes, if you wouldn’t mind sharing.

Posted by: Steven Sam on February 19, 2014 9:06 PM | Permalink | Reply to this

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