The n-Category Café

Category theory offers a clean definition of generation: if $C$ is some category of algebraic objects and $F \dashv U$ is a free-forgetful adjunction with $U : C \longrightarrow \mathrm{Set}$, then it makes sense to say that a subset $S \subseteq U X$ generates $X$ if the adjunct arrow $F S \rightarrow X$ is epic.

Certainly $R$-modules fit into this setup nicely, and groups, commutative rings, etc. What about simplicial sets? It makes sense to say that some simplicial set $X$ is “generated” by its 1-simplices, for example: this is saying that $X$ is 1-skeletal. But simplicial sets come with many sorts of generator…Ah, and they also come with many forgetful functors, given by evaluation at the various objects of $\Delta^{op}$.

Let’s assume we’re in a context where there are many forgetful functors, and many corresponding notions of generation. In fact, for concreteness, let’s think about cosimplicial vector spaces over the rational numbers. A cosimplicial vector space is a functor $\Delta \longrightarrow \mathrm{Vect}$, and so for each $d \in \Delta$ we have a functor $U_d : \mathrm{Vect}^{\Delta} \longrightarrow \mathrm{Set}$ with $U_d V = V d$ and left adjoint $F_d$. We will say that a vector $v \in V d$ sits in degree $d$, and generally think of $V$ as a vector space graded by the objects of $\Delta$.

Definition A cosimplicial vector space $V$ is generated in degree $d \in \Delta$ if the component at $V$ of the counit $F_d U_d V \longrightarrow V$ is epic. Similarly, $V$ is generated in degrees $\{d_i \}$ if $\oplus_i F_{d_i} U_{d_i} V \longrightarrow V$ is epic.

Example Let $V = F_d \{ \ast \}$ be the free cosimplicial vector space on a single vector in degree $d$. Certainly $V$ is generated in degree $d$. It’s less obvious that $V$ admits a unique nontrivial subobject $W \hookrightarrow V$. Let’s try to find generators for $W$. It turns out that $W d = 0$, so no generators there. Since $W \neq 0$, there must be generators somewhere… but where?

Theorem (Wrangling generators for cosimplicial abelian groups): If $V$ is a cosimplicial abelian group generated in degrees $\{ d_i \}$, then any subobject $W \hookrightarrow V$ is generated in degrees $\{d_i + 1 \}$.

Ok, so now we know exactly where to look for generators for subobjects: exactly one degree higher than our generators for the ambient object. The generators have been successfully wrangled.

The preorder on degrees of generation $\leq_d$

Time to formalize. Let $U_d, U_x, U_y: C \longrightarrow \mathrm{Set}$ be three forgetful functors, and let $F_d, F_x, F_y$ be their left adjoints. When the labels $d, x, y$ appear unattached to $U$ or $F$, they represent formal “degrees of generation,” even though $C$ need not be a functor category. In this broader setting, we say $V \in C$ is generated in (formal) degree $\star$ if the component of the counit $F_{\star} U_{\star} V \longrightarrow V$ is epic. By the unit-counit identities, if $V$ is generated in degree $\star$ , the whole set $U_{\star} V$ serves as a generating set.

Definition Say $x \leq_d y$ if for all $V \in C$ generated in degree $d$, every subobject $W \hookrightarrow V$ generated in degree $x$ is also generated in degree $y$.

Practically speaking, if $x \leq_d y$, then generators in degree $x$ can always be replaced by generators in degree $y$ provided that the ambient object is generated in degree $d$.

Suppose that we have a complete understanding of the preorder $\leq_d$ , and we’re trying to generate subobjects inside some object generated in degree $d$. Then every time $x \leq_d y$, we may replace generators in degree $x$ with their span in degree $y$. In other words, the generators $S \subseteq U_x V$ are equivalent to generators $\mathrm{Im}(U_y F_x S \longrightarrow U_y V) \subseteq U_y V$. Arguing in this fashion, we may wrangle all generators upward in the preorder $\leq_d$. If $\leq_d$ has a finite system of elements $m_1, m_2, \ldots, m_k$ capable of bounding any other element from above, then all generators may be replaced by generators in degrees $m_1, m_2, \ldots, m_k$. This is the ideal wrangling situation, and lets us restrict our search for generators to this finite set of degrees.

In the case of cosimplicial vector spaces, $d + 1$ is a maximum for the preorder $\leq_d$ with $d \in \Delta$. So any subobject of a simplicial vector space generated in degree $d$ is generated in degree $d + 1$. (It is also true that, for example, $d + 2$ is a maximum for the preorder $\leq_d$. In fact, we have $d + 1 \leq_d d+2 \leq_d d + 1$. That’s why it’s important that $\leq_d$ is a preorder, and not a true partial order.)

Connection to the preprint arXiv:1508.04107

In the generality presented above, where a formal degree of generation is a free-forgetful adjunction to $\mathrm{Set}$, I do not know much about the preorder $\leq_d$. The paper linked above is concerned with the case $C = (\mathrm{Mod}_R)^{\mathcal{D}}$ of functor categories of $\mathcal{D}$-shaped diagrams of $R$-modules. In this case I can say a lot.

In Definition 1.1, I give a computational description of the preorder $\leq_d$. This description makes it clear that if $\mathcal{D}$ has finite hom-sets, then you could program a computer to tell you whenever $x \leq_d y$.

In Section 2.2, I give many different categories $\mathcal{D}$ for which explicit upper bounds are known for the preorders $\leq_d$. (In the paper, an explicit system of upper bounds for every preorder is called a homological modulus.)

Connection to the field of Representation Stability

If you’re interested in more context for this work, I highly recommend two of Emily Riehl’s posts from February of last year on Representation Stability, a subject begun by Tom Church and Benson Farb. With Jordan Ellenberg, they explained how certain stability patterns can be considered consequences of structure theory for the category of $\mathrm{FI}$-modules $(\mathrm{Vect}_{\mathbb{Q}})^{\mathrm{FI}}$ where $\mathrm{FI}$ is the category of finite sets with injections. In the category of $\mathrm{FI}$-modules, the preorders $\leq_n$ have no finite system of upper bounds. In contrast, for $\mathrm{Fin}$-modules, every preorder has a maximum! (Here $\mathrm{Fin}$ is the usual category of finite sets). So having all finite set maps instead of just the injections gives much better control on generators for subobjects. As an application, Jordan and I use this extra control to obtain new results about configuration spaces of points on a manifold. You can read about it on his blog.

For more on the recent progress of representation stability, you can also check out the bibliography of my paper or take a look at exciting new results by CEF, as well as Rohit Nagpal, Andy Putman, Steven Sam, and Andrew Snowden, and Jenny Wilson.