The n-Category Café

Alex and I started off thinking about various sorts of things you might do with operads in $\mathbf{Cat}$, and were eventually forced into what we currently call an action operad. This is an operad $G$ whose job it is to act on the objects of other operads. The key examples to keep in mind are the terminal operad (each set is just a singleton), the symmetric operad (the $n$th set is the $n$th symmetric group), and the braid operad (the $n$th set is the $n$th braid group). The technical definition involves an operad $G$, a group structure on each set $G(n)$, a map of operads $\pi:G \rightarrow \Sigma$ to the symmetric operad which is levelwise a group homomorphism, and a final condition (when it makes sense) relating operadic composition, $\mu$, with group multiplication:

$$\mu(g; f_1, \ldots, f_n) \cdot \mu(g'; f_1', \ldots, f_n') = \mu (g g'; f_{\pi(g')(1)}f_{1}', \ldots, f_{\pi(g')(n)}f_{n}').$$

These ideas have cropped up before, for example in Nathalie Wahl’s thesis or this preprint of Wenbin Zhang. Once you have this definition, you can define operads which are equivariant with respect to $G$: you have an operad $P$, a group action of $G(n)$ on $P(n)$ for each $n$, and some equivariance conditions that generalize the equivariance conditions for a symmetric operad.

This isn’t the only thing you can do with an action operad, you can also think about the 2-monad on $\mathbf{Cat}$ whose algebras are strict monoidal categories where $G(n)$ acts naturally on $n$-fold tensor products. If you do this with the symmetric operad, you get symmetric strict monoidal categories (or permutative categories, if you are a topologist); if you do this with the (ribbon) braid operad, you get (ribbon) braided strict monoidal categories; and if you do this with the action operad of all terminal groups, you get back plain old strict monoidal categories. The only other “naturally-occurring” example of an action operad that I know of is the operad of $n$-fruit cactus groups, $J_{n}$. These groups come up in the representation theory of quantum groups, particularly the theory of crystals, and the monoidal structure you get out here is something Drinfeld called a coboundary category. I can give you a generators-and-relations definition of these groups, at which point I would have completely exhausted my understanding of this operad. The best reference that I know of is the paper Crystals and coboundary categories by Henriques and Kamnitzer.

What does this have to do with my question about operads of finite groups? Well, as it turns out, the structure map for an action operad $\pi:G \rightarrow \Sigma$ only has two options: it can be surjective, or it can be the zero map (i.e., everything maps to the identity permutation). Furthermore, that condition I wrote down relating group multiplication and operadic composition says that giving an action operad with $\pi$ the zero map is equivalent to giving an operad in which the operadic composition maps all preserve group multiplication. Alex and I already showed that the operadic composition of all identity elements is the identity element in the target, in other words an action operad with $\pi$ the zero map is just an operad in the category of groups using the cartesian product.

You can take kernels of maps between action operads, so in particular given any action operad $G$ you can take the kernel of $\pi:G \rightarrow \Sigma$; this gives you an operad in groups. For the examples above, you get finite groups when $G$ is the terminal operad or $G = \Sigma$ (as the terminal operad is obviously the kernel in the case of $G = \Sigma$), but for the rest of the examples you get an operad in the category of groups, but most of those groups are infinite. The groups involved are the so-called pure versions: pure braids, pure ribbon braids, and pure $n$-fruit cacti. One can then think of an action operad with surjective map $\pi$ as being an extension of the symmetric operad by an operad in the category of groups (the “pure” version), and that action operad is finite if and only if the operad in the category of groups is one containing only finite groups. We can translate this back into thinking about monoidal categories by then noting if we have some notion of strict monoidal category in which $n$th tensor powers come equipped with a natural action of a finite group $G(n)$ for all $n$, then we must be able to dig up an operad in the category of finite groups.

Now let’s talk concrete examples: what operads do I know in the category of finite groups? Well, there is obviously the terminal operad, but I can go very slightly further in that I can tell you how to construct some new ones. Here are two methods you can use to construct operads of finite groups.

• Let $A$ be a finite abelian group. Then there is an operad $\underline{A}$ where $\underline{A}(n) = A^{n}$ (the power here is a cartesian power). The operadic composition map $$A^{n} \times A^{k_{1}} \times \cdots \times A^{k_{n}} \rightarrow A^{\Sigma k_{i}}$$ takes the vector $(a_{1}, \ldots, a_{n})$ in the first coordinate and duplicates the $i$th coordinate $k_{i}$ times, then adds the result to the vector you get by just concatenating the $n$ vectors in the $A^{k_{i}}$. Here $A$ must be abelian as it appears as $\underline{A}(1)$, and $G(1)$ must be abelian for any action operad by an Eckmann-Hilton argument.
• Now let $G$ be any finite group, but in fact you will see that we don’t use anything interesting about the group structure here. There is an operad $G^{c2}$ with $$G^{c2}(n) = G^{\binom{n}{2}}$$ given the pointwise group structure. You should think of this as the set of functions from $\{ (i,j): 1 \leq i \lt j \leq n \}$ to $G$. Operad composition is a map $$G^{\binom{n}{2}} \times G^{\binom{k_{1}}{2}} \times \cdots \times G^{\binom{k_{n}}{2}} \rightarrow G^{\binom{\Sigma k_{i}}{2}}.$$ If we are given $1 \leq a \lt b \leq \sum k_{i}$, we must give back an element of $G$. If there is some $r$ such that $$\sum_{i=1}^{r-1} k_{i} \leq a \lt b \lt \sum_{i=1}^{r} k_{i},$$ then we use the function coming from $G^{c2}(r)$ evaluated on $$1 \leq a +1 - \sum_{i=1}^{r-1} k_{i} \lt b+1- \sum_{i=1}^{r-1} k_{i} \leq k_{r}.$$ If not, then there exist $r \lt s$ such that $a$ lies in the $r$th “interval” as we had before and $b$ lies in the $s$th interval, so then you use the function coming from $G^{c2}(n)$ evaluated on $r \lt s$.

I am reasonably content with the first of these constructions, I understand how to do the second but don’t really know where it comes from, and I have some ideas about how one could try to insert the finite group of their choice as $G(0)$ but haven’t checked the details, and know basically nothing else. Furthermore, I don’t know how these interact with each other, or how you can form extensions of $\Sigma$ with them outside of some obvious constructions.

Those are just the two straightforward approaches that I know of to construct operads in the category of finite groups. You can also try to construct these operads from operads of topological spaces or simplicial sets, but once again I don’t know of an example that produces finite things (apart from ones giving the groups above). Do you know any others?