### Generalized Multicategories

#### Posted by Mike Shulman

Geoff Cruttwell and I have just finished an improved version of our paper “A unified framework for generalized multicategories,” which can be found here. Although the main purpose of the paper is, as the title says, to describe a framework which unifies the many existing approaches to generalized multicategories, we deliberately started out from basics so that it would also be suitable as an introduction to the subject. So whether or not you’ve ever encountered generalized multicategories before, if you have a chance to look over the paper, we’d appreciate any feedback. It’s a little long, but not as long as it looks, since a lot of space is taken up by diagrams.

(Coincidentally, the timing of this post is quite good; this paper is another example of the importance of double categories, proarrow equipments, and lax functors. The notion of “representability” for generalized multicategories is also closely related to the “algebraic/non-algebraic” dichotomy under discussion here.)

I won’t say much more here, because I want you to go read the paper, but let me try to whet your appetite a bit. In particular, if you’ve never seen generalized multicategories before, I’d like to try to convince you that they’re interesting. (I gave a very different sort of introduction back here. To start with, an *ordinary* multicategory is like a category, except that the source of an arrow can be a finite list of objects instead of a single object. A nice example is the multicategory of vector spaces, in which a “multiarrow”
$(V_1, \dots, V_n) \to W$
is just a multilinear map. The composition of multiarrows can be visualized by drawing them as trees with $n$ inputs and one output, and then “plugging” outputs into inputs. There are some nice ASCII pictures at week 191.

Now, a *generalized multicategory* is what we get when we replace the “finite lists” by “something else.” It turns out that a lot of interesting and well-known things pop up when we make clever choices of the “something else.” For example:

If we replace “finite lists” by “finite lists whose elements can be permuted,” we get

*symmetric multicategories*. These are actually what John was talking about in week 191, or rather their one-object version called*operads*.Of course, if we replace finite lists by “single objects,” then we just get out

*categories*again.If we replace them by “finite lists whose elements can be discarded or duplicated,” we get (multi-sorted)

*Lawvere theories*, along with some variants that I was talking about back here.If we replace them by “objects labeled by an integer,” we get

*$\mathbb{Z}$-graded categories*, i.e. categories in which every morphism has an degree $\in\mathbb{Z}$ and composition is additive on degrees.If we internalize in globular sets and replace finite lists by “pasting diagrams,” we get (multi-sorted) Batanin/Leinster

*globular operads*.And if we enrich over truth values and replace finite lists of objects by

*ultrafilters*of objects, we get…*topological spaces*! The idea here is that if $U$ is an ultrafilter of objects, then there’s an arrow $U\to x$ (necessarily unique, since we’re enriched over truth values) if and only if $U$*converges*to $x$. This is the Australian definition of a topological space.

In general, the basic data for a “type of generalized multicategory” consists of a *monad* $T$. The idea is that if $X$ is the set of objects, then $T X$ is the set of “possible sources” for a multiarrow. (For ordinary multicategories, $T$ is the “free monoid” monad.) The problem is that $T$ is often a *lax* functor—but as we’ve recently seen there is no good 2- or 3-category of lax functors, so it’s unclear what a “lax monad” should mean. Different authors have come up with different ideas of what the relevant monads should look like, but usually only relative to a particular bicategory or class of bicategories. What one would really like, however, is a framework that captures all the above examples at once.

Geoff and I realized that this problem can be solved by using double categories, or more precisely proarrow equipments, since for those we do have a 2- or 3-category of lax functors. Double categories also solve another problem: if we just consider $T$ as a monad (lax or not) on a bicategory, then it’s not immediately clear how to define *functors* and *transformations* between generalized multicategories, but if we use double categories, both fall out automatically.

The final nifty thing I want to mention is the idea of *representability*. In the multicategory of vector spaces, we have *universal* multiarrows $(V_1,\dots, V_n) \to V_1\otimes \dots \otimes V_n$. This, in turn, implies that $Vect$ is not just a multicategory but a *monoidal* category. In general, any pseudo $T$-algebra has an underlying generalized multicategory relative to $T$, and a generalized multicategory comes from a $T$-algebra if and only if its multiarrows have this “representability” property. I believe this was first written down in some generality by Claudio Hermida. It also makes sense in the general double-categorical context as long as our double categories are actually proarrow equipments.

Because of this close connection between algebras and multicategories for a monad, Geoff and I proposed to call generalized multicategories defined relative to the monad $T$ *virtual $T$-algebras*. This is most advantageous when working with a monad $T$ whose algebras are called (say) “widgets,” but for which $T$ itself has no name other than “the free widget monad,” since it is more convenient and descriptive to say “virtual widget” than “free-widget-multicategory.” In fact, for the theory of generalized multicategories it turned out to be necessary to use, not ordinary double categories, but *virtual double categories*, a.k.a. fc-multicategories.

(**Edit:** There’s a confusing subtlety here in that some notions of multicategory can be defined starting from two different monads $T$, using slightly different definitions of “generalized multicategory.” For instance, ordinary multicategories are related both to the “free monoid” monad and to the “free monoidal category” monad, while virtual double categories are related both to the “free category” monad and to the “free double category” monad. Tom Leinster’s terminology “$T$-multicategory” uses the former kind of definition, hence the name “fc-multicategory” with “fc” standing for the “free category” monad. Geoff and I use the latter kind of definition, since it turns out to include more examples; hence “virtual double category” rather than “virtual category.” There also seems to be some dispute about the appropriateness of “virtual;” see the comments below.)

You can have some fun working out what representability means in all the above examples, but here’s one I didn’t mention before. Suppose we fix a category $A$ and work internal to $Cat^{ob(A)}$, with $T$ the monad whose algebras are functors $A^{op}\to Cat$. Then a virtual $T$-algebra turns out to be just a category $C$ equipped with a functor $C\to A$, and the “representability” property turns out to say precisely that this functor is a fibration. Thus, the equivalence between fibrations and pseudofunctors is a special case of representability for generalized multicategories.

## Re: Generalized Multicategories

Nice stuff, Mike!

Somebody should start internalizing this nice general theory into the $(\infty,1)$-context:

as you know (but I’ll repeat it for the record), there are currently two different definitions of $(\infty,1)$-operad in the literature: one of them $\infty$-categorifies the definition of ordinary operads in terms of its category of operators, the other defines the expected notion of nerve of an $(\infty,1)$-operad in terms of dendroidal sets.

It would be nice to have in addition a definition in terms of $(\infty,1)$-monads. I suppose if we’d had a notion of

proarrow equipment of an $(\infty,1)$-category(which sounds easier) orframed $(\infty,2)$-category(which sounds harder), we’d be in business, by directly applying your theory.We had a discussion about this already last time. But I am not sure: have you seriously thought about the notion of $(\infty,1)$-proarrow structures?