By John Gowers

2019-01-11 13:21:23 8 Comments

We all know that

A monoidal category is a bicategory with one object.

How do we fill in the blank in the following sentence?

A multicategory is a ... with one object.

The answer is fairly clear: it'll be a bit like a bicategory, but instead of being able to compose $1$-cells straightforwardly, all we can do is specify, for any 'composable' sequence $$ A_0 \xrightarrow{f_1} \cdots \xrightarrow{f_n} A_n\,, $$ of $1$-cells and any $1$-cell $g \colon A_0 \to A_n$, the set of $2$-cells $$ f_1,\dots,f_n \Rightarrow g\,. $$ What is the name of such a thing? A natural example, for a given SMCC $\mathcal V$, has $\mathcal V$-enriched categories as the objects, profunctors $\mathcal C^{op}\otimes \mathcal C\to \mathcal V$ as the $1$-cells and, for any collection $$ \mathcal C_0,\cdots,\mathcal C_n $$ of categories and profunctors $F_i\colon \mathcal C_{i-1}^{op}\otimes \mathcal C_i\to \mathcal V$, $G\colon \mathcal C_0^{op}\otimes \mathcal C_n \to \mathcal V$, the set of extranatural transformations $$ \phi_{b_1,\cdots,b_{n-1}} \colon F_0(a,b_1) \otimes \cdots \otimes F_n(b_{n-1},c)\Rightarrow G(a,c) $$ as the $2$-cells $F_1,\cdots,F_n \Rightarrow G$.

Of course, if $\mathcal V$ is cocomplete, then this is equivalent to the usual bicategory of $\mathcal V$-enriched profunctors.


@Peter LeFanu Lumsdaine 2019-01-11 23:11:18

Another established structure very close to what you want is opetopic bicategories, which are equivalent to classical bicategories but formulated opetopically; see e.g. §3 of Cheng 2003, Opetopic bicategories: comparison with the classical theory, and the subsection Non-algebraic notions of bicategory at the end of §3.4 of Leinster 2003, Higher operads, higher categories.

Concretely, in an opetopic bicategory $B$, you have a graph of 0-cells and 1-cells like in a normal bicategory; the source of a 2-cell is not just a 1-cell but a composable string of 1-cells; 2-cell composition looks just like what you’d expect; and the 1-cell composition condition says that for every composable sequence of 1-cells, there’s a universal 2-cell out of them, for a certain sense of universality.

Keeping all of this except the last condition — call such a thing an opetopic bicategories minus 1-cell composition — seems to give exactly what you’re asking for. In the one-object case, the 1-cell composition condition is exactly what Leinster calls representability of a multicategory (Def 3.3.1, ibid.) — so adding this back recovers the equivalence between monoidal categories and one-object bicategories:

(monoidal category) = (multicategory with representability) = (one-object opetopic bicategory minus 1-cell composition, with 1-cell composition) = (one-object opetopic bicategory) = (one-object bicategory)

Comparing this with Simon Henry’s answer, I would expect (opetopic bicategories minus 1-cell composition) should be fairly concretely equivalent to (fc-multicategories with only identity vertical 1-cells); indeed, Leinster hints at such a connection in the subsection mentioned above, though he doesn’t spell it out precisely.

@Simon Henry 2019-01-11 13:47:31

This has been called a "fc-multicategory" by Tom Leinster, for example here.

I think this as also been called a "Hypervirtual double category" here, but I don't remember if this is exactly the same notion or if there are some additional assumption in the second link.

@John Gowers 2019-01-11 13:56:36

fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.

@Simon Henry 2019-01-11 13:59:25

Oh, yes you are write I missed that: the structure you are considering is a fc-multicategory with only identity vertical 1-cells.

@Roald Koudenburg 2019-01-11 16:17:56

Shulman and Cruttwell have been using the term "virtual double categories" for fc-multicategories. Hypervirtual double categories generalise these by also including cells with empty horizontal targets (besides the usual target: a single horizontal morphism). It looks like that, when restricting to a single identity as the set of vertical morphisms, a hypervirtual double category consists of a multicategory C equipped with a module C -|-> 1, in the sense of 2.3.6 of link, where 1 is the terminal multicategory.

@Roald Koudenburg 2019-01-11 16:19:23

This is interesting, it makes me wonder if there is a similar characterisation of general hypervirtual double categories.

@Tim Campion 2019-01-11 17:13:21

Here's the paper where Cruttwell and Shulman introduced the "virtual" terminology. "A unified framework for generalized multicategories"

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