I started comma double category. Since I care about equipments more than double categories in general, and because it actually is an instance of a comma object, I made the article mostly about virtual double categories. I wrote down a couple of conjectures about when the comma has units and composites, but haven’t verified them yet and not sure when I will.

]]>I asked this question on mathoverflow but I think I should have just posted here in the first place.

A $T$-multicategory in the sense of Crutwell-Shulman, where $T$ is a monad on a virtual double category $C$ is a monoid in the “horizontal kleisli category”, i.e., an object of objects $O$, a horizontal arrow of arrows $A : O \to T O$ with composition and identity cells. Is there a good way to define a co-presheaf on such a generalized multicategory?

This seems to give a natural notion of $T$-presheaf as just a bimodule of the monad with a terminal object, i.e. a horizontal arrow $P : 1 \to T O$ with a composite $P;A \Rightarrow P$ that is compatible with composition and identity. If you look at what this means for specific cases like $T =$ free symmetric monoidal category monad on the virtual equipment of cats, functors and profunctors this looks like the definition of presheaf you would come up with, with $P : 1 \to T O$ giving an abstract notion of map from a list of objects of $O$ to $P$ and so can be used to define universal properties like the product, and it looks like you can use the language of cartesian cells to define the right notion of representability.

On the other had, just taking the dual doesn’t look like the right thing. What I’m hoping would happen is that if I try to define the universal property of a coproduct using such a copresheaf I would be “forced” to make it a distributive coproduct. However, if we say a copresheaf is a bimodule $Q : O \to T 1$, this doesn’t look right because for our example $T$, this would give us an abstract notion of maps $Q^n \to A$ for each $A \in O$, but it seems to me that the right notion of copresheaf (based on the type theory) would be to give an abstract notion of $A_1,...,Q,...A_n \to B$.

In particular for $A,B \in O$ if I try to define a copresheaf $A+B$ by

$(A+B)^n \to C = \Pi_{m+l=n} A^m,B^l \to C$this only gives me a distributive coproduct-like behavior when $A+B$ repeated is the only thing in the domain, and it looks like for a representing object

$(A+B),C \to D = (A,C \to D)\times (B,C \to D)$will not in general be true.

]]>In “Direct Models of the Computational $\lambda$-calculus$ pdf link, Fuhrmann introduces the informal concept of a “direct model” of a programming language as a semantic notion (category-like structure) whose operations directly correspond to the syntactic constructions of the syntax. Informally, a direct model “doesn’t compile”.

For instance, they call the interpretation of call-by-value languages in the Kleisli category of a monad $T$ on a category $C$ “indirect” because you “compile” the terms of the language into morphisms of $C$ and operations in $C$ (like composition) don’t correspond to something in the syntax. Instead, they propose a “direct model” with thunk-force categories that axiomatizes the structure oft he Kleisli category of a monad directly, so that for instance, composition in the thunk-force category directly corresponds to a let-binding in the original syntax. Then the Kleisli category construction is seen as a functor from categories with a monad to thunk-force categories. Munch-Maccagnoni’s duploids are also presented in this vein.

This seems very similar in spirit to multi-categorical approaches to syntax, as I believe the original motivation for multi-categories was similarly to have a category-like structure with closer correspondence to syntax, and then to move constructions like the category of contexts completely over to the semantic realm.

What do peopl that are interested in multicategories think about this comparison? Do you have a similar term to “direct model”? I think it would be nice to have a page to collect examples of this idea and connect them.

]]>Hey everyone,

I feel like this must be written down somewhere, and you all are probably the most knowledgeable about such things. Given a monoidal category, if I take its underlying multicategory, take the free symmetric multicategory thereon, followed by the category of operators, does this thing admit a Grothendieck opfibration to the category of operators of the free symmetric multicategory associated to the associative operad (i.e. the category of operators of the multicategory with one object and mapping sets given by the symmetric groups)? This seems certainly true, but I’m more interested in finding a place where this is written down so I can cite it directly (trying to avoid going into too significant detail on multicategories in a paper I’m writing).

Thanks for any kind of references to check out!

-Jon

]]>