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Stream: theory: category theory

Topic: iterating Hom with vdcs

Noah Chrein (Jan 04 2025 at 19:50):

Is there a sense in which n-categories form some higher virtual double category? One which, upon application of hom, lands us in n-1 categories which also form a higher virtual double category?

Is there a formal way to parse the promorphisms as
$C^\text{op}\times D \to \text{Cat}_{n-1}$
where $\text{Cat}_{n-1}$ is also one of these higher virtual double categories?

So that one has access to deeper layers of hom
$\text{Cat}_n(C_0,D_0)\;\;:\;\;\text{Cat}_{n-1}$
$\text{Cat}_n(C_0,D_0)(C_1,D_1)\;\;:\;\;\text{Cat}_{n-2}$
$\text{Cat}_n(C_0,D_0)(C_1,D_1)(C_2,D_2)\;\;:\;\;\text{Cat}_{n-3}$
... and so on?

There are tons of coherence issues, and I am not asking for an equipment just yet, so no need for higher day conv. I really want to know if anyone has investigated what this kind of "higher" vdc would be, I have a few ideas.

Noah Chrein (Jan 05 2025 at 19:41):

edited the above, it was poorly worded.

Nathanael Arkor (Jan 05 2025 at 23:32):

Certainly no such account exists in the literature. It's possible to form a structure like a VDC in which one does not have a set of 2-cells, but rather an n-category, and (n + 1)-categories and their distributors form such a structure. But this only contains the data of "promorphisms" at the top level, so isn't what you're after.

Nathanael Arkor (Jan 05 2025 at 23:45):

@CB Wells' thesis The Metalanguage of Category Theory may be closer to what you are looking for (for n = 3).

Noah Chrein (Jan 07 2025 at 01:59):

Thank you for the reference, I will respond more completely to your previous idea soon. My initial thought, though, is that I am not asking for the cells to hold the lower categorical data, but rather something like restrictions along generalized elements:

Screenshot 2025-01-06 at 8.55.25 PM.png

I have more to say but I will write it out later.

Noah Chrein (Jan 07 2025 at 02:04):

I'm always weary of introducing too many details about unfinished work into posts like this but in the image above one can think of the (one object) n-category

$\bullet \overset{\mathbb C(A_0,B_0)}{\rlap{+}\to} \bullet$

as the loopspace of the (n-1)-category

$\mathbb C(A_0,B_0)\overset{\mathbb C(A_0,B_0)}{\rlap{+}\to} \mathbb C(A_0,B_0)$