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Stream: theory: category theory

Topic: Trimble n-categories, "weakly enriched"

Patrick Nicodemus (Jan 27 2024 at 23:42):

I have been reading the n-lab page on Trimble n-categories and I noticed that a morphism of n-categories is "strict" according to this definition. So you have weakly associative and unital composition but a strict interchange law.

Patrick Nicodemus (Jan 27 2024 at 23:42):

Eugenia Cheng and Tom Leinster have given a notion of $(\mathcal{V},P)$ enriched categories for $P$ an operad on $\mathcal{V}$. If you could prove inductively that Trimble $n$-categories are enriched over themselves then it might be possible to modify the definition of Trimble $n$-category so that it is fully weak and so includes, for example, tricategories.

Patrick Nicodemus (Jan 27 2024 at 23:42):

One thing I thought of was that if $\mathcal{C},\mathcal{D}$ are two $(\mathcal{V},P)$ enriched categories then you could define a weak functor $\mathcal{C}\to\mathcal{D}$ in terms of a function $F_0 : \mathcal{C}_0\to\mathcal{D}_0$, and, for all sequences of objects $x_0,\dots, x_n$, a $\mathcal{V}$-morphism

$Q_{n} \otimes \mathcal{C}(x_0,x_1)\otimes\dots\otimes \mathcal{C}(x_{n-1},x_n)\to \mathcal{D}(F_0(x_0),F_0(x_n))$

where $Q$ is a kind of generalized operad controlling all the different ways that the functor $F$ can be applied together with the composition actions in $\mathcal{C}$ and $\mathcal{D}$ coordinated by $P$. In my intuition, $Q = \mathcal{P}\circ\mathcal{P}$, i.e. an element of $Q$ represents compositions in $\mathcal{C}$ (the first $\mathcal{P}$), then the application of the weak functor, then composing everything in $\mathcal{D}$ (the second $\mathcal{P}$)

Patrick Nicodemus (Jan 27 2024 at 23:43):

I am not familiar with enough differerent kinds of operads and I struggle to recognize a familiar definition here, $Q$ seems like it is some kind of typed or colored operad. Does anyone recognize a known concept in this construction? In particular it would be nice if there were elegant rather than ad-hoc characterizations of the kinds of coherence laws $Q$ should have to satisfy with respect to the action of $\mathcal{P}$ on $\mathcal{C}$ and $\mathcal{D}$.

Patrick Nicodemus (Jan 27 2024 at 23:46):

I spent a while working it out but I don't think I have time to get sucked into this particular rabbit hole right now :laughing:

Patrick Nicodemus (Jan 28 2024 at 00:26):

One hint is that in Cheng's "Comparing operadic theories of $n$-category" she reduces the notion of a $(\mathcal{V},P)$ category to an instance of a "generalized operad" with respect to a monad, that might be a useful starting point.

Todd Trimble (Jan 28 2024 at 23:48):

Patrick Nicodemus said:

I spent a while working it out but I don't think I have time to get sucked into this particular rabbit hole right now :laughing:

Neither do I. I've thought about the general issue, certainly, but the margin here is too small to contain my thoughts on the matter.

(Seriously: it would take me a while. I've been meaning to write some things down though. But it's vaguely similar to what I gather you're suggesting.)