What are the higher morphisms between enriched higher categories?

2012-09-01T06:21:42Z

This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ... My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me force you to work in my favorite model, I ask only that you say with some precision what model you are using, and that your model is not too strict.

You may assume that I have, and am comfortable with, some chosen symmetric monoidal $n$-category $(S,\otimes)$. I am trying to understand the notion of "($n$-)category enriched in $S$". By this I would like to mean the following. Roughly, $A$ is enriched in $S$ if $A$ has a set of $0$-morphisms, and between any pair of $0$-morphisms $X,Y \in A$, there is an $S$-object $A(X,Y)$ of 1-morphisms between them. The composition should be a morphism $A(X,Y) \otimes A(Y,Z) \to A(X,Z)$ in $S$.

I am fairly satisfied that, at least in my example, I can write all of this down explicitly. Recall that $S$ comes with a chosen object $1 \in S$. Then the corepresentable functor $S(1,-) : S \to (n-1)\text{-Cat}$ is symmetric monoidal in an essentially unique way. The usual thing is to use this functor, and define the de-enrichment $A_\delta$ of $A$ to be the $n$-category with hom-$(n-1)$-categories given by $A_\delta(-,-) = S(1,A(-,-))$. Certainly in my example I can work out this $(n-1)$-category. (De-enrichment is often a highly lossy operation, and I am OK with that.)

Suppose that $A$ and $B$ are both $S$-enriched categories. This is where I start to run into trouble. I understand what an $S$-enriched functor $A \to B$ is. But I'm having trouble figuring out what is the correct definition of "natural transformation", and in general of the higher morphisms.

Question: Given a symmetric monoidal $n$-category $S$, what is the $(n+1)$-category of $S$-enriched ($n$-)categories? In particular, what are the higher morphisms?

I recognize that this "$(n+1)$-category of $S$-enriched categories" is likely itself enriched in $S$-enriched categories. But I am interested simply in writing down its de-enrichment — I'm looking for it just as an $(n+1)$-category.

A final remark: The notion of enrichment in this question is not the same as in n-categories enriched in an (n+1)-category. That question concerned $n$-categories in which the collection of $n$-morphisms was an object of the enriching category. I would like the collection of $1$-morphisms to be an object of the enriching category.

Answer by Mike Shulman for What are the higher morphisms between enriched higher categories?

2012-09-02T00:17:17Z

For $F,G:C\to D$, an $S$-enriched transformation $F\to G$ will consist of, for each $x\in C$, a morphism $1\to D(F x, G x)$ in $S$, together with for each $x,y\in C$, an equivalence between the composites $$ C(x,y) \to D(F x, F y) \to D(F y, G y) \otimes D(F x, F y) \to D(F x, G y)$$ and $$ C(x,y) \to D(G x, G y) \to D(G x, G y) \otimes D(F x, G x) \to D(F x, G y)$$ plus some higher coherence cells until you run out of room. Does that help?

Answer by Mike Shulman for What are the higher morphisms between enriched higher categories?

2012-09-10T03:15:41Z

Another way to get the definition of an $S$-transformation, if you know what an $S$-functor is and also the tensor product of $S$-categories (so $S$ must be symmetric) and $S$ has an initial object preserved by $\otimes$ in each variable, is as an $S$-functor

$$ C \otimes \mathbf{2} \to D$$

where $\mathbf{2}$ is the $S$-category with two objects $a$ and $b$, hom-objects $1$ from $a$ to $a$, $b$ to $b$, and $a$ to $b$, and the initial object of $S$ from $b$ to $a$.

What are the higher morphisms between enriched higher categories? - MathOverflow

What are the higher morphisms between enriched higher categories?

Answer by Mike Shulman for What are the higher morphisms between enriched higher categories?

Answer by Mike Shulman for What are the higher morphisms between enriched higher categories?