What are the higher morphisms between enriched higher categories? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T01:03:26Z http://mathoverflow.net/feeds/question/106096 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106096/what-are-the-higher-morphisms-between-enriched-higher-categories What are the higher morphisms between enriched higher categories? Theo Johnson-Freyd 2012-09-01T06:21:42Z 2012-09-10T03:15:41Z <p>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.</p> <p>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 <em>enriched in $S$</em> 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 $S$.</p> <p>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 <em>de-enrichment</em> $A_\delta$ 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.)</p> <p>Suppose that $A$ and $B$ are both $S$-enriched categories. This is where I start to run into trouble. I understand what an <em>$S$-enriched functor</em> $A \to B$ is. But I'm having trouble figuring out what is the correct definition of "natural transformation", and in general of the higher morphisms.</p> <blockquote> <p><strong>Question:</strong> Given a symmetric monoidal $n$-category $S$, what is <em>the $(n+1)$-category</em> of $S$-enriched ($n$-)categories? In particular, what are the higher morphisms?</p> </blockquote> <p>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.</p> <p>A final remark: The notion of enrichment in this question is not the same as in <a href="http://mathoverflow.net/questions/102492/n-categories-enriched-in-n1-categories" rel="nofollow">n-categories enriched in an (n+1)-category</a>. 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.</p> http://mathoverflow.net/questions/106096/what-are-the-higher-morphisms-between-enriched-higher-categories/106145#106145 Answer by Mike Shulman for What are the higher morphisms between enriched higher categories? Mike Shulman 2012-09-02T00:17:17Z 2012-09-02T00:17:17Z <p>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?</p> http://mathoverflow.net/questions/106096/what-are-the-higher-morphisms-between-enriched-higher-categories/106771#106771 Answer by Mike Shulman for What are the higher morphisms between enriched higher categories? Mike Shulman 2012-09-10T03:15:41Z 2012-09-10T03:15:41Z <p>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</p> <p>$$C \otimes \mathbf{2} \to D$$</p> <p>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 $a$.</p>