Timeline for Equivalences of $n$-categories
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 27, 2021 at 16:54 | vote | accept | Student | ||
| Nov 27, 2021 at 15:39 | comment | added | Theo Johnson-Freyd | To summarize: the cartesian closedness of $n\mathrm{Cat}$ is what knows that $n\mathrm{Cat}$ is not just an $(\infty,1)$-category, but rather an $(n+1)$-category. | |
| Nov 27, 2021 at 15:38 | comment | added | Theo Johnson-Freyd | Of course, for $n$-categories $X,Y$, by $\hom(X,Y)$ I mean the collection (homotopy type, $n$-groupoid, whatever) of functors $X \to Y$ and just the natural isomorphisms between functors, natural isomorphisms between isomorphisms, etc. So it is just a space. However, by fiat $A^B$ is another $n$-category. Indeed, it is an $n$-category whose groupoid of objects is the space of functors $\hom(A,B)$. The (higher) morphisms in $A^B$ are interpreted as not-necessarily-invertible (higher) natural transformations. | |
| Nov 27, 2021 at 15:35 | comment | added | Theo Johnson-Freyd | As for recovering the (n+1)-dimensional nature of the collection of all n-categories: By "exponential objects", I meant that $n\mathrm{Cat}$ is "Cartesian closed". The Cartesian product $\times$ (which is of course something that you can inquire about in any $(\infty,1)$-category) exists, of course, but moreover it has an adjoint: for any $n$-categories, there exist "exponentials" $A^B$ so that for every $C$, the homotopy types $\hom(C, A^B)$ and $\hom(C \times B, A)$ are (canonically and coherently) equivalent. | |
| Nov 27, 2021 at 15:32 | comment | added | Theo Johnson-Freyd | @Student The fundamental reason that $(\infty,1)$-categories suffice for organizing and expressing weak $n$-categories is because the fundamental axioms of a higher category — that it have some $k$-morphisms, and some operations that compose those — only ever use spaces (of morphisms) and 1-morphisms between spaces (composition data) and, vitally, equivalences (isomorphisms, homotopies, coherences, whatever you want to call them) between various orders of operations. | |
| Nov 27, 2021 at 15:28 | comment | added | Theo Johnson-Freyd | @DavidRoberts Yes, of course I should have. For other people reading these comments, David's point is that Riehl and Verity are developing a way of talking about very nice (much nicer than just presentability) $(\infty,1)$-categories of categories which is "synthetic" in the sense of being defined from axioms, rather than "analytic" in the sense of being modelled by model categories. The collection of $(\infty,n)$-categories is an example of an $\infty$-cosmos, but lots of collections are not. | |
| Nov 26, 2021 at 16:27 | comment | added | Student | It seems to be a great step to reduce the need of $(\infty,n+1)$ categories to just the $(\infty,1)$ ones (also as Marc Hoyois indicated in the comments). Do you have a reference for this ("looking at exponential objects")? | |
| Nov 26, 2021 at 2:36 | comment | added | David Roberts♦ | It is worth mentioning Riehl and Verity's $\infty$-cosmoi here. One might have extra axioms (something like "well-pointedness") that pick out the example of $(\infty,n)$-categories. | |
| Nov 26, 2021 at 0:59 | history | answered | Theo Johnson-Freyd | CC BY-SA 4.0 |