Timeline for Internal $2$-categories
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2021 at 21:08 | answer | added | varkor | timeline score: 5 | |
| Sep 6, 2019 at 10:51 | comment | added | Mike Shulman | The nLab page is fairly nebulous, but once you fix particular meanings of "n-category" and "k-category" you can make the definition precise. See the references linked there: Batanin gave one precise notion of $(n\times 1)$-category, and Douglas-Henriques study, IIRC, 2x1-categories (closely related to the definition you wrote down) and perhaps also mention 1x2-categories. | |
| Sep 6, 2019 at 9:46 | comment | added | Alec Rhea | @DavidRoberts Interesting, so this would be an example of a naturally occurring $2$-category internalized at the $1$-categorical level -- in light of Kevin's comment above this type of internalization is perhaps all we should expect to occur naturally. | |
| Sep 6, 2019 at 9:34 | comment | added | Alec Rhea | @MikeShulman Much appreciated, it looks like this might be the higher existing notion that trivializes the above example. I notice a lack of a 'definition' section -- has "someone" (wink) written it down and not latexed it up (completely understandable as the above took some time and is presumably much shorter), or is the notion still too nebulous for a precise definition? All the examples on the page are just $2$-categories internal to $1$-categories, but the references look more promising. | |
| Sep 6, 2019 at 9:27 | comment | added | Alec Rhea | @KevinCarlson It was clever not to write down the domain of that map ;). The pointer is appreciated -- regarding the last sentence, you're saying we can interpret any statement about bicategories inside a category with finite limits, so we don't 'gain anything' by internalizing them in a higher setting? Nick Gurski gave a fully algebraic formalism for weak $3$-categories in his 2009 thesis (and proved a corresponding coherence theorem for weak $3$-functors), so presumably the same would be true at the $3$-categorical level? | |
| Sep 5, 2019 at 22:19 | comment | added | Kevin Carlson | @MikeShulman Ah, so the isomorphism in $\mathfrak C$ is a reason why $(fg)h$ and $f(gh)$ are "equal", as opposed to an isomorphism in $C$ between two things that might have been more strictly equal, interesting. | |
| Sep 5, 2019 at 21:09 | comment | added | Mike Shulman | You might be interested in ncatlab.org/nlab/show/%28n+%C3%97+k%29-category and the references at the bottom. | |
| Sep 5, 2019 at 21:08 | comment | added | Mike Shulman | @KevinCarlson Both choices are sensible, though in general distinct. One might say that one gives an "internal 2-category" and the other an "internal bicategory", with the warning that an internal 2-category is not "fully strict" when internalized in a 2-category rather than a 1-category. | |
| Sep 5, 2019 at 15:27 | comment | added | Arnaud D. | Dominique Bourn has written a few papers on (strict) $n$-groupoids internal to a category with finite limits (or sometimes an exact category). You can find some titles and short summaries on his webpage. | |
| Sep 5, 2019 at 0:30 | comment | added | David Roberts♦ | For a semilocally 2-connected space $X$, the fundamental bigroupoid $\Pi_2(X)$ lifts to be internal to $\mathbf{Top}$. | |
| Sep 4, 2019 at 20:28 | comment | added | Kevin Carlson | It seems odd to ask for an isomorphism $(fg)h\cong f(gh)$ in $\mathfrak C$. Better to ask for them to be isomorphic in $C$, so that the map $\left(\circ_C\circ (\circ_C\times 1),\circ_C\circ (1\times \circ_C)\right)$, with codomain $\mathbf{1-Hom}_C\times \mathbf{1-Hom}_C$, is supposed to factor through $(\mathbf{dom},\mathbf{cod}):\mathbf{2-Hom}_C\to \mathbf{1-Hom}_C\times \mathbf{1-Hom}_C$, giving the associator, etc. The theory of bicategories is essentially algebraic and can be modeled in any cat with finite limits. | |
| Sep 4, 2019 at 14:11 | history | asked | Alec Rhea | CC BY-SA 4.0 |