Timeline for Limits and colimits in a 2-category vs. in an infinity category: the non- (2,1)-case
Current License: CC BY-SA 3.0
9 events
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| Feb 24, 2018 at 6:45 | comment | added | David Carchedi | @DavidJordan This is implicit in Lemma 4.2.4.3 of HTT, if you apply it to a $(2,1)$-category, since it says they have the same universal property. | |
| Sep 29, 2017 at 11:15 | comment | added | David Jordan | Out of curiosity, do you know a clear reference for the implicit statement in your question that for $(2,1)$-categories the $(2,1)$-colimit and the $(\infty,1)$-colimit coincide? | |
| Aug 21, 2014 at 12:47 | comment | added | David Carchedi | @Zhen: There's lots of examples like this, I was just hoping there was some relation between limits and colimits in the bicategory and in the $(\infty,1)$-category. They certainly won't be the same- that would be way too strong. I'm just wondering if there's any relation at all. shrug? | |
| Aug 21, 2014 at 11:20 | comment | added | Zhen Lin | Hmmm. Consider the free 2-cell. If I'm not mistaken, its homotopy coherent nerve is a simplicial set with two vertices, two non-degenerate edges (which are parallel), and some number of 2-simplices saying that the two non-degenerate edges are homotopic. So after taking Joyal-fibrant replacement, one should have a quasicategory that has an initial object – but this is not the case in the original 2-category. | |
| Aug 21, 2014 at 10:58 | comment | added | David Carchedi | I'm hoping perhaps that some weighted limits and colimits in the bicategory can be used to get my hands on limits and colimits in the $(\infty,1)$-category, or something along these lines. | |
| Aug 21, 2014 at 10:57 | history | edited | David Carchedi | CC BY-SA 3.0 |
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| Aug 21, 2014 at 10:56 | comment | added | David Carchedi | @Zhen: I mean bicategorical, not $2$-categorial, so I will edit. And yes, of course $(2,1)$-categorical limits and colimits are the same if I construct the associated $\left(\infty,1\right)$-category simply by taking the maximal groupoid at the level of mapping spaces, but that's precisely what I don't want to do. My construction isn't "incorrect", I'm just after a different thing. I've constructed a bicategory where the mapping categories are supposed to model the homotopy type (via their classifying space) of the mapping spaces in the $(\infty,1)$-category I'm really after. | |
| Aug 21, 2014 at 8:04 | comment | added | Zhen Lin | Traditionally, 2-categorical (co)limits simply refers to the special case of $\mathcal{V}$-enriched (co)limits for $\mathcal{V} = \mathbf{Cat}$, and these coincide with ordinary (co)limits in nice situations. You are probably thinking of bicategorical (co)limits, which are invariant under equivalence – but (under the assumption of existence) these will also be bicategorical (co)limits in the underlying (2, 1)-category. I would expect these to coincide with (∞,1)-(co)limits, but I think your construction of the associated (∞,1)-category is not correct | |
| Aug 21, 2014 at 1:29 | history | asked | David Carchedi | CC BY-SA 3.0 |