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Suppose that $\mathscr{C}$ is a $2$-category (or more generally a bicategory) which is not a $\left(2,1\right)$-category. Is there any relation between limits and colimits in $\mathscr{C}$ (in the weak $2$-categorical sense), and limits and colimits in the associated $\left(\infty,1\right)$-category? By the associated $\left(\infty,1\right)$-category, I mean any fibrant replacement of its homotopy coherent nerve (viewing $\mathscr{C}$ as a simplicial category) in the Joyal model structure, or in the bicategory case, the Duskin nerve of $\mathscr{C}$.

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  • $\begingroup$ 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 $\endgroup$ – Zhen Lin Aug 21 '14 at 8:04
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    $\begingroup$ @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. $\endgroup$ – David Carchedi Aug 21 '14 at 10:56
  • $\begingroup$ 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. $\endgroup$ – David Carchedi Aug 21 '14 at 10:58
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    $\begingroup$ 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. $\endgroup$ – Zhen Lin Aug 21 '14 at 11:20
  • $\begingroup$ @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? $\endgroup$ – David Carchedi Aug 21 '14 at 12:47

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