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Do we have a discussion anywhere that 2-limits in the (2,1)-category of categories as defined in the 2-category-literature do coincide with the coresponding limits computed inside the $(\infty,1)$-category of $(\infty,1)$-categories?
I thought we had, but maybe we don’t. If not, I’ll try to add some discussion.
This holds for limits but fails for colimits, right?
(And this is because truncation is a left adjoint to the inclusion $(n,1) Cat \hookrightarrow (\infty,1) Cat$.)
That is why the derived intersection theory is (read: could be) better than the nonderived.
Right, but if one truncates one degree “too high” it should work for colimits, too.
This was actually the motivation for the post: somebody asked me how to precisely understand the statement at Grothendieck construction that we may compute the pushout of categories (the restriction to 1-catgeories of the pushout that Harry keeps talking about in the other thread) as a “weak pushout”.
(Well, unfortunately weak limit points to something diferent.)
I still haven’t gotten around to write more about this.
Right, but if one truncates one degree “too high” it should work for colimits, too.
Oh! I didn’t know this (although now that I think about it, I knew several examples of it). Now that’s interesting!
Right, but if one truncates one degree “too high” it should work for colimits, too.
I don’t understand what you mean.
I mean this:
given a pushout diagram for $n$-groupoids we can either form the $(\infty,1)$-pushout or form the (n+1,1)-pushout and then embed that into to the $(\infty,1)$-context. The result is the same-
The reason is intuitively that there has to be room for “one extra homotopy” only: we are suspending once. Formally, it can be seen from the fact that we are computing an ordinary pushout of the mapping cone of one of the two maps, which, as it involves forming a cyclinder, adds in non-degenerate cells of only one dimension higher.
That’s of course the reason, for instance, why it is sufficient to work with quotient stacks of a group acting on some, say, manifold, instead of having to invoke quotient $\infty$-stacks of the same situation.
For more complicated colimit diagrams of course one needs to go higher up in dimension, and generally all the way to $\infty$. For instance the colimit of a simplicial diagram of 0-groupoids is in general not an $n$-groupoid for any finite $n$.
(I am saying this just since I am at it, not because I think that you don’t know it.)
But I should have stated my original motivation here more specifically: at Grothendieck construction there is this discussion of the left adjoint to “$\int$” in terms of a pushout of categories as $(\infty,1)$-categories, and a vague remark that one may think of this as a pushout as $(2,1)$-categories, or something like that. I should eventually fill in details for what this means.
For more complicated colimit diagrams of course one needs to go higher up in dimension, and generally all the way to ∞
Ah, that’s why I was confused. I didn’t realize you intended the remark “truncating one degree higher is enough” to only apply to pushouts (pushouts not having been mentioned previously in this thread!), and I knew it wasn’t true for colimits in general. Thanks for clarifying.
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