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Denis Nardin
Denis Nardin
12:15
Suppose I have a cosimplicial diagram of Kan complexes X. Is the totalization (i.e. the simplicial set of maps Δ^*→X) equivalent to the homotopy limit? Or do I need more fibrancy conditions on X?
Tom Bachmann
Tom Bachmann
12:27
@DenisNardin I would expect that you need the latching maps (or whatever they are called) to be fibrations, but maybe this is not necessary...
Denis Nardin
Denis Nardin
Hmm... maybe you're right
I think I said something false in class about the geometric realization of simplicial spaces :(
Oh well I'll fix it in the notes
 
4 hours later…
Charles Rezk
Charles Rezk
16:42
@AdrianClough @DenisNardin Thanks. I decided that the thing I actually wanted to know is that, for general $C$, the category $P_\Sigma(C)$ (the sifted colimit completion of $C$) is such that $X\in P_\Sigma(C)$ iff $C/X$ is sifted. And I think I can prove it now.
 
2 hours later…
Ian Coley
Ian Coley
18:35
@DenisNardin ncatlab.org/nlab/show/totalization#AsTheHomotopyLimit so being fibrant in the Reedy model structure would mean (I think) that all the matching maps are fibrations. See the second bullet point under 'remarks' ncatlab.org/nlab/show/Reedy+model+structure. Everything is written in terms of cofibrations/cofibrancy and I'm not the best at dualizing what I don't know
 
4 hours later…
Aaron Mazel-Gee
Aaron Mazel-Gee
22:23
@ManuelRivera it sounds like maybe you're describing the adjunction between monoid-objects and group-objects in affine schemes? (after taking opposites, of course.)
@SaalHardali as a small comment (basically echoing @DenisNardin) that you may have already realized, the original argument works as long as the operad is "quadratic", i.e. every $n$-ary operation for $n>2$ can be expressed as a composite of binary operations.

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