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Higher vector spaces
Well, there is a trivial observation: an idempotentcomplete category $\mathcal{C}$ is (equivalent to) the category of finitely presented $R$modules if and only if $\mathbf{Ind}(\mathcal{C})$ is (equivalent to) the category of $R$modules. 
Oct 15 
revised 
$(LLP(Epi), Epi)$ is a WFS on any variety of algebras
added 1 character in body 
Oct 15 
answered  $(LLP(Epi), Epi)$ is a WFS on any variety of algebras 
Oct 15 
comment 
$(LLP(Epi), Epi)$ is a WFS on any variety of algebras
$U$ should be only faithful, not fully faithful. 
Oct 15 
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Cocomplete but not complete abelian category
This looks like the direct limit of an $\mathbf{On}$indexed sequence of cocomplete abelian categories along colimitpreserving exact functors, so it should be just abstract nonsense that it is a cocomplete abelian category. 
Oct 13 
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Homotopy uniqueness of some coends
Essential uniqueness here is referring to the fact that some classifying space is contractible. I wouldn't read too much into it. 
Oct 13 
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Homotopy uniqueness of some coends
But the diagrams in question are indeed cofibrant? 
Oct 11 
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Question regarding 2mathematics: Can you stackify a 2functor without prestackifying it first?
This seems to be a folklore result. It is alluded to in [Higher topos theory, §6.5.3]. 
Oct 9 
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The Karoubi model structure on Cat
Yes, it is a left Bousfield localisation of the canonical model structure, almost by definition. The generating cofibrations are therefore the same as for the canonical model structure. 
Oct 8 
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The Karoubi model structure on Cat
I describe the model structure here. There is just one extra generating trivial cofibration, namely the inclusion of the free idempotent into the free split idempotent. 
Oct 2 
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When is the product of an infinite family of simplicial sets also a homotopy product?
Here's a simple observation: let us say that a simplicial set $X$ has property $C_n$ if two vertices are in the same connected component of $X$ if and only if there is a path of length $\le n$ connecting them. So for example Kan complexes have property $C_1$. Then for every positive integer $n$, $\pi_0$ preserves products of families of simplicial sets having property $C_n$. This seems to suggest that it suffices to satisfy the Kan condition up to some finite subdivision... 
Sep 19 
awarded  Enlightened 
Sep 18 
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Commutation of simplicial homotopy colimits and homotopy products in spaces
What exactly are you asking, though? If you're asking whether the functor $\prod : \mathcal{S}^{I} \to \mathcal{S}$ preserves homotopy colimits, then the answer is no even for $I = 2$: after all, $(1 + 1) \times (1 + 1) \ne (1 \times 1) + (1 \times 1)$. But if you're asking whether it preserves homotopy colimits in each variable, then the infinite case reduces to the finite case and there's no problem. 
Sep 10 
answered  Categories of spans from categories of fibrant objects 
Sep 3 
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Is there an analog of the BarrattEccles construction for grouplike E_∞spaces and E_∞ring spaces?
@QiaochuYuan It seems to me that there is a lot of interest in enriched algebraic theories coming from the theoretical computer scientists. 
Sep 2 
accepted  Aspheric functors and Grothendieck fibrations 
Aug 31 
asked  Aspheric functors and Grothendieck fibrations 
Aug 30 
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Is there a general notion of semigroup action?
@QiaochuYuan It's a nice trick for $\mathbf{Set}$ but it doesn't work properly in general: you would need every subobject to have a complement in the strong sense. 
Aug 26 
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Modern versions of Verdier's hypercovering theorem?
Thanks for your answer! It took a long time to check all the details, but now I understand what's going on. There is indeed a formula in terms of the simplicial hom spaces, as you said. 
Aug 26 
answered  Modern versions of Verdier's hypercovering theorem? 