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5h

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A model category of abelian categories?
One needs a little bit more than chosen (co)limits to define strict abelian categories. However, §8 of the paper does sketch an extension which might be relevant. 
2d

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Reflective Localizations vs. categories of local objects
Yes, that's one way of looking at it. Another way is to observe that we have a calculus of right fractions, so we have a formula for the homsets of the localisation, and it is essentially the plus construction. 
May 24 
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Reflective Localizations vs. categories of local objects
@NicolasSchmidt I have found a counterexample. Take $\mathcal{W}$ to be the class of $J$dense monomorphisms in the category of presheaves on a small site $(\mathcal{C}, J)$; then the $\mathcal{W}$local objects are the $J$sheaves (which are, of course, reflective) but localisation with respect to $\mathcal{W}$ is not reflective. 
May 19 
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A generalization of the SpanierWhitehead construction
I don't think it's a good idea to do (1) as a full subcategory. Having $F \dashv S \dashv F$ should really be thought of as extra structure on $S$, not a property. 
May 18 
awarded  Popular Question 
May 17 
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The classifying space of an infinite totally ordered set is contractible
It's probably not true without hypotheses on the diagram. It suffices that the functor $\mathbf{Top} \to \mathbf{sSet}$ preserve the colimit in question. I guess that happens if the arrows in the diagram are closed $T_1$embeddings. 
May 17 
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The classifying space of an infinite totally ordered set is contractible
@ToddTrimble Filtered colimits (of simplicial sets) preserve weak homotopy equivalences, hence filtered colimits of weakly contractible simplicial sets are weakly contractible. Since the space in question is the geometric realisation of a simplicial set, this proves the claim. 
May 15 
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Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces
For the argument I was thinking of, I really want $\pi_0 : \mathbf{Top} \to \mathbf{Set}$ to be a left Quillen functor. But perhaps we can work around that problem by first passing through the Quillen equivalence $\mathbf{Top} \to \mathbf{sSet}$. 
May 15 
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If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?
@TimCampion That is what Łoś's theorem tells us, yes. 
May 15 
revised 
If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them?
added 227 characters in body 
May 15 
answered  If two structures are elementarily equivalent, is there a zigzag of elementary embeddings between them? 
May 15 
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Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces
Morally, $\pi_0 (B \mathcal{D})$ should be the coequaliser of $\pi_0 (\operatorname{mor} \mathcal{D}) \rightrightarrows \pi_0 (\operatorname{ob} \mathcal{D})$; however, $\pi_0 : \mathbf{Top} \to \mathbf{Set}$ is not a left adjoint unless $\mathbf{Top}$ means something like locally connected spaces. 
May 15 
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Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces
Neither: two objects become identified if there is a zigzag of morphisms connecting them. 
May 13 
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A model category of abelian categories?
I wonder if there's an actual proof in the literature somewhere? Anyway, Ignacio López Franco suggested to me an argument for reducing the monadicity of $\mathfrak{M}$ to the monadicity of categories with finite (co)limits, so I can moreorless believe that there is a 2monad of the desired form. But it would be nice to have some intuition for when a 2category of categorieswithstructure is 2monadic. 
May 12 
awarded  Nice Question 
May 12 
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A model category of abelian categories?
But is there such a finitary 2monad? I'm not even sure the underlying ordinary adjunction here is monadic. 
May 12 
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A model category of abelian categories?
Interesting. If the answer to Q1 and Q2 are both yes, then I would expect to get some kind of calculus of fractions. (Notice that the purported model structure in Q1 has the property that all objects are fibrant.) 
May 12 
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A model category of abelian categories?
Question 3 is the most interesting part, and questions 1 and 2 seem like the natural way to proceed. 
May 11 
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Reflective Localizations vs. categories of local objects
It's true if the components of the adjunction unit are in $\mathcal{W}$, but I don't know what happens in general. 
May 11 
revised 
A model category of abelian categories?
added 38 characters in body 