5,543 reputation
11430
bio website dpmms.cam.ac.uk/~zll22
location
age
visits member for 4 years, 5 months
seen 10 mins ago

5h
comment 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
comment 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 hom-sets of the localisation, and it is essentially the plus construction.
May
24
comment 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
comment A generalization of the Spanier-Whitehead 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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 more-or-less believe that there is a 2-monad of the desired form. But it would be nice to have some intuition for when a 2-category of categories-with-structure is 2-monadic.
May
12
awarded  Nice Question
May
12
comment A model category of abelian categories?
But is there such a finitary 2-monad? I'm not even sure the underlying ordinary adjunction here is monadic.
May
12
comment 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
comment 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
comment 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