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I'm a mathematician interested in algebraic topology, category theory, homological algebra, and K-theory.


20h
answered An example of two cofibrant dg categories whose tensor product is not cofibrant
20h
reviewed Approve What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?
1d
comment Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$
But you should have a way of controlling that, right? The third object is not far from the first two ones, and you say you can extend the nat. trans. from those two ones.
1d
comment Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$
What about restricting and then extending?
2d
comment Categorical proof subgroups of free groups are free?
What's the meaning of 'categorical proof'?
Jan
24
awarded  Popular Question
Jan
24
comment Question about Lusternik-Schnirelmann Category?
How do you happen to know that the equation holds?
Jan
19
comment Is the functor of points of a scheme cofinally small?
I might be very confused w.r.t. the first question, but what about the set $\{B\}$ if $\operatorname{Spec}(B)\rightarrow X$ is an atlas?
Jan
17
comment Homological algebra is linearized homotopical algebra?
If you work with simplicial sets, rather than simplicial complexes, the idea is clear, as the comments suggest. Given a simplicial set $X$ we can take the free abelian group functor $\mathbb Z[X]$, which yields a simplicial abelian group. From this simplicial abelian group you can define a chain complex with the same sequence of abelian groups and alternating sums of face operators as differentials $d=\sum_{i=0}^n(-1)^id_i$.
Jan
17
comment Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
Shit, maybe I linked to the wrong paper, there were 50% possibilities! Anyway, I hope you've found what you were looking for @DmitriPavlov !
Jan
16
comment Which properties of a variety are detected by its derived category of coherent sheaves?
@waikit as Adeel says, although at the higher categorical level $D(QCoh(X))$ is the free cocompletion of $D^b(Coh(X))$ at the triangulated level there's no notion of cocompletion. However there are still open questions in this direction, like the Margolis Conjecture, which is the analogue of this in stable homotopy theory mathoverflow.net/questions/67227/…
Jan
15
answered Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
Jan
12
comment Which properties of a variety are detected by its derived category of coherent sheaves?
3. is slightly different: $X$ is regular iff $D^b(Coh(X))\subset D(QCoh(X))$ is the subcategory of compact objects.
Jan
11
awarded  Yearling
Jan
9
comment Quillen adjunction betwen simplicial presheaves and cochain complexes
@Cepu since $L$ and $R$ are arbitrary, I can't imagine how the specification of the model categories might help. Maybe if you specify what functors you have in mind people can help.
Jan
9
comment Quillen adjunction betwen simplicial presheaves and cochain complexes
@GeoffroyHorel Chain complexes is not simplicial either since Dold-Kan does not strictly preserve tensor products.
Jan
9
comment Is there a “simplification” functor in algebraic topology?
Rethinking about Eric's example, I think it also works to answer negatively my question in a previous comment: there's no map $f$ such that $f$-local spaces are simple spaces.
Jan
8
comment Is there a “simplification” functor in algebraic topology?
@TylerLawson the extension need not be unique, though, so the inclusions $S^1\vee S^n\rightarrow S^1\times S^n$ would not become invertible after simplification (if it happens to exist as a Bousfield localization).
Jan
8
comment Is there a “simplification” functor in algebraic topology?
Eric Wofsey's excellent example reminds me of Mislin's example showing that localizations in the homotopy category need not exist. But one can ask about homotopy localizations (using mapping spaces rather than sets of homotopy classes). Is there any map $f$ such that $f$-local spaces are precisely the simple spaces?
Jan
8
revised Is there a “simplification” functor in algebraic topology?
two misprints