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bio website math.mit.edu/~hoyois
location Cambridge, MA, USA
age 27
visits member for 2 years, 11 months
seen Nov 24 at 1:53

I'm a postdoc at MIT.


Nov
24
comment When is a topological space the homotopy colimit of an open covering?
As a follow-up to my first comment, the proof of Lemma A.4.14 in HA shows the following. If $X$ is locally contractible, then $Map(Sh(X),K)\to Map(Sing(X),K)$ is an equivalence provided that the constant sheaf with fiber $K$ is hypercomplete. In particular, $Sing(X)$ and $Sh(X)$ have the same pro-truncated reflections. I suspect they're not the same in general because the usual definition of "locally contractible" for a space corresponds to the topos being "locally $\infty$-connective" rather than actually "locally contractible".
Nov
23
comment When is a topological space the homotopy colimit of an open covering?
I think it would be unfair not to call Lurie's theorem itself a higher Seifert-van Kampen theorem. Note that the result I've quoted is only a special case of half of the theorem. The other half (in the same special case) says that the actual colimit of the diagram of simplicial sets $Sing(C_U)$ is weakly equivalent to $Sing(X)$.
Nov
23
awarded  Nice Answer
Nov
23
reviewed Approve suggested edit on Given a Levy Exponent find the jump-measure and drift
Nov
23
revised When is a topological space the homotopy colimit of an open covering?
added 252 characters in body
Nov
23
answered When is a topological space the homotopy colimit of an open covering?
Nov
23
comment When is a topological space the homotopy colimit of an open covering?
Well, not only is the shape homotopy invariant (Higher Algebra, A.2.10), but your question is actually answered completely in that appendix, see Remark A.3.8: the functor $Shv(X)\to \infty Gpd$ induced by the "underlying homotopy type" functor preserves all colimits!
Nov
23
comment What does an endomorphism in a triangulated category give rise to?
What I mean by $\phi$-completion here is the inverse limit of the $cofib(\phi^n)$. For example, if $\phi$ is multiplication by $p$ on $\mathbb{Z}$, you get $\mathbb{Z}_p$.
Nov
23
comment When is a topological space the homotopy colimit of an open covering?
This is true in particular when the topos-theoretic shape functor $Sh$ agrees with the classical "underlying homotopy type" functor, since it's always true that $Sh(X)$ is the colimit of $Sh(C_U)$. Lurie shows that $Sh(X)$ is weakly equivalent to $X$ when $X$ is paracompact and homotopy equivalent to a CW complex (Higher Algebra, A.1.4). I'm not sure if you can generalize this to "locally contractible". It's not even clear to me that the shape of a contractible space is contractible...
Nov
23
awarded  Custodian
Nov
23
reviewed Approve suggested edit on Calculating the quotient group $\mathbb{Z}\times\mathbb{Z}/<(1,1),(1,-1)>$
Nov
21
answered What does an endomorphism in a triangulated category give rise to?
Nov
17
comment What does an endomorphism in a triangulated category give rise to?
No matter what the degree of $\phi$ is, it gives rise to a filtered object. If you apply a homological functor $H$ to it (e.g. map another object into it), you get two spectral sequences, potentially converging to $H$ of the limit and colimit of the filtered object: ncatlab.org/nlab/show/…
Nov
14
comment Is the site of (smooth) manifolds hypercomplete?
@DmitriPavlov I believe so, provided your topological manifolds are paracompact (so that homotopy dimension = covering dimension, HTT §7.3).
Nov
2
comment Why higher category theory?
It's not true that maps inducing the same map on all homotopy groups are homotopic! See mathoverflow.net/questions/20275/…
Aug
31
answered Aspheric functors and Grothendieck fibrations
Aug
4
comment Unbounded derived category that is not left-complete
Infinite products are not necessarily exact in a Grothendieck category, as Neeman's example shows.
Aug
3
comment Unbounded derived category that is not left-complete
Neeman gives examples of Grothendieck categories where $D(A)_{\geq 0}$ is not closed under countable products: arxiv.org/pdf/1103.5539v1.pdf. One such is the category of representations of $\mathbb{G}_a$ over a field of positive characteristic.
Apr
28
comment Blow-ups in Motivic Homotopy Theory
Ah, that's a great example. In this case the blow-up square is trivially a pushout (without suspending), but there's no reason for it to remain a pushout when you remove a point... I missed this point before.
Apr
24
comment Blow-ups in Motivic Homotopy Theory
If you remove a hyperplane section from the exceptional divisor, then I agree that it looks like the map could be an $\mathbb{A}^1$-equivalence, but you need a finer topology than the Nisnevich one. Maybe this will be useful: Unstable motivic homotopy categories in Nisnevich and cdh-topologies.