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bio website math.mit.edu/~hoyois
location Cambridge, MA, USA
age 28
visits member for 3 years, 4 months
seen May 15 at 18:54

I'm a postdoc at MIT.


May
15
answered higher Eilenberg-Moore-toposes of left exact derived comonads
May
15
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Indeed, Theorem 6.3.8 does the trick! So coalgebras satisfy descent, but do they form an accessible ∞-category?
May
15
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Yes, sorry, that's what I meant. The reflection part follows from conservativity of the forgetful functor, which is left adjoint, so I think the only thing to check is the preservation of pullbacks.
May
14
comment higher Eilenberg-Moore-toposes of left exact derived comonads
Do you know that the forgetful functor from coalgebras preserves and reflects colimits and pullbacks? If so, the characterization of ∞-topoi by descent (prop 3 here) immediately implies that coalgebras form an ∞-topos.
May
2
comment a (pseudo)adjunction for the functor sending a category C to PSh(C) the category of presheaves
The forgetful 2-functor from large categories with small colimits to large categories has a genuine left adjoint which sends a small category C to PSh(C). But this left adjoint is not a right adjoint: PSh(*) is not final, since it has many colimit-preserving endofunctors.
May
1
comment Why are unramified maps not required to be locally of finite presentation?
@jmc Well, you usually want closed immersions to be proper, that's what I had in mind. A specific situation where properness turns out to be appropriate is in the proper base change theorem (in $\ell$-adic cohomology, say): there's no need for finite presentability there.
May
1
comment Why are unramified maps not required to be locally of finite presentation?
You similarly don't want to require proper maps to be of finite presentation.
Mar
17
answered When is the pullback in Chow groups defined?
Mar
16
answered Does the kan extension preserves contractible presheaves?
Feb
16
comment What are iterated cobar constructions?
A modern treatment of this is also in section 5.2.3 of math.harvard.edu/~lurie/papers/higheralgebra.pdf
Jan
25
awarded  Enlightened
Jan
25
awarded  Nice Answer
Jan
25
revised Category of motivic spectra
added 940 characters in body
Dec
28
awarded  Yearling
Dec
10
answered Localizations of model categories and $\infty$-categories
Dec
8
comment Topological retraction vs categorical retraction
This only shows that $A'$ is a topological retract of $X$, but $A$ need not be. See Mark Grant's counterexample!
Dec
7
comment unbounded derived category of a $\infty$-topos
Hypercompleteness is defined in HTT 6.5.2. The point is that $Shv^{hyp}(X)$ is the localization of $Shv(X)$ at morphisms which induce isomorphisms on homotopy groups. From this it's easy to show that $Shv^{hyp}(X,\mathfrak{C})$ is the localization of $Shv(X,\mathfrak{C})$ at the quasi-isomorphisms.
Dec
6
comment unbounded derived category of a $\infty$-topos
I agree with Dylan: for general $X$, $D_\infty(Ab(Sh(X)))$ and $Sh_\infty(X,D_\infty(Ab))$ are not equivalent. That's the whole point of Lurie's remark: proper base change holds for the latter but not the former. $D_\infty(Ab(Sh(X)))$ is equivalent to the $\infty$-category of hypercomplete $D_\infty(Ab)$-valued sheaves.
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 $Sing(X)$.