bio | website | math.mit.edu/~hoyois |
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location | Cambridge, MA, USA | |
age | 28 | |
visits | member for | 3 years, 4 months |
seen | May 15 at 18:54 | |
stats | profile views | 1,147 |
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 |
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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 $Sing(X)$. |