Homotopy Theory

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7:52 AM

@PaulVanKoughnett The claim does hold for non-hypercomplete sheaves in this case, see Lemma C.3 in mathematik.ur.de/hoyois/papers/lefschetz.pdf.

Also, I don't think finite limits need to exist to get the conclusion for hypercomplete sheaves. Tom's example does in fact satisfy the desired conclusion, even for non-hypercomplete sheaves: étale sheaves on Spec(R) are just spaces with C_2-action.

Yai0Phah

Simplicial ideals, as far as I see, are defined to be simplicial submodules of the simplicial commutative ring. I don't know whether it could be characterized homotopy-invariantly.

Denis Nardin

8:08 AM

@Yai0Phah What application of the idea of simplicial ideal are you thinking of? I think that when doing DAG people usually take as closed subschemes the maps of simplicial rings A→B that are surjective on π_0 and this does not really coincide with the notion of simplicial ideal you seem to be positing

5 hours later…

Tom Bachmann

12:52 PM

@MarcHoyois in what sense does my example satisfy the desired conclusion? Surely C_2-spaces is not the same as spaces?

Denis Nardin

1:24 PM

@TomBachmann If I understand correctly, the subsite spanned by Spec C is the category of free C_2-sets with the obvious coverings, and its sheaves are indeed C_2-spaces

Tom Bachmann

Oh of course. Man I had this completely backwards ^^.

Yai0Phah

2:17 PM

@DenisNardin I don't know whether only pi_0 suffices. For example, if B is a commutative A-algebra, and I is the kernel of the multiplication map $B\otimes_AB\to B$, then I/I^2 is isomorphic to Kähler differentials. It generalizes to the simplicial case, obtaining cotangent complexes. And one usually also consider the I-adic filtration. And for example, Bhatt has a paper identifiying derived de Rham with Hartshorne's algebraic de Rham.

Denis Nardin

@Yai0Phah I think in those situation people choose I to be the derived kernel (i.e. the fiber of the map B⊗_AB→B), which is homotopically meaningful. But I'm not an expert of these things.

Yai0Phah

2:35 PM

@DenisNardin I would be seriously suspicious about this, because in that way you cannot diatinguish topological cotangent complexes and algebraic ones if you succeed to develop things in this way.

Denis Nardin

@Yai0Phah I'll stop blabbering because I'm not an expert, but my understanding is that if you're working with simplicial rings, you're automatically working with algebraic cotangent complexes (to get the topological one you need to use E_∞-rings)

Yai0Phah

I meant you take I to be the fiber, then you realize I/I^2 by the cofiber of some multiplication map. I mean, through this you did not really take advantage of the simplicial structure, but only playing games with the underlying $E_\infty$ structure.

3 hours later…

Aaron Mazel-Gee

5:36 PM

i have a random tex question, which i'll post to tex.SE if nobody here can help. i have two different computers, which i believe both have the same installation of TeXShop and mactex or whatever the back-end stuff is called (yes clearly i am a computer whiz).

however, when i compile the same tex code on them, the labels in my tikzcd diagrams get different placements! (so for instance i'll put [pos=0.6] on one computer to center it, but then it's too far out on the other computer.) has anyone experienced this before, and or does anyone know how to fix this?

or, just let me know if you feel this is sufficiently off-topic -- but it's about commutative diagrams after all ;o)

Denis Nardin

@AaronMazel-Gee You don't have the same version of the software, clearly. It's still not supposed to happen (TeX ought to be perfectly backwards compatible) but there could be a bug in one of the two versions..

Beyond that, I cannot help you sorry :(

Aaron Mazel-Gee

5:57 PM

@DenisNardin okay, thanks for the input! that is helpful (i am not being facetious). i will double-check the version numbers then.

1 hour later…

Eugene Rabinovich

7:16 PM

Hi All, I am new to this chat room, but I have come upon a question that I'm sure someone here can answer. I am in a situation where I have a manifold M (with boundary), and three (pre)-sheaves of L_\infty algebras on M; let's call them A,B,C, and they fit into a diagram (with all maps strictly intertwining brackets) A\to C\leftarrow B. I further know that A,B, and C satisfy Čech descent (i.e. the natural map from the Čech complex for a cover of U to A(U) is a quasi-isomorphism) for arbitrary covers. And I know that the map B\to C is, open by open, surjective. I would like to say that this …

Denis Nardin

7:47 PM

@EugeneRabinovich Yes, sheaves and hypersheaves on M are the same thing. This is because the ∞-topos Sh(M) has locally finite homotopy dimension. This is because for sheaves over paracompact spaces the homotopy dimension coincides with the (Lebesgue) covering dimension (this is theorem 7.2.3.6 in higher topos theory), so in particular $Sh(\mathbb{R}^n)$ has homotopy dimension n. Finally, if an ∞-topos has locally finite homotopy (cont.)

dimension, then it is hypercomplete (this is proposition 7.2.1.10 in Higher Topos Theory)

Eugene Rabinovich

8:01 PM

@DenisNardin Thanks! I figured it should be possible to combine results from HTT, but I don't know this stuff.

Denis Nardin

@EugeneRabinovich No problem, glad of having been of help. And since I forgot to write it earlier, welcome to the chatroom! :)

2 hours later…

Paul VanKoughnett

9:56 PM

@MarcHoyois Thanks, Marc -- I realized shortly after asking the question that Tony and Mauro's proof works identically in that case. But it seems like you don't need their subcanonicality hypothesis?

@AaronMazel-Gee Just move the other computer 0.6 to the right.

2 hours later…