@frogeyedpeas I don't think there's a consensus royal road to homotopy theory, but one solid starting point is Peter May's A Concise Course in Algebraic Topology, though it is tough unless you have some algebraic topology background

@ArunDebray my goal is to eventually understand homotopy type theory what would you recommend.

@frogeyedpeas I know nothing about homotopy type theory

sorry!

apply the nerve, and we get a map from the nerve of the category of simplices of X, to the nerve of the category of simplicial sets weakly equivalent to F. Then, the nerve of the category of simplices of X is w.e to X, and the nerve of the category of simplicial sets weakly equivalent to F is weakly equivalent to BhAut(F) .

I agree that the iso between the nerve of the category of ssets w.e. to F is weakly equivalent to BhAut(F) is plausible, but what is a way to make it rigorous? I would want to say that it is enough to just look at one object, and then we would be done. Can one use some homotopy finality / cofinality?

That's not how the Grothendieck construction works in this context. The natural output is a map of simplicial sets from X to some model for the homotopy coherent nerve of the subcategory of ssets weakly equivalent to F. I think in this case you can model that homotopy coherent nerve by the nerve of a category built from zig-zags of weak equivalences between ssets weakly equivalent to F, and you can model X as the nerve of its category of simplices, but the former is not a Kan complex

so I think you end up with a zigzag between the nerve of the category of simplices of X and this other nerve (which is the nerve of a category more complicated than the subcategory of stuff eqvt to F and weak equivalences between them).

In particular this does not arise as the nerve applied to any obvious functor from the category of simplices to the category you mention.

Maybe the category of ssets is nice though and you can magically forego the need for zig-zags here if like... you restrict to fibrant F'a and X's or something? But it's definitely not obvious (at least to me)

i'm confused about something basic: suppose i have a thom isomorphism MO ^ X --=-> MO ^ T, T some thom complex formed from X, but suppose only that i know it's an isomorphism of MO-modules in the homotopy category. i want to push it forward along a map MO --> E, also a map of rings in the homotopy category. i can use the unit and multiplication maps to get a map E ^ X --> E ^ MO ^ X --=-> E ^ MO ^ T --> E ^ T. what is the two line proof that this map is the isomorphism of E-modules that i want?

i'm sure it's easy, but i keep messing it up

@Twistediso I think that BhAut(F) is given by (a fibrant replacement of) the nerve of simplicial sets weakly equivalent to X and weak equivalences between them, not all maps. This shouldn't break your argument though.

Thank you for pointing that out. I agree with your remark!

By the way, the zig-zag that Dylan mentions should be given by X←N(Δ/X)→N(wF)←BhAut(X)

where the backwards maps are w.e.

This zig-zag, seems to me, to be the maps you get from my argument. But I might be wrong

I think I agree with you.

@EricPeterson : there is a typo in the foreword of your book : chormatic instead of chromatic :)

oh, thank you; i'll fix it for matt

If $X \rightarrow Y$ is a Nisnevich cover of a smooth $\mathbb{C}$-schemes, is $X(\mathbb{C}) \rightarrow Y(\mathbb{C})$ necessarily surjective on integral homology? (Better yet, $MU_{*}$?) Are these some assumptions on $X, Y$ that would make it true?

@Twistediso can you give the full reference for that statement in Goerss-Jardine? I can't find it, and there are lots of "5.2"s because there are lots of chapters... I guess I'm slightly surprised that you really get such an honest functor to simplicial sets. I imagine on objects $\Delta^n \to X$ you're assigning the fiber of like... the zero vertex? or the last vertex? That's fine, but now how do you proceed, especially if $Y \to X$ is some arbitrary map as you say.

because it feels like you'd need to start choosing lifts of paths (= 1-simplices) and do so in a strictly compatible way. and that sounds hard.

so @DenisNardin the zig-zag I was actually referring to was the zig-zag that goes $N(\Delta/X) \to X \to BhAut(F) \leftarrow N(wF)$ where the middle arrow is the straightening that I already know about... Maybe you're saying I get to choose a section of that last arrow and then the composite will be a functor of the form @Twistediso says exists? I guess that doesn't sound so crazy...

and probably Twistediso will tell me in a moment that there's a perfectly reasonable definition of the functor in question which foregoes that straightening discussion.