@JonathanBeardsley thanks for encouraging me to email salch, i learnt a lot by talking with him!

@JonathanBeardsley |Sing(X)| is one way to do a CW replacement for X

a very very huge one

If you think about it, when you take a realization of a simplicial set S, you can think of it as attaching dimension by dimension new simplices, possibly along degenerate boundaries

in this case, topological simplices

and the n-simplex is homeomorphic to the n-disk

so it's a cw complex

actually going the other way around is a way to build a very very big Kan complex for any simplicial set S (so you get a fibrant replacement haha)

You can think of this filtration by dimension idea more generally in reedy categories (they are defined to have the desired latching and matching objects, modeled on the simplex category)

I'm not sure what's a better way to make a CW complex out of a cgwh space, but there is a much smaller and much more combinatorial way to get a Kan complex out of a simplicial set than Sing(|X|) called Ex^\infty (right adjoint to the Sd^\infty infinite barycentric subdivision functor)

Goerss and Jardine have the best textbook/reference on simplicial homotopy theory I think

it's a more modern treatment than reading Dan Kan's original papers, and it also covers a whole lot of simplicial stuff

quick question about tensors... say I have R-->A a morphism of commutative ring spectra. denote by $\otimes_R$ the tensor of the category of commutative R-algebras over unbased spaces, and similarly for A. What can I say about $Y\otimes_A (A\otimes_R X)$ for X, Y spaces? the fact that the forgetful functor from A-CAlg to R-CAlg does not take tensors to tensors is not helping

Are Grothendieck opfibrations stable under pullback? Also, is the functor of post-composition with an opfibration itself an opfibration?

answer to the first question is definitely, but I don't know the source

and I don't feel like working it out by hand so it's only a definitely maybe

I think the proof amounts to the fact that it is defined by a lifting property

@HarryGindi are things defined by lifting properties generally pullback stable?

@HarryGindi This is a mistake I used to make until recently. Ex^infty is not a right adjoint: it does not commute with infinite products. In fact if it did, then an infinite product of simplicial sets would have the right homotopy type; but we know that this is not the case.

Indeed there cannot exist a model structure for homotopy types on sSet (and not even on Cat) such that all the objects are fibrant. This is remark, for instance, in the introduction of Thomason's paper "Cat as a closed model category"

@AndreaGagna I forget then how it's defined, is it the colimit of the Ex^n?

I thought that directed colimits commute with finite products in sSet

oh, infinite products

I know that there is no model on sSet where all are fibrant

I guess you're saying that the infinite product is not a homotopy limit in sSet

(and it would be if all of the objects were fibrant)

@Arrow

err, @Arrow yes, things defined by right lifting properties are generally pullback stable

with grothendieck fibrations, the proof is ever so slightly more fiddly, but the same idea is at work

For right lifting properties this is like saying the right class of a weak factorization system is pullback stable, right? I think this is what makes fibrations pullback stable, but wasn't sure about opfibrations. Maybe I'll just try to write things down.

the trick here is that for general grothendieck fibrations

you have to look only when the top of the lifting diagram is given by the inclusion of cartesian arrows

it's not a full weak factorization system

that's why lurie introduces marked simplicial sets

that way he can define it as a WFS for when the inclusion of the cartesian edges that you're trying to lift a 2-cell for is actually an RLP with respect to a full class of maps

rather than a sort of subset of the lifting diagrams

Thank you

np =]

Where are fibrations "needed" in descent theory? The nlab entry on pseudofunctors says Grothendieck and Gabriel "replaced pseudofunctors in the treatment of descent by more invariant fibered categories" in SGA1, but I don't understand anything in SGA. On the other hand, the category of descent data of an arrow w.r.t a pseudofunctor is given by pseudo-limit and I don't understand what is lacking about this approach.

Assuming the "reasons" are the usual for preferring fibrations (as mentioned in Bénabou's 1985 paper), how to define the category of descent data of an arrow w.r.t a fibration using a universal construction? It's no longer possible to take the "image" of the Čech nerve, and without a cleavage, I don't see how to "lift" the Čech nerve either...