Sorry, messing up on the interface from my tablet...
Some of you might have seen this come up on MSE earlier. Is there any way to choose coherent cubes in a quasicategory with all faces fixed, given enough information about how these faces interact in the homotopy categories of squares and of arrows in the quasicat?
I'm thinking of something like choosing six squares and morphisms between three pairs of them in the homotopy category of squares, such that there are only three different induced squares in the homotopy category of arrows, and only one induced homotopy commutative cube. Could any extra data like this let me choose a coherent cube that has all six of my chosen squares as faces? Sorry if this doesn't make sense, I'll explain more carefully, but maybe this is obviously impossible already.
@KevinCarlson such questions are governed by the (joyal) acyclic cofibrations. that is, you can do this iff you can include from the given diagram shape to the desired one as a categorical equivalence
@JuanVilleta-Garcia if you're referring to the mistake i'm thinking of, this is explained and corrected in dugger's "classification spaces of morphisms in model categories" (or something like that), and also fixed (but not explained) in a paper of mandell
@JonBeardsley this sounds wacky, i've never heard of it before
i'm more and more getting the sense that as much as we tout that "model categories are good for computations", they're actually mostly just a crutch for performing homotopy-invariant operations in a (now) overly complicated way
to be clear, this is entirely a philosophical point. i'm not saying anyone should or shouldn't use model categories, i'm just asking about the (theoretical) computational limitations of $\infty$-categories
@JonBeardsley homotopy categories are terrible for computations! e.g. most co/limits don't exist, etc. etc.
@AaronMazel-Gee thanks. I have trouble seeing this as an extension problem...if this is asking for an extension along $f:B\to \mathbf{cube}$, then it seems $B$ should be be built in part from copies of an $A$ such that maps from $A$ to any $D$ are arrows in $Ho(D^\square)$. I don't feel like such an $A$ should exist, since $Ho$ is a left adjoint. Am I missing something?
@KevinCarlson no, sorry you're right -- i wasn't reading what you wrote quite correctly. the answer might be partly related to this (joyal acyclic cofibrations), but i get the sense it has something to do with "obstruction theory" on the various hom-spaces; certainly i think the question is really an "invariant" one (i.e. it doesn't specifically have to do with quasicategories, only "$\infty$-categories" and their homotopy categories)
@JonBeardsley i meant "nifty", but i suppose "infty" would also be appropriate!
@AaronMazel-Gee I'd say model categories can be useful for "computations" of the form "the colimit of this diagram is this other thing" - sometimes there's a simple non-fibrant model for the homotopy colimit that maps to the thing you want, and you might be able to show it's a weak equivalence e.g. by writing down some horrible filtration by pushouts of generating trivial cofibrations.
@RuneHaugseng yes, i suppose i've actually done a few model categorical computations of that flavor myself -- all ultimately having to do with (bi)simplicial sets. out of curiosity, do you have any particular examples?
@AaronMazel-Gee there are parts of the "by-hand" construction that are actually formalized by model categorical arguments. If you look at the construction by Bruner in the H_{\infty}-ring spectrum book he uses that things are cellular. I guess that is what you were referring to.
@AaronMazel-Gee This isn't how I interpret the statement "model categories are for computations" (which is how I phrase the quote and I think it is a more accurate statement.) I see it as telling you how you know which was to resolve an object when you are making a computation. ...
Why projective resolutions? Well, they are part of a model structure which tells you that when you are deriving a functor the structure you want is provided by a projective resolution and gives you things like the comparison lemma.
@SaalHardali Yeah, if $A \in M_{nn}(\mathbb{C}), n \in \mathbb{Z}^+$, then tr$(A)$ $\neq$ tr$(A^*)$. I should have thought about it instead of asking impulsively.
@AaronMazel-Gee I'm confused, mostly about why anyone would bestow the title of maestro of nifty examples on me, but also a little about what you actually wanted an example of. Is it of a simplicial object in the homotopy category (of spaces) with no geometric realization?
If it is that, then maybe this works (I'm not completely sure). I remember that when Emily Riehl taught the Categorical Homotopy Theory course that turned into a book someone in the class (I think it was Michael Andrews) proved that the the degree 1 and degree -1 maps S^1--> S^1 don't have a coequalizer. I think the example made into the book.
OK, so we have maps without a coequalizer, maybe we can make a simplicial object where those are d_0 and d_1 from 1-simplices to 0-simplices.
I mean, since Delta^n is just a point in the homotopy category, I think geometric realization of a functor Delta^op --> homotopy-category is just its colimit. So we want a simplical object with no colimit.
I think given any pair of parallel arrows f, g : A-->B you can make a reflexive pair with the same colimit: (f,id), (g,id) : A+B --> B (that now have a common section B --> A+B given by the canonical map to the coproduct).
That means that given any pair without a coequalizer we can convert it to the [0], [1] portion of a simplicial object that also has no colimit. Finally, I think you can actually perform the extension from a functor Delta_{<=1}^op to a full (1-skeletal) simplical object in the homotopy category (is that right? I think you just need coproducts to add the degenerate higher simplices and those you do have).
If all of that works that should give you a simplicial object in the homotopy-category whose geometric realization would be the coequalizer of those two maps S^1-->S^1.
@SeanTilson yes, i totally agree that model categories are for resolutions. and projective resolutions are a great example (which lead to actual computations), good point! as for cellular spectra though, i can't imagine that this is actually necessary, at least for an $E_\infty$ ring spectrum. you might be right that $H_\infty$ ring spectra are a funny in-between case, though
@JuanVilleta-Garcia in the final (non-appendix) section of my paper "quillen adjunctions present adjunctions of quasicategories", i essentially spell out mandell's argument in a bit more detail, in a way which should hopefully make clearer what's going on. (at least, it made it clear to me back when i wrote it! but it's been a while...)
@OmarAntolín-Camarena haha no, i meant "examples of computations that are done using model categories that cannot be done homotopy-invariantly (i.e. using $\infty$-categories)"
but i like your suggestion! i'd certainly agree that a reflexive pair should be extendible to a (1-(co?)truncated) simplicial object with the same (possibly nonexistent) colimit
ooh although to define a simplicial object in a category with finite limits and colimits, you can just factor the latching-to-matching objets. but ho(Spaces) doesn't have those...
hmm, so abstractly we're talking about a left kan extension along the (full) inclusion ${\bf\Delta}^{op}_{\leq 1} \hookrightarrow {\bf\Delta}^{op}$, and i think this won't exist (unless maybe your value at $[0]^\circ$ is empty, or something)
on the other hand, there may well exist some extension which in this case happens to have the same colimit. but i guess it wouldn't be for purely formal reasons (namely that left kan extensions commute (when they exist))
@SeanTilson i guess what i'm trying to say is that i don't see spectra truly getting resolved pretty much ever. R-modules on the other hand, totally. somehow it feels like the point is that "chain complexes" are in and of themselves a "true thing", whereas "point-set presentations of spectra" are not (or rather, have not mostly been, in practice -- of course this is in a sense a generalization of the previous)
i too gave a topics class this semester, starting from not very much and working up to the MU<6> --> K_Tate orientation. the notes are still a work in progress, in the senses that (1) there are some errors, (2) there are organizational issues, and (3) they're incomplete after chapter 5. nonetheless: github.com/ecpeterson/FormalGeomNotes/blob/master/main.pdf
@AaronMazel-Gee From my own work, Prop 5.9 in arxiv.org/abs/1412.8459 has an operadic example, with a particularly horrendous filtration (hopefully better explained in the new version of the paper, which I've been too lazy to put on arXiv yet)
i remember reading somewhere that notions of module over an algebra over an operad O are parameterized by right O-modules, or something, but i can't find where i read this. anyone know a reference for a statement like this?
@QiaochuYuan i don't know a reference, but that's a nice perspective. i guess the "standard" thing must just be where you just consider O as a module over itself?
