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Sometimes I doubt how $\infty$-categorical $(\infty,1)$-categories really are. I mean, they seem a lot more like “fluffy” ordinary categories than honest algebraically-defined bicategories are. If anything, they seem more like a stepping stone between 1-categories and $(\infty,n)$-categories. In principle, shouldn’t $(\infty,\infty)$-categories (for lack of a better name) be as “easy” to work with as strict $\infty$-categories as soon as we have a good understanding of the one-dimensional case?
Mike said recently (on the café) that the tools used for $(\infty,2)$-categories are not well-developed. Shouldn’t they be the same tools we use in strict 2-category theory, since all of the nastiness should be handled at the bottom level? I ask this because I read a while ago in the HTT introduction that generalizing to (oo,n)-categories should be in principle as easy as generalizing to the strict n-category case, but from what I’ve read elsewhere (including Mike’s post on the café), there are significant complications past the (oo,1)-categorical machinery to get even to the case n=2.
Maybe someone can set me straight on all of this, but if we can’t really do a naïve enrichment (in the appropriate (oo,1)-categorical sense), then is the theory of (oo,1)-categories really higher category theory at all?
Is it because the machinery for strict oo-categories isn’t well-developed either?
In principle, shouldn’t $(\infty,\infty)$-categories (for lack of a better name) be as “easy” to work with as strict $\infty$-categories as soon as we have a good understanding of the one-dimensional case?
That would certainly be nice. I think one of the goals of higher category theory is to come up with definitions and tools that make that as close to true as possible. I don’t see any reason to expect it to be true “in principle,” however. Certainly I feel that $(\infty,\infty)$-categories are conceptually as natural and important as anything else, but I don’t see any a priori reason for that to translate into their being as easy to work with as anything else is.
But maybe I’m being too persnickety about your choice of words.
Mike said recently (on the café) that the tools used for $(\infty,2)$-categories are not well-developed. Shouldn’t they be the same tools we use in strict 2-category theory…?
There’s certainly a good point to be made that there are holes in the existing theory of $(\infty,1)$-categories which, if plugged, would make one sort of $(\infty,2)$-category theory easy to obtain automatically by generalizing what we do for strict 2-categories. If we had a good theory of categories (weakly) enriched in a monoidal $(\infty,1)$-category (or, better, an “$(\infty,1)$-cosmos”), including in particular the appropriate notion of weighted limits, then applying that theory to categories weakly enriched in the $(\infty,1)$-category of $(\infty,1)$-categories, and making use of analogies to Cat-enriched category theory as a road to strict 2-categories, would go a long way towards a theory of $(\infty,2)$-categories.
One would probably need to do a little more work at least, though. For instance, I think one would want ways to produce such $(\infty,2)$-categories, starting for instance from any of the various possible “point-set-level” models such as quasicategory-enriched categories or 2-fold complete Segal spaces.
I read a while ago in the HTT introduction that generalizing to (oo,n)-categories should be in principle as easy as generalizing to the strict n-category case
There goes this “in principle” again. (-: I think the direction to go is certainly clear if one wants to approach $(\infty,n)$-categories via $(\infty,1)$-categories (rather than defining them directly) but the technical difficulty may still be considerable. If we had a theory of $(\infty,1)$-categories that was as easy to use as the theory of 1-categories is, then maybe passing from it to a theory of $(\infty,2)$-categories and $(\infty,n)$-categories would be as easy as passing from 1-categories to strict n-categories is.
Actually, that’s one reason I’m so big on derivators right now; it seems to me that they have the potential to make $(\infty,1)$-category theory not much harder than 1-category theory. I have all sorts of ideas for how to define things like enriched derivators, 2-derivators, and monoidal derivators, and while I hadn’t thought about it too much before, there really ought to be a notion of a “derivator enriched in a monoidal derivator.”
Actually, that’s one reason I’m so big on derivators right now; it seems to me that they have the potential to make (oo,1) -category theory not much harder than 1-category theory. I have all sorts of ideas for how to define things like enriched derivators, 2-derivators, and monoidal derivators, and while I hadn’t thought about it too much before, there really ought to be a notion of a “derivator enriched in a monoidal derivator
That’s pretty awesome. I hope you keep us informed!
Also, from what you’re saying, it seems like Grothendieck was on to something after all! Question: How much of the theory of derivators is covered in his book on them? Is there a lot of conjectural material like PS, or is it an attempt at rigour like SGA?
How much of the theory of derivators is covered in his book on them?
I don’t know! I haven’t read it yet. Reading French is enough of a chore for me that so far (lacking a big chunk of time to sit down and plow through that and the other literature in French on derivators) I’ve found it easier to play around with things and figure them out myself. (Which is probably is more educational in the long run anyway, at least up to a point.)
By the way, it occurs to me that instead of a theory of enriched $(\infty,1)$-categories, it could also suffice to use a theory of internal ones, which might be easier. One could then define the $(\infty,1)$-category of $(\infty,n)$-categories inductively as internal categories in $(\infty,n-1)$-categories satisfying some “reducedness” condition, akin to completeness of a Segal space. I think I remember Joyal talking about taking that route starting with quasicategories, several years ago at the Fields institute, but of course I don’t know whether he’s carried it any further.
Of course that whole approach might also be doable in the language of derivators. One more thing for someone to think about. (-:
Of course that whole approach might also be doable in the language of derivators. One more thing for someone to think about. (-:
I wish I were at a level where I could pursue this seriously =\
Question: To what extent does the theory of derivators allow us to work with (oo,1)-categories? You said that they’re like a GUI for it, but given all of the stuff you suspect there to be, is it possible that they’re actually an equally good model for them? I mean, they seem like they’re a whole lot easier to work with (we don’t need to show that the left/right hom objects and power objects are well defined up to a contractible space of choices, for example). Would you describe the “compression” of the data of honest (oo,1)-categories to derivators to be lossy or lossless?
Also, by the way, is there an nLab page that describes the differences between internalization and enrichment at a deeper level than the little blurb over at internalization.
is it possible that they’re actually an equally good model for them?
Well, as I mentioned way down on this post, there’s a paper which suggests that the 2-category of locally presentable derivators is equivalent to the homotopy 2-category of the (∞,2)-category of locally presentable (∞,1)-categories. So at least insofar as locally presentable ones go, the compression could be called “lossless” in the sense that a locally presentable (∞,1)-category can be recovered from its underlying derivator up to equivalence.
However, there is definitely loss in other places. There’s loss in passing from the (∞,2)-category of locally presentable (∞,1)-categories to its homotopy 2-category. And I don’t see any reason to believe that non-locally-presentable (∞,1)-categories could be recovered from derivators. In particular, (∞,1)-categories without limits and colimits don’t even give rise to derivators; they give rise to prederivators, but a prederivator is a very flabby structure. My current feeling is that as long as we want to be working in a particular bicomplete (∞,1)-category, its underlying derivator is perfectly adequate, but as soon as we start talking about more than one (∞,1)-category, or constructing new ones from old ones, or about (∞,1)-functors, then derivators are likely to turn out inadequate.
For that, we need to work in the 2-derivator Ho((∞,1)-Cat)!
is there an nLab page that describes the differences between internalization and enrichment at a deeper level than the little blurb over at internalization.
I don’t know of one.
Also, from what you’re saying, it seems like Grothendieck was on to something after all! Question: How much of the theory of derivators is covered in his book on them? Is there a lot of conjectural material like PS, or is it an attempt at rigour like SGA?
Of course he was on to something :) Working on it from 1983-91 is a lot of work. But seriously, what transcription you can get from Maltsiniotis’ Derivateurs page is pretty rigorous - the conjectural stuff is in PS, but the derivators manuscript is seriously written (quite terse, actually) with proper numbered sections, definitions, lemmas etc. The only problem is that it isn’t all there yet (13 chapters out of at least 19). The article by Maltsiniotis is good for basics, but at 27 pages it is really only an introduction to AG’s document, which was about 2000 pages long.
Have you read any of it (Grothendieck’s book)?
One comment: so far the most successful approach to “real weak n-categories” has been in terms of n-fold $(\infty,1)$-categories: namely n-fold Segal spaces.
Another comment:
Concerning $(\infty,\infty)$-categories: it would help to understand fibrant replacement in Verity’s model structure for weak complicial sets.
A crucial question seems to to be: given a stratified simplicial set $S$, does its fibrant replacement $S\to \hat S$ fail to produce a weak complicial set in which all cells that are equivalences are marked as thin? And if it does, to which extent so.
@Mike
I’ve had a flick through. The French is a little more idiosyncratic than usual, and there are editorial insertions where the sentences are hastily written, but from a first glance it looks ok. I tried downloading all the pdfs from the Derivateurs page and joining them togather so as to save others the effort, but it made the file huge (I mean huge). In the end it came to 733 pages - longer than PS and less fluff, so there’s a whole lot of content.
As an aside, I found the idea that there is a derivator HOT which is essentially a presheaf, whose value on the point is Ho(Top), very cool.
@DR: Yes, is this related to the theorem that $W_\infty$, the class of Quillen weak equivalences for SSets, is the trivial localizer?
@Harry,
I wouldn’t call $W_\infty$ the trivial localiser - it is the smallest one, though, as you are no doubt aware. Actually HOT sends a small category $D$ to the homotopy category of the category of diagrams of simplicial sets of shape $D$, with the Reedy model structure. One could say that this uses the localiser $W_\infty$, and that there would be other derivators using other, larger, localisers. Then $HOT(\ast) = Ho(sSet) = Ho(Top)$ Another notation I’ve seen is $\widehat{\ast}$, which is probably the most understated notation for anything I’ve seen in maths.
I’m not really fond of the notation HOT. I’d rather just call that derivator $Ho(Top)$ or $Ho(sSet)$ or $Ho(\infty Gpd)$, since it’s essentially a boosted-up version of the homotopy category.
BTW, diagrams of shape $D$ do not admit a Reedy model structure for arbitrary categories $D$, only for Reedy ones (unless the meaning of “Reedy model structure” is being generalized a bunch somehow). But since the homotopy category only depends on the weak equivalences, which are always levelwise, for purposes of defining a prederivator it doesn’t matter which model structure on diagrams we pick. (For constructing left and right homotopy Kan extensions, the projective or injective ones might be useful respectively.)
@DR: How about the “indiscrete localizer”?
@Mike
you’re right: I meant the localisation at the level-wise weak equivalences.
Can someone remind me what “a localizer” is?
Let C be any presheaf category. Then we automatically say “let the cofibrations be exactly the monomorphisms”. Then a localizer W is a class of maps of C containing all trivial fibrations that satisfies 2 out of 3, and such that $Cof\cap W$ is closed under transfinite compositions and pushouts.
It is a theorem of Cisinski that the “indiscrete localizer” for any presheaf category is proper (when $W(\emptyset)$ is considered as the class of weak equivalences, it generates a right-proper closed model category (Astérisque 308 chapter 1 section 5 gives the proof of a condition for properness that gives this as an easy corollary)). It is also a theorem of Cisinski that any localizer generated by a set of maps (an accessible localizer) admits a closed model structure. A reference is given that using a large enough large cardinal axiom, we can show that all localizers are accessible, although Ast308 does go into this.
We should create a page Cisinki theory. Andre Joyal has one, but it is just a stub so far.
Yes, this is the thing I wanted to write up on a personal web, then transfer to the lab, although I think we should probably call it Cisinski-Grothendieck homotopy theory. As the abstract of Ast308 says, it finishes a lot of the work done by Grothendieck in Pursuing Stacks and Les Dérivateurs and puts the work of Thomason into a systematic framework.
Just go ahead and do it on the nLab. What can go wrong? Nothing.
Alright, I’ll start writing up the page Cisinski-Grothendieck homotopy theory at some point later.
So a basic localizer is a red herring, since Cat is not a presheaf category? Who chose these words? (-:
Well, this seems to be a sort of an incomparable case. All of ch.1 of Ast308 is for presheaf categories. Everything is put in context around ch. 3. Basic localizers and presheaf localizers are related by the nerve and SSet. This was one of the problems I was running into while naming cylinder on a presheaf.
I suspect we’ll be better off by just naming everything as Cisinski does, then putting “Cisinski-Grothendieck” in front of it, so a Cisinski-Grothendieck cylinder will be a cylinder in the sense of Ast308.
Well, this seems to be a sort of an incomparable case.
I’m not sure what that means. If a basic localizer is not a localizer, then “basic” is a red herring adjective by definition, isn’t it?
It is also a theorem of Cisinski that any localizer generated by a set of maps (an accessible localizer) admits a closed model structure.
That just sounds like the specialization of Smith’s theorem to presheaf categories, as a special case of locally presentable ones. Am I missing something that makes this a deeper result?
That appears to be the case. Cisinski proves that we can give the model structure with the Lawvere Cylinder. It may be that my French is very bad, and I misread which was a generalization of the other one (since Cisinski does in fact metion Jeff Smith’s theorem right afterwards) (Theorem 1.4.3).
Are you using “indiscrete localizer” for the smallest localizer on a given presheaf category? That seems odd to me; I think of the indiscrete topology on a set as the largest one (even though it has the fewest open sets), since it is the terminal object in the poset of topologies. How about calling it left-determined?
Or the “minimal localizer” – I think I’ve heard that before.
I used indiscrete localizer because it’s generated by the empty set. i don’t really care what we call it, but David didn’t like “trivial localizer”.
Actually a localiser is just a class of arrows of a category: you don’t need a presheaf category. (See Maltsiniotis’ article Quillen’s Adjunction theorem for derived functors, revisited, for example: Definition 1 is “A localizer is a pair $(C;W)$, where $C$ is a category and $W$ a class of arrows in $C$ such that the localized category $W^{-1}C$ is locally small”) For Grothendieck Cat was the ’best’ category to work with, followed by presheaves on a test category, which were used to link to existing theory (simplicial sets) and (in my opinion, which doesn’t count for much) to provide a bit of geometric comparison.
Actually that $W_\infty$ is the smallest (=minimal) basic localiser was only conjectured by Grothendieck, and then proved by Cisinski. Let me point to basic localizer.
Ah, that’s better. So in the context of presheaf categories we should say something like “presheaf localizer”?
And now I’m confused about what $W_\infty$ means – back in #11 Harry seemed to be talking about the weak homotopy equivalences in sSet, not about a class of equivalences on Cat.
I suppose it could mean either, but Grothendieck used it for $Cat$. I haven’t checked what it means in Cisinski, but via the Quillen equivalence $N:Cat \leftrightarrows sSet$ the two versions are mapped to each other, so the overloading of the symbol isn’t too bad.
Maybe, but when you start talking about its being the smallest (blank), then of course it depends what category you’re in and what other axioms you impose on it.
There’s also a bit of interplay between the $Cat$ approach and the presheaf approach, namely looking at the localiser $W_D$ on a presheaf category associated to a given basic localiser $W$ (i.e. one on $Cat$). I can’t remember the details, but it’s along the lines of ’a map in $PSh(D)$ is in $W_D$ if the such-and-such functor is in $W$’. Sorry for being cryptic, my brain is a bit full at the moment :S
Well, I suppose by saying $W_\infty$ is the smallest basic localiser, you are firmly putting it in $Cat$. I suppose then one could accurately say that the coresponding class of arrows on $sSet$ is the smallest localiser on $sSet$…
I suppose then one could accurately say that the coresponding class of arrows on sSet is the smallest localiser on sSet …
If I understand the definitions correctly, then it isn’t. The paper on left-determined model categories shows (remarks?) that the usual model structure on simplicial sets is not left-determined.
I don’t know what left-determined means, but it’s clear from ch.2.1 of Ast308 that the usual model structure on SSet has the minimal localizer as its class of weak equivalences.
I don’t have Ast308, but it is also clear that the weak equivalences for the usual model structure on sSet are not the smallest class of weak equivalences for a model structure on sSet whose cofibrations are the weak equivalences, since the weak equivalences in the Joyal model structure are a smaller such class. So if “localizer” means “class of weak equivalences for a model structure on a presheaf category whose cofibrations are the monomorphisms,” then the weak equivalences for the usual model structure on sSet are not the minimal one on sSet. (I think neither are those for the Joyal model structure minimal, although that isn’t as obvious.)
It’s on Cisinski’s website: http://www-math.univ-paris13.fr/~cisinski/ast.pdf
Perhaps you could figure out what the issue is then.
It’s on Cisinski’s website
It’s on the nLab
The link is, anyway. The actual paper is still hosted on his website =p.
(I think neither are those for the Joyal model structure minimal, although that isn’t as obvious.)
can’t we take weak equivalences = isomorphisms in any complete cocomplete category and get a model structure (cof and fib are, I believe, all arrows). This qualifies as the smallest localiser… actually, take you take weak equivalences = identities? So certainly when we talk about the ’smallest localiser’ on sSet, it needs to be not just any old localiser, but one linked to a basic localiser (i.e. on Cat) somehow.
Ah, having a bit of a read, I see that a localiser as defined by Cisinski is not the same as a localiser as I define above, namely a class of arrows $W$ in a category $C$ such that $W^{-1}C$ is locally small. In particular, it seems to explicitly be a class of arrows in a presheaf category, and satisfies all sort of conditions (essentially such that $W$ and monomorphisms are the weak equivs and cofibrations of what we call the Cisinski model structure). In that case, the silly counterexample I wrote in #41 is not applicable.
Then Harry in #17 is right, if we replace localiser by Cisinski localiser (or I replace it by Grothendieck localiser).
Hmm… is it true then that the weak cofibrations of the Joyal model structure are not closed under pushouts (images directes) and transfinite compositions?
Edit: Hmm… It seems that the trouble lies in Cisinski’s definition of an “anodyne structure” for a cylinder. The Joyal model structure doesn’t have this property (1.3.5) for the standard functorial cylinder on SSet, $X\times \Delta^1$. If we take the cylinder to be $\Delta^2$, something rather different happens.
EDIT: Alright, I’ve figured it out. The localizer $W_\infty$ is not the minimal localizer for sSet, but it is the minimal localizer for the cylinder $X\times \Delta^1$.
The honest minimal localizer is given by $W(\mathfrak{L},\emptyset)$, where $\mathfrak{L}$ is Lawvere’s cylinder (the cylinder given by the cartesian product with Lawvere’s interval object).
Hey, no fair, you can’t link to a nonexistent nLab page as if it explains something! (-:
Getting back to Grothendieck’s tome… I’ve skimmed through the first chapter. So far it reads somewhat more like notes to himself than a finished work for other people to read – he seems to keep changing the axioms, states a bunch of guesses or conjectures without proof, and claims a fair number of things that are noted to be false in footnotes or transcriber’s notes.
Lawvere’s segment is apparently jus the subobject classifier, according to Finn Lawler.
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