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This looks like an interesting program being rolled out by Clark Barwick and colleagues. So far there’s an introduction
and the first Exposé
It sounds like it’s worth a dedicated entry, so I’ll start one Parametrized Higher Category Theory and Higher Algebra. I guess we can then spin off uncapitalized entries for the concepts.
They want to “untether equivariant homotopy theory from dependence upon a group” by generalizing away from the orbit category of a group to ’atomic, orbital $\infty$-categories’.
I find the use of the term ‘parametrized’ as being very opaque and vague. I wish they had thought up a better terminology. I also think that it is not really infinity category theory, but that is another matter.
In the talk I linked to, Barwick says ’parametrized’ is in honour of Peter May’s Parametrized Homotopy Theory. Is the latter typically limited to parametrization over a homotopy type? Here it’s over an orbit category, and then further to some generalization known as ’atomic orbital $\infty$-categories’.
They explain this in point 4 of p. 6 of that introduction. Given the discussion we’re having elsewhere about the legitimacy of retaining a name when generalizing, it seems to be becoming quite a theme.
Of course, possibly the original name “parametrized” was not very clear either. (I don’t think May-Sigurdsson invented that use of the word, though.) Generalization can also be an opportunity to fix bad terminology…
“Relative”? Since that is what it is called in algebraic geometry, and that came earlier. Though on consideration, it seems less outright informative than parametrized!
But do we have an nLab angle on all this? It seems to aim at quite some unified account. Do the instances of parametrised spectra and orbit categories suggest cohesion’s in the air?
Just looking at their definition now, I think the correct term to use would have been “fibered” or “indexed”, those being the standard name for the corresponding 1-categorical notion.
Tim, why do you say it isn’t $\infty$-category theory? I haven’t actually read any of their stuff beyond the definition, but it’s well-known in 1-category theory that fibered/indexed categories are a perfectly good “category theory” with basically all the same theorems as ordinary category theory.
Mike. It is partially an old prejudice on my part. I never really got a feel for things like stable homotopy theory and that sort of approach to algebraic K-theory (and was force fed it at algebraic topology conferences in the UK in the 1970s!). I think I helped found infinity category theory back in the 1980s so am slightly aggrieved to see the term applied to what is equivariant homotopy theory (an area I have worked in.)
The subject matter is a neat generalisation of equivariant homotopy theory and an application to derived algebraic geometry, but is not that central to what I feel is infinity category theory, although I know that that term is not well delineated in its use. The boundaries between homotopy theory and infinity category theory are very blurred. (For that matter I do not know how to say what category theory is anyhow …! A rose by any other name etc. ….)
What do you feel is $\infty$-category theory?
I’m not sure as yet. Its links with fairly classical homotopy theory get in the way of having an independent idea of its structure as a subject area. I feel that homotopy theorists ask a different type of question from category theorists, although there is a big overlap. Perhaps I would say there is some infinity categorical homotopy theory that is more homotopy theory in its ‘aims and objectives’ than category theory, but the boundaries are not really important. I would resist the idea that homotopy theory and infinity category theory coincide. I think it is more a question of they interact strongly. What do you think, Mike?
I agree with Tim that it does not seem to me that very much new ’category theory’ has been done in $(\infty,1)$-category theory as it stands today. An enormous amount of translation of classical category theoretic concepts: yes. Really new category theoretic ideas: not as I see it. Of course the translation of category theoretic concepts to a homotopical setting is a very powerful idea in itself, and has led to progress in homotopy theory; but that does not make it a new kind of category theory.
Richard
I would count the work of Riehl and Verity on infinity cosmoi as being category theory, and new category theory at that.
To me $(\infty,1)$-category theory means doing with $(\infty,1)$-categories the sorts of things that 1-category theorists do with 1-categories: limits, colimits, Kan extensions, adjunctions, generators, factorization systems, monads, monadicity, local presentability, etc. etc. Of course there are some new phenomena, such as the possibility of descent along arbitrary colimits, but for the most part it seems to me that the point is that nearly all classical category-theoretic concepts translate. In fact, this to me is what it means to be “a kind of category theory”, analogous to enriched category theory or indexed category theory: all the basic concepts of category theory are available. And it sounded to me from the introduction (which, again, is about all I read) that Barwick et. al. are going to show that “indexed $(\infty,1)$-categories” are also a kind of category theory in this sense — and then apply them in various ways, analogous to the way that classical homotopy theorists apply ordinary $(\infty,1)$-categories.
Someone ought to spend a few years writing a graduate text “$(\infty, 1)$-Categories for the Working Mathematician”.
My personal opinion is that the tools needed to write such a book don’t exist yet. The “working mathematician” shouldn’t have to bother about things like umpteen million different kinds of fibrations of simplicial sets. The Riehl-Verity programme is a significant step forwards; it may or may not be the eventual answer.
I think, Richard, you misunderstood what I was saying slightly. There is clearly new category theory being done in the context of infinity categories, but there is also a lot of good homotopy theory of infinity categories being done. The former informs the latter (perhaps more than the other way around). I agree the the list of topics that Mike mentions constitutes category theoretic content in infinity category theory and I would also say that there can be ‘categorical content’ in a lot of the new homotopy theory that uses infinity categories. I would not wish to be prescriptive, but found that certain well meaning homotopy theorists have a tendency to declare they were doing categoy theory although their motivation was more about calculating some famous invariant in a new setting rather than understanding the mechanisms involved more for their own sake (which would be more to the taste of a categorist).
There are questions in the interface between the two areas that perhaps defy classification as one or the other’s area. I think some of Whitehead’s problems from his ICM discourse in 1950 come into that class, e.g. suppose $X$ is a $n$-dimensional polyhedron and we model its homotopy type by, say, an infinity groupoid, can we see that $n$-dimensionality in the specification of the infinity groupoid? I deliberately do not say what sort of infinity groupoid / homotopy type model I want to use. Any thoughts anyone (but if you have any start a new thread otherwise this one will go wandering off away from the topic title!!!)
Returning to what this program’s about, some selections from the introduction:
In their landmark solution of the Kervaire Invariant Problem [25], Mike Hill, Mike Hopkins, and Doug Ravenel developed a perspective on equivariant stable homotopy theory that centered on the study of indexed products, indexed coproducts, and indexed symmetric monoidal structures (incorporating their multiplicative norms). They argued that these structures were fundamental to the basic structure of equivariant stable homotopy theory.
In 2012, Hill presented (partly jointly with Hopkins) a sketch of a program to rewire huge swaths of higher category theory in order to embed these structures into the very fabric of the homotopy theory of homotopy theories
In this text, we completely realize Hill’s vision
One is thus led to ask whether one might untether equivariant homotopy theory from dependence upon a group.
one finds that the unstable results continue to hold when $\mathbf{O}_G$ is replaced with any base $\infty$-category $T$. The stable results require only very mild conditions on $T$; in effect, one requires the analogue of the Mackey decomposition theorem in $T$ (“$T$ is orbital”) and a condition that no nontrivial retracts exist (“$T$ is atomic”).
Do these “mild conditions” tell us something about the relationship beween HoTT and linear HoTT?
Tim: I was agreeing with the suggestion from #11 that $(\infty,1)$-category does not yet have a clear independent structure as a subject area; and giving my own reasons for why I think a point of view of that kind is reasonable. My apologies if I misinterpreted that suggestion.
Mike: I agree completely that your description corresponds to how $(\infty,1)$-category is today. This is what I was getting at. My point is that there are not really new categorical ideas here: there are new applications of 1-categorical ideas, or 2-categorical ideas in the case of Riehl and Verity, but not new categorical concepts as such. By a categorical idea here, I am referring to something like a limit, or an adjunction, or a comma category, or a monad, or a pro-functor. Such notions for $(\infty,1)$-category theory all really come from 1-category theory. Whereas I think that everybody expects there to be truly different categorical phenomena in higher category theory. For instance, even a weak 6-groupoid, never mind a weak $(6,1)$-category, when understood algebraically, should have an extraordinary amount of structure, that one can probably do all kinds of remarkable ’purely categorical’ things with; but we understand nothing of this.
David: As I just wrote, I would say that Riehl and Verity’s work is an application of 2-category theory, which is not what I was getting at.
So in summary, I see $(\infty,1)$-category theory as category theory (mostly 1-category theory) as we already knew it, fused with homotopy theory in a new (or not so new, nowadays!) way.
Could people please take this elsewhere. I’d just like to hear discussion about what Barwise et al. are doing.
I am myself not keen on a long continuation of the above discussion, but given that nobody replied to the original post for 3 days, or has since, I don’t think it’s unreasonable that this side discussion sprung up. And a few thoughts on the Barwick et al paper did crop up in the course of the side discussion, which is surely preferable to no discussion at all.
It’s always much easier to argue about terminology than to actually read mathematics…
Mike: :-) and I do not know how to do a ‘thumbs up.’ Back to discussing the content rather than the terminology. …
I added some more detail about the program.
Reflecting on Mike at #14, I wonder why they don’t work up an $\infty$-version of Enriched indexed categories.
They seem to have given a lot of thought to the indexing category having the twin properties of being orbital (making the Beck-Chevalley condition hold, at least some of the time) and being atomic (having trivial retracts). I wonder if the latter condition has something to do with Mike’s EI-inverse condition.
I wonder why they don’t work up an $\infty$-version of Enriched indexed categories.
Does it seem like something that they would want, based on what they’re doing? It could be they’re just not aware of the idea.
(BTW, your link is broken for me; the paper on the TAC website is here.)
Well your list of “category-like objects which appear simultaneously enriched and internal” includes some of the areas they’re considering later in the program. But more broadly, I wonder if the choice of ’parametrized’ is a sign of a disconnect of one community from the category theoretic community. The first expose is just about what might have been ’indexed higher category theory’. No references are made to indexed or fibered category theory.
(I fixed my link to the one you give. Thanks.)
I wonder if the choice of ’parametrized’ is a sign of a disconnect of one community from the category theoretic community.
Yes, I think it certainly is. To me it indicates that they didn’t even take five minutes to talk to a category theorist about their ideas, since any category theorist could have told them that the standard terms are ’indexed’ or ’fibered’ and given them literature references. I might use a stronger word than “disconnect” for that.
Hi Clark; thanks for dropping by! I am duly chastened; you are of course right that contacting you directly would be more productive than bitching in an online forum. (-:
I’m glad that your friend directed you here, and that you were willing to overlook my “anger” enough to reply politely. (I wouldn’t say I was angry, although I admit to being slightly annoyed.) I’m also glad to hear that you’re aware of classical indexed category theory and intend to add citations in the next revision, and I’m happy to hear your thought processes that went into the name “parametrized”.
In the hope that you’re even feeling generous enough to listen to me further, let me try to convince you to stick with “indexed” (or, if you prefer, “fibered”). While I certainly agree about the importance of May-Sigurdsson, and I don’t know anything about the applications you’re referring to, I don’t think I would find any such argument compelling enough to warrant changing a terminology that is absolutely standard in category theory. Despite the arguable inappropriateness of the name “limit”, for instance, it would be ridiculous to try to introduce a different word for it in $(\infty,1)$-category theory. As Jaap van Oosten said, the only thing worse than bad terminology is continually changing terminology. It’s bad enough that we already have “indexed” and “fibered” meaning essentially the same thing; we don’t need a third word that also means the same thing!
Using standard terminology also makes it easier to build bridges between communities. If you call it “indexed $(\infty,1)$-category theory”, then a category theorist who knows something about indexed category theory can look at the title of your paper and say “oh, that sounds interesting, let me see whether it can do anything for me or I can do anything for it”. It also means that theorems from category theory “look the same” when translated to $(\infty,1)$-category theory, making it easier for a category theorist to ease into the latter: “everything works the same as you’re used to” is easier to deal with than “everything works the same as you’re used to, but we use a bunch of different words for things.” And conversely, it means that someone working on indexed $(\infty,1)$-category theory and needing inspiration from 1-category theory will automatically know where to look, namely indexed 1-category theory.
In other words, the effect of renaming a standard thing is to create a disconnect, even if that isn’t your goal.
Well I blame the author of
a ’monoidal fibration’, meaning a parametrized family of monoidal categories…
But seriously, seeing that 1-categorical issues have been so thrashed over, it does seem reasonable to stick with its terminology as far as possible.
For another (but related) case, I wonder if the ’flagged $\infty$-categories’ of Fibrations of ∞-categories are really new under the sun. Might the 2-category equipped with proarrows literature be relevant there?
Naturally, Mike was already onto this ’flagged’ case a few months ago, here, and that’s pleasing to see my hunch about equipments was on track. Is the suggestion at Conduche functor that the full pro-arrow treatment would have been better?
I take your point, and I think it might be a good idea for us to try to use the indexed terminology in addition to the parametrised one in the first couple of Exposés … if we can find a way to do so gracefully.
But I dispute the claim that we’re REnaming anything; we’re rather naming a new thing that happens to be a natural generalisation of two existing things with names:
(1) indexed 1-category theory (the case when the base is a 1-category), and
(2) parametrised homotopy theory (the case when the base is an ∞-groupoid).
In fact, it’s kind of the pushout of these two theories.
It seems to me that all the arguments you gave for using ’indexed’ instead of ’parametrised’ apply equally well for the opposite view if you wanted to represent a community of mathematicians that knows more about parametrised homotopy theory than indexed 1-category theory: If we call it ’parametrised (∞,1)-category theory’, then a homotopy theorist who knows something about parametrised homotopy theory can look at the title of our paper and say ’oh, that sounds interesting, let me see whether it can do anything for me or I can do anything for it’. It also means that theorems from homotopy theory ’look the same’ when translated to (∞,1)-category theory, making it easier for a homotopy theorist to ease into the latter: ’everything works the same as you’re used to’ is easier to deal with than ’everything works the same as you’re used to, but we use a bunch of different words for things’. And conversely, it means that someone working on parametrised (∞,1)-category theory and needing inspiration from homotopy theory will automatically know where to look, namely parametrised homotopy theory.
So we can name things after (1) and risk annoying/confusing folks who are happier with (2); we can name things after (2) and risk annoying/confusing folks who are happier with (1); or we can call everything something totally crazy and annoy/confuse everyone. How do we choose?
One idea is to look at the relative sizes of the communities, and try to piss off the fewest people. But I’ll be honest: I have no idea which is larger, and I wouldn’t know how to sort something like that out.
A better idea, in my view, is to look carefully at what we’re doing and see which leg of the pushout our stuff more closely resembles — (1) or (2). Here’s where things get more interesting, because the details of our project beyond the first couple of Exposés really matter. The bases in which we’re most interested — and have the most to say — are atomic orbital ∞-categories that admit a conservative functor to a poset (so that they are ’EI ∞-categories’). Examples include: the orbit category of a group, the category of finite sets and surjective maps, any ∞-groupoid, the cyclonic orbit category, and the total ∞-category of any G-space. Morally, these ∞-categories have an ∞-groupoid direction and a poset direction. If the poset direction is trivial, we’re clearly in (2) territory, and if the ∞-groupoid direction is 1-categorical, we’re in (1) territory.
Now ’EI ∞-categories’ essentially are stratified spaces. That is, the equivalence of homotopy theories between spaces and ∞-groupoids (via Sing) extends to an equivalence of homotopy theories between stratified spaces over a fixed poset P and ∞-categories with a conservative functor to P (via the exit-path ∞-category). So, in reality, the bases we’re most interested in give us a theory that looks more like a stratified version of (2). The role played by locally constant sheaves of spaces or ∞-categories in (2) is played by constructible sheaves of spaces or ∞-categories in our theory. More complicatedly, the role played by locally constant sheaves of spectra or stable ∞-categories in (2) is played by constructible bisheaves of spectra or stable ∞-categories in our theory. From this point of view, our theory appears to have more in common with (2) than (1).
I’d say that if we were interested largely in results that work for completely general bases, it would be about even whether we’d want to call them ’indexed’ or ’parametrised’ — maybe you could argue for a slight edge to the former. But since the bases that we’re most focused on are of such a very special kind, I’d say that ’parametrised’ gets the edge.
But this is all a judgement call, and that’s my main point: I don’t think there is a genuinely ’correct’ term to use.
Isn’t parametrized homotopy theory the case where the base and the total category are $\infty$-groupoids? Unless I’m confused, the categories of parametrized spaces over a base $B$ that appear in parametrized homotopy theory are not themselves indexed categories over $B$ qua $\infty$-groupoid, so the “category theory” that’s being used in parametrized homotopy theory is not what I would call “indexed $(\infty,1)$-category theory” (though it is enriched indexed $(\infty,1)$-category theory…). And in general, allowing non-invertible morphisms (passing from groupoids to categories, or $\infty$-groupoids to $(\infty,1)$-categories) tends to introduce orders of magnitude more new phenomena than passing from 1-categories to $(\infty,1)$-categories, so it seems to me that the notion of “indexed $(\infty,1)$-category” is much closer in spirit and behavior to indexed 1-categories than it is to parametrized spaces.
Re #32, I wouldn’t necessarily say “better”; it would say more, but it would also be a lot more work, starting with defining “$\infty$-double categories”.
Maybe I was unclear. Parametrised homotopy theory (over a fixed space X) deals with a number of different homotopy theories, not only the homotopy theory of parametrised spaces. There is also, as I mentioned, the homotopy theory of parametrised spectra, as well as parametrised module categories, etc., etc. To rephrase one big insight of the May-Siggurdson work in overly terse ’modern’ language, all of these homotopy theories are naturally parametrised ∞-categories over X (i.e., locally constant sheaves of ∞-categories on X, or equivalently functors from ∏∞(X)^{op} to Cat∞), the theory of which doesn’t involve any new phenomena beyond ordinary ∞-category theory. (For example, indexed products and coproducts over an ∞-groupoid simply reduce to the usual notion.) Again, my point is that if you perform the natural generalisation of this story to a stratified space Y, then these are what we have called parametrised higher categories (i.e., locally constant sheaves of ∞-categories on Y, or equivalently functors from ∏∞(Y)^{op} to Cat∞, where ∏∞(Y) is the entry-path ∞-category of Y).
In other words, I claim that the move from May-Siggurdson parametrised homotopy theory to our stuff is exactly the passage from the study of these locally constant sheaves of ∞-categories on a space to analogous constructible sheaves of ∞-categories on a stratified space. This is — for me at any rate — a nontrivial fact. I therefore believe that our choice of terminology is reflective of one of the main insights we’ve made into the behaviour of these objects, so I think it’s worth taking this point more seriously.
I’m extremely uncomfortable participating in internet debates, and in these circumstances, even more so. So I’d like to leave it there if I may.
All: I do hope you’ll read our papers as they appear (hopefully with more alacrity now), and I do hope you’ll see past potential terminological disputes to what I’d like to think are some interesting and fun results.
@Clark - a question if I may: I’m surprised to see you use the entry path $\infty$-category in your above message, rather than the exit path $\infty$-category, can you give a pointer than explains this choice?
Thanks for your reply, Clark! I totally understand that you want to cut this short; I regret very much that my rude and unprofessional comment created an uncomfortable environment for the conversation.
Before you go, may I make a compromise suggestion? What I care most about is not what you call your subject, but what you call the basic object (a functor $C^{op} \to (\infty,1)Cat$). Such objects and their theory are important in their own right, before specializing to the EI case from which your motivation for “parametrized” seems to come. So what about this:
It seems to me that this would give all the advantages of both sides that we argued for in #30 and #33. The word “parametrized” would be there to grab and assist the homotopy theorists in the title of your overall work, whereas your first couple of Exposes that are mainly about indexed $(\infty,1)$-category theory more generally could also mention that in the title to grab and assist the category theorists. And the word “indexed” would be used in theorems about categories, which are the ones that will “look like” theorems from 1-category theory; whereas the word “parametrized” would be used in all the cases where May-Sigurdsson used it, so that those theorems will “look like” theirs. (I don’t think that May-Sigurdsson ever spoke about “parametrized categories”; as far as I remember they only applied the word “parametrized” to the objects of their categories.)
David: of course the exit-path ∞-category and the entry-path ∞-category are opposites. There’s a whole chain of op-choices we had to make, and in the end we could only find one that minimised the dualisation-contortions we had to tolerate when we started contemplating parametrised symmetric monoidal structures. It turns out that that was to encode indexed/parametrised ∞-categories as cocartesian fibrations to the opposite of the orbital ∞-category. When the orbital category is an entry-path-category of a stratified space Y, that’s the same as constructible sheaves of ∞-categories on Y.
A good example of this is the category of finite sets and surjective maps, which is the entry-path category of the Ran space of R∞. (One of Saul Glasman’s theorems — which I still find mindblowing — can thus be re-expressed as saying that constructible bisheaves of spectra on the Ran space of R∞ are the same thing as weakly analytic functors Sp → Sp.)
Mike: No worries. Thanks. Please forgive my Internet-reticence. Your proposal is very reasonable and a good deal more refined and principled than what I was suggesting in #33. I will propose this to my coauthors.
constructible bisheaves of spectra on the Ran space of R∞ are the same thing as weakly analytic functors Sp → Sp.
Hi, could you possibly say what a weakly analytic functor means here? The result does sound a bit mindblowing, but maybe the mindblowing-ness is being hidden in the definition of these functors…!
Maybe that’s nonstandard … by ’weakly analytic’ I just meant functors that are the limits of their Taylor towers.
In arXiv:1507.01976 it’s being used as expected in terms of convergence of the Goodwillie-Taylor tower
$F \cong lim_n P_n F.$It looks like arXiv:1610.03127 is the place for that result being redescribed in #39.
We drop the ’weakly’ at analytic (∞,1)-functor.
Thanks! I think maybe then I do not find it mind-blowing, because somehow cubes are built into the Ran space in a way which it is not too difficult to imagine gives rise to the Goodwillie layers, and so by using $R^{\infty}$ and imposing a sheaf condition, it is not too surprising to me that one is imposing convergence. Establishing the technical details is of course a different challenge altogether, but still at least the picture seems quite natural.
Clark: Nothing to forgive. Thanks for dropping by, and for your openness to the suggestion!
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