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While discussion rages about motivation and understanding in another thread, I would like to settle a question about the interrelatedness of various subjects which impinges on my personal motivations and understanding of the topics in the title.
Specifically, I would like to know if I am correct in my assessment that the landscape of those topics’ connections to each other is like this:
“An -groupoid (in the weak sense, which is the most important sense) and a space up to homotopy are the exact same thing. The study of homotopy simply is the study of -groupoids; they are two names for the same thing. Homotopy spaces were traditionally constructed out of topological spaces (in the subframe of powersets sense), and studying the relationship to topology may still be interesting and illuminating, but homotopy done in this way is equivalent to any other way one might approach -groupoids.
The study of -groupoids naturally leads one to consider categories of -groupoids, and thus -categories (or -topoi more specifically). [Analogously, the study of continuous functions up to homotopy leads one to consider a -category built out of the category of topological spaces (in the subframe of powersets sense)]. Homotopy type theory is a convenient internal language for describing -categories, generally with additional structure of the sort one asks for in an -topos, in the same sense in which familiar type theories are the internal logics of locally cartesian closed categories or topoi or such things. Thus, homotopy type theory is ALSO in some sense just another name for the study of higher-dimensional categories of a particular sort, only perhaps with more emphasis on how these may be syntactically presented.
Still, it would not be morally wrong to say -category theory, homotopy theory, and homotopy type theory are all fundamentally the same subject, just as investigated out of different traditions.”
Is that correct or am I missing some sense in which these actually should be considered substantively different subjects that happen to be tightly related?
Hi Sridhar,
that sounds exactly right!
You may be pleased to hear that, indeed “homotopy theory” is used by many people as a plain synonym for “-category” (I think the entry homotopy theory discusses it this way.).
Famously Julie Bergner’s work comes to mind, where she used to speak of “The homotopy theory of homotopy theories”. This is synonymous to “The -Category of -categories”.
Concerning topological spaces as models for homotopy types: I like to point out the following basic lore of category theory:
The way you think of presenting the objects in a given category means nothing. The nature of the objects is only determined by what the morphisms are like.
I can define a category whose objects are defined to be flavors of green cheese, and if only I define the morphisms appropriately, I can make it equivalent to any finite groupoid whose is the number of flavors that I could think of.
Analogously, the fact that there is a way to present the standard homotopy category (by which I mean ) by objects which carry a “topology” means intrinsically very little if the morphisms don’t respect that topology. And they don’t. (It still means a lot “extrinsically”, namely for the practice of actual computations, but that’s how it goes with presentations.)
A side note: I believe the phrase “homotopy theory of homotopy theories” was first used by Charles Rezk, in relation to complete Segal spaces.
On the main point, I agree to a large extent, but it can sometimes be hard to tell the difference between “the same subject, investigated out of different traditions” and “different subjects that happen to be tightly related”. There’s a good case to be made that abstract homotopy theory and -category theory are (or should be, or will eventually be) the same subject, although I could definitely imagine someone (like a traditional algebraic topologist) arguing that they are merely closely related.
But I don’t think I would say that homotopy type theory is the same subject as -category theory, or even -topos theory — just as I don’t think I would say that extensional type theory is the same subject as category theory. Type theory of any sort has many different (but related) concerns to category theory, even if there is a close relationship, and certainly not everything you can say in one subject can be expressed in terms of the other.
But, as I said, the main point is right; this is just nitpicky.
Ah, alright, good to hear. Still, wanting to understand the nitpick nonetheless, what might be examples of things which could be said in the subject of -category/topos theory but not in homotopy type theory or vice versa?
There is, perhaps, a point that non-compact phenomena in topology although ‘homotopic’ are not yet fully understood from the -category/topos theoretic viewpoint. (Suppose you have an open manifold, saying if you have a ’potential’ boundary making the result into a closed manifold uses proper homotopy theory.) The problem with the question you pose in 1, is thus slightly ill posed because you are making an assumption about what homotopy theory is (and then risk a tautology which is not that helpful). Defining homotopy theory that way may be self defeating if stuck to too rigidly. The paradigms of the two subjects may be different and so one ends up in a philosophical and sociological argument. This can be useful as discussion of methodology, motivations, etc is important .
There are topologists who would not like this approach! And they would at least remind you that weak -groupoids only classify topological spaces up to weak homotopy type. (The two appearances of ‘weak’ here act in opposite directions.)
..and, continuing from Toby, weak homotopy type only works well with CW complexes. Not all reasonably defined spaces are CW-complexes and the extra structure of a CW-complex, i.e. its filtration and the attaching maps etc., means that they are more than just spaces. Note that CW-complexes were introduced by Henry Whitehead in Combinatorial Homotopy theory, and the combinatorics is nearly what is needed to define infinity groupoids! (As I said this quickly starts to look inwards and to go towards tautologies!)
There are topologists who would not like this approach!
But also nothing in the above is about topology. There is a big difference between topology and homotopy theory.
It is just a coincidence that topological spaces may serve to present homotopy types. But so do, say, posets. You wouldn’t care to say “Poset-theorists would not like this approach.”
There is a big difference between topology and homotopy theory.
Most topologists would say that homotopy theory was a part of algebraic topology. To go against that seems a bit odd.
Homotopy theory is more than just the theory of CW complexes or spectra.
Is there any work treating the proper homotopy theory from the -point of view.
Not really to my knowledge… although I have recently been thinking of exactly that question! but I have not got far.
I suppose the work on model category structures etc in the proper case may give something. A related question would be about strong shape theory (in all generality).
@Tim, Toby: I specifically referred to abstract homotopy theory, which I understand to refer to the study of model categories and suchlike structures, not restricted to plain topological spaces and weak homotopy type. Similarly, -category theory obviously studies -categories other than . There is a perfectly good model category of topological spaces and strong homotopy equivalences, which form a perfectly good -category. (Which isn’t to say there aren’t other reasons that a classical algebraic topologist might object to the identification of the two subjects!)
@Sridhar: Well, my previous comment is sort of an example: homotopy type theory works much better with -toposes, but -category theory can easily study -categories with less structure. We can omit things like the univalence axiom and dependent products from HoTT to obtain a theory that applies more generally, but it also becomes harder to do anything useful that way.
Also (and this is something that’s bitten me several times already), HoTT can only express things “internally”, which in particular means things that are true+coherent “locally”, i.e. in every slice category. If you have, for instance, a reflective subcategory of an -topos that is not an exponential ideal, then it’s hard if not impossible to express that in the internal HoTT.
On the other hand, type theory of any sort has concerns about, and can yield conclusions about, things like constructivity and computability, which are largely absent from -category theory and homotopy theory. We don’t fully understand HoTT yet from a computability point of view, but progress is being made.
@ Mike: As a reposte to your comment, and as I think I am the only one of us who has published a book with ‘Abstract Homotopy Theory’ in its title, I would prefer to say that there is an almost alarming sense in which what you say is true. This is not to be against model categories or whatever but rather to say that there may be too much of convergence of the two subjects, so that they may each dominate the other, whilst I think that the two traditions they represent have different outward looking aspects that are, for the present, being drowned out by the attraction of the other of this symbiotic pair.
Some time ago (1990s perhaps) when working on algebraic models for homotopy types I realised that in one direction one ended up with the data to specify an algebraic model for a homotopy type was identical with the combinatorial data needed to specify a CW complex, so that I felt that I was chasing my tail and getting more or less nowhere. I now think that view is over pessimistic, but here is a question. Apart from HoTT what do we as a group of researchers in this area see as the future directions for abstract homotopy theory etc.? I like the idea of using the insights of the modern (∞,1)-topos approach to try to develop more of Whiteheads algebraic homotopy programme, to understand simple homotopy in this context and to get more explicit and ’constructive’ ideas about Grothendieck’s dream as I think that it has essentially been done but that the answer is not clear.
too much of convergence of the two subjects
I don’t think that there can ever be too much convergence of two subjects. From every convergence of subjects we learn more about each. The more the better.
What I have witnessed are cases where researchers in one subject fought the convergence with another subject because they felt that this diminished the recognition of their work, which now was being related to other people’s work in the parallel subject. But I think this is an unjustified and unhelpful worry. On the contrary, the more convergence, the better for everyone.
My meaning is not at the immediate cientific level at which I nearly agree with you, but at a slightly different perhaps more methodological level. The paradigms and motivations of two subjects may be different but the subject matter or a large part of it, may be converging. The convergence then tends to accentuate the immediate overlap between the two subjects causing a ’gravitational pull’. This then can lead to researchers looking towards the ’black hole’ of the convergence rather than building using analogies in those parts that the comparison between the two theories is not yet done. If you like, in our case, saying that the two subjects of abstract homotopy theory and (∞,1)-topos theory are well nigh the same should encourage the building of new areas in both subjects by using the transfer between them, always with an eye on sets of problems or problem types to keep away the demons of the ’wrong’ type of abstraction. For instance, simple homotopy theory provides a set of constructive moves that generate some homotopy equivalences and algebraic K-theory provides a set of algebraic tools for handling some of the resulting calculations. What would be the potential in (∞,1)-topos theory for similar machinery. What would it tell us about the situations where (∞,1)-toposes are being applied? Would it be useful? If yes, then great. If no, what is the barrier for its usefulness? Can we deconstruct one or other of the theories and hence build up further knowledge of their foundations. (This is just a thought experiment here but I do wonder if HoTT will meet some simple homotopy theoretic blockages, that is results that correspond to Whitehead torsion obstructions being non trivial.)
There is another slight disadvantage in too much convergence. The big hope for using categorical models for homotopy types (from the homotopy theorists POV) was perhaps to enable calculations in the non-stable case. If the (∞,1)-topos view corresponds exactly to the abstract homotopy point of view then that going down the (∞,1)-topos track is not that immediately fruitful. (I once heard John Jones (Warwick) give a talk in which he claimed to show that one object that no one had wanted to calculate was isomorphic to another that no one knew how to calculate. He joked that this was a considerable achievement! It was. Initially it looks silly but the reasons behind the isomorphism opened up a whole new set of methods between the two theories.) Calculation with (∞,1)-toposes is not that obvious, and hence my second set of directions would be to look back at the aims of Whitehead’s algebraic homotopy theory, and also at Baues’ work in the area and to interpret some of that in terms of (∞,1)-groupoids. Ronnie et al’s work on strict infinity groupoids is a good base on which to build, but I do not know of a really satisfying (∞,1)-topos treatment of Whitehead products interpreting them in a sensible way in infinity category theory. Again does it tell us useful things about that theory.
To play devils advocate, you might ask : if the two subjects do converge completely how will it help either? (I am not saying it will not, and think and hope that it will, but merely what we might reasonably expect as a payoff from the convergence.)
I don’t think that there can ever be too much convergence of two subjects.
Mathematically, virtually, not – the more connections in existence the better – but socially, and in practice, people forgetting many points of view and tools just because of embracing more modern twin subject, can overlook something essential. As a social phenomenon this negative aspect is the reductionist spirit.
people forgetting many points of view and tools just because of embracing more modern twin subject, can overlook something essential
People can always overlook something essential. That’s a problem in the people, not in the theory.
And in the examples that I can think of (including the one we are talking about) it was strikingly the opposite: people have overlooked something essential for decades. And when they finally realized it, it turned out that there was a big convergence of subjects.
Yes, but Tim is saying that it is an obvious danger that some of the things from homotopy theory not existing in infinite categories but existing in algebraic topology will be forgotten, because of reductionist spirit and this is dangerous. Tim said and I said it is not about theory, but about our attitude about it, and you disagree by emphasising it is not important to think even of the phenomenon we described!
Luckily we all run a wiki to counteract these dangers.
This discussion makes me think of the following analogy.
(Warning: In another thread we are just seeing that my analogies don’t work for everyone.)
Here is the analogy:
when I was a young kid, I used to eventually create a huge mess in my room. I felt very cosy in that mess. After all, it was my personal mess, every item associated with pleasent feelings and memories, they laid around in the pattern in which I happened to have run across them, and so to me it all looked like a perfectly enjoyable state of affairs.
One day I would come home and find my room all tidied up. Some parent or grand parent had decided that despite the bad pedagogical side-effect of cleaning up after me, there was a point at which the mess could not be tolerated anymore. Also, guests were expected to arrive (not to say: expected to converge ) soon.
What a shock! I could not recognize this as my room anymore. This wasn’t the place where I felt at home! Even though now every single item was neatly stowed away in a labeled drawer, with the drawers neatly arranged into a systematic pattern (this is for the sake of the analogy now, not for the sake of realism…), I couldn’t find my own stuff anymore! At least not as quickly as I did before.
I wished those guests had never been converging to our place, forcing us to clean it all up.
people have overlooked something essential for decades.
I was a proponent of Kan complex weak infinity groupoid from about 1980 onwards. I got a lot of things wrong, of course, and some things right, but I could not get funding and that was the fault of the ‘nomenclatura’ of the time. Theories exist but it is the people who have to write them out. There is a lot of good algebraic topology and it is not always like the mess in your bedroom, Urs. :-)
My main point is to try to get people to try to apply the modern methods on some of the older problem areas. Calculating some invariants in a new way, or better still seeing if the modern methods allow new calculations. Let us look out as well as looking into the eyes of the beloved twin subject. ;-)
There is a lot of good algebraic topology and it is not always like the mess in your bedroom,
No doubt it is good, but the concepts are sometimes not put into their natural abstract order. Then a new generation comes and tries to learn it. In the course of it they organize it. Because there is way too much to learn not to try to bring some order into it. I don’t think anyone who did good work in algebraic topology or homotopy theory needs to worry about or reject the observation that this work can be nicely put into a clean general context. One may find this more or less useful. But I think there is no need to fight it.
I keep looking for the following cartoon, but I can’t find it anymore. Joseph Polchinski, string theorist and discoverer of the theory of “D-branes” once adapted it from a Gary Larsson cartoon.
The original cartoon first showed a dog looking up to his owner, with the owner saying something like: “Ginger, you have been a bad bad dog, a very bad dog Ginger. Don’t do this again, Ginger. If Ginger does this again he will be…” etc.
The next slide would show the very same scene, with now almost everything of the text removed, subtitled: “What the dog hears”: “Ginger, …………………..Ginger. ………… Ginger. …. Ginger ……..”.
Now, as you may know, after Polchinki’s discovery of D-branes, mathematically inclined people understood that a huge bit of the theory (the “topological version”) finds its natural formulation in the context of derived categories of coherent sheaves. Subsequently, there are lots of conferences that discuss D-branes in this way.
So Polchinski edited the cartoon: now he was the dog, and the owner was the mathematical physicists. The subtitle was something like: “What Polchinki understands in a conference on D-branes” and the slide would show: “Polchinski, …………………..Polchinski. ………… Polchinski. …. Polchinski ……..”
Elsewhere he and his students would be more explicit about his feelings towards the situation, which were and are negative. The feeling is that the convergence of the theory of D-branes with that of derived categories of coherent sheaves was a step in the wrong direction.
But if convergencies like this would be fought against less by established researchers, much of mathematics (and mathematical physics, in this case) would be a better place, I think.
@ Urs. There is only one point I think where I disagree with you on this and that is the use of the word ’natural’. You and I would agree to 95% at least on what was natural, but people coming from a different direction may not agree with us, and I doubt we have the ’right’ to insist that we are right (I would also hope they do not insist that they are right … in my experience they often do! but aren’t! if you see what I mean.)
I am not advocating fighting the convergence but I do think that by itself it does have a negative side that if we are not careful can have a bad effect and a negative impact on careers, etc. (Especially since there is a vicious reaction to category theory in some quarters.) It should and does, however, give a marvelous opportunity to clean up some bits of algebraic topology that are not always written to the level of exactitude and clarity that could be hoped for, and the nLab is a good place to air those new viewpoints.
Let me mention one area that that may be feasible, namely classification of homotopy n-types for low values of n. The available truncated simplicial models need refining and thinning down to allow one to see what is the structure that makes them ’tick’. Then how does that tie in with weak n-categories and weak n-groupoids and their structure. We know this works but the methods of attack available may not be up to calculation of invariants. Such a problem need both homotopy theoretic and categorical insights to be solved, so convergence but not merger, unless it is to include the insights from both ’traditions’.
So you see I am an advocate of Whitehead’s algebraic homotopy programme, but using the modern categorical viewpoint as a starting position.
It seems like we are all talking past each other. I think all of us here agree that convergence of fields is good in the big scheme of things. At the same time, it’s entirely understandable that people who are used to the old approach to a field (like Urs’ messy room) will find the transition difficult and the new language offputting. And if those people get replaced by new people who only know the new ways of thinking about things, then important ideas, problems, and techniques may also get lost — especially if the younger generation gets the impression that the new language is so much better than the old one that it doesn’t seem worth their while to learn anything that was done in the old language. Imagine if Urs’ parents were then the ones who had to live in in his newly tidied up room; even though they were the ones who’d tidied it up, they wouldn’t know where to find everything (especially not at first), and they probably wouldn’t even know what there was to be found.
In some ways, I think the more abstract -categorical language is (or at least can be) more friendly to classical algebraic topology than the intermediate stage of abstract homotopy theory which appears to care only about model categories. Model categories are great for some things, but a lot of the time they are inconveniently restrictive, and it seems like the perspective that we are just interested in presenting an -category in whatever way seems best should be more flexible and open to using other, older, techniques as well.
when I was a young kid, I used to eventually create a huge mess in my room. I felt very cosy in that mess. After all, it was my personal mess, every item associated with pleasent feelings and memories, they laid around in the pattern in which I happened to have run across them, and so to me it all looked like a perfectly enjoyable state of affairs.
I share similar experiences :)
after Polchinki’s discovery of D-branes
One should mention that what are now called the infinity categories of D-branes were described beforehand in some circumstances (in A- and B-model) by Kontsevich in his 1994 proposal for homological mirror symmetry. You are referring to a B-model side as the derived category of coherent sheaves, but as you know by 2004 Kontsevich and Costello, the entire SCFT picture can be reconstructed from the A-infinity description; this the picture is not any worse. (I think this is just an explanation for what you wrote implicitly in 24 and I agree here with the moral of your story in 24).
In some ways, I think the more abstract (∞,1)-categorical language is (or at least can be) more friendly to classical algebraic topology than the intermediate stage of abstract homotopy theory which appears to care only about model categories. Model categories are great for some things, but a lot of the time they are inconveniently restrictive, and it seems like the perspective that we are just interested in presenting an (∞,1)-category in whatever way seems best should be more flexible and open to using other, older, techniques as well.
Yippee Mike. That is exactly what I feel.
That is exactly what I feel.
Ah, great. But what Mike just said is: the theory of model categories is best understood only after seeing its convergence with the abstract theory of -categories, where one understands that the whole theory is just a presentation for something else.
I think this same statement applies to other aspects of homotopy theory / algebraic topology, too. For instance every now and then one sees on MathOverflow voiced the statement that the convergence of notions of higher groupoids is a step in the wrong direction, and that algebraic models are much better and should be preferred.
Here, too, I think it is clear: the role and use and importance of algebraic models becomes more pronounced after we understand them from the bigger perspective of -category theory, where – as with the model categories in the above example – we understand that there is a simple abstract concept with many different more rigid constructions presenting it (or not presenting it, sometimes).
I feel that the algebraic topologists who are publically (and by private emails and other means…) fighting the rise of -category on the basis that it somehow ignores the insights that they spent their career on, are seeing an enemy where there is actually a friend: the way to promote their work (which should be promoted) is not to argue that it is “better” than the new theory. Because it’s neither better not worse: it is a presentation of a special case of the new theory. They should go around and say: look, the objects that you youngsters are now all excited about I have shown decades ago how to present, construct and analyze in detail, for important special cases.
I agree with that as well! but the new methods and insight are just the start, so convergence gives opportunities. I think that the opposition to the -category methods is partially that the algebra developed to handle those objects is not yet well enough developed.
Some years ago I gave a talk in Aberdeen about homotopy coherence and also about crossed ’gadgetry’. There were some in the audience who had reacted badly in the past to such areas (which we can think of as a halfway house to -category stuff). I made a great play on the links with the classical simplicial group methods, and how the crossed stuff gave neat new ways of encoding old ideas and of pushing them further. I was told afterwards that that was the first time that X, a well known algebraic topologist of the older school, had understood the links between what we were trying to do at Bangor and the classical stuff. Before that he had been suspicious and hostile. I know that that view was silly but it is a fact of life that it (still) exists. Some of the die-hards are in positions of making judgements on the -category approach and as the newer approach we have to make a determined effort to do the translation, and to show the advantages that we feel it gives. Otherwise it gets damned as innovation for innovation’s sake. The convergence is mathematically good overall, but we have to show that to the ’disbelievers” and therein lies one of the dangers. We have to show that it is a better way of approaching homotopy theoretic problems.
Just another way of putting things but perhaps one difference in approach or viewpoint that may be important is that ’classically’ there is a lot of homotopy theory that is involved with individual homotopy types. The aspect of the -category approach is as yet very underdeveloped.
are seeing an enemy where there is actually a friend.
That is a very nice way of putting it, and is quite accurate.
On a personal level I have been heard to state that showing that some situation gave a model category was sometimes seen as a be-all-and-end-all, almost an end in itself. It was to me a pity when the individual objects were forgotten in favour of the big picture. (The big picture was important but was not everything.) A model category structure was a neat way of encoding a lot of individual results (analogues of Hurewicz theorems etc perhaps), but then needed using. The same criticism can be leveled at the -category approach in some of its current manifestations, but clearly this will change using slice categories etc over time.
(I must now stop and do some combinatorial homotopy theory!!!)
They should go around and say: look, the objects that you youngsters are now all excited about I have shown decades ago how to present, construct and analyze in detail, for important special cases.
Yes, I agree. But I think the ’youngsters’ are not entirely blameless as well (and here I include myself, although I certainly do my best); they should take more time to learn the old language and results.
Coming back to this discussion after 6 years, I wonder if views have changed much. I was prompted to do so having read David R.’s g+ post on Clark Barwick’s The future of Homotopy Theory.
I noted there the proximity of Clark’s
I think of homotopy theory as an enrichment of the notion of equality, dedicated to the primacy of structure over properties
and Mike’s Homotopy Type Theory: A synthetic approach to higher equalities
Further proximity, in the latter
we may consider HoTT/UF to be a synthetic theory of structures.
With regard to Clark’s recommendation (6)
We need to write more and write better. We should write to communicate our own ideas and we should write to communicate our colleagues’ ideas. We should celebrate, openly and in writing, advances from writing teams that are perhaps quite different from our own,
of course homotopy theorists are very welcome to do their writing on the nLab.
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