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Definition : Categorical algebra is not category theory. It is the study of algebra using categories as a tool. The past thirty years have given us a new approaches and new structures to study. These include the study of operads, multicategories, clubs, algebraic higher categories, and other generalized abstract nonsense several steps beyond the familiar and easy to define notion of a monoid (the fundamental "unit" of algebra, so to speak... pun intended =p).
The so-called (by me, I'm coining the term right now) study of "classical modern algebra", if we reduce it, can be seen as the study of monoids and related concepts like monads, monoidal categories, groups, rings, modules, etc. (of course, we're oversimplifying a bit, but the monoid-based algebraic structures are by far the most studied by mathematicians). (I am including higher categorical generalizations to some extent, since the fundamental idea of a monoid is not really altered in a fundamental way by vertical categorification. Monads are also included, since they're nothing more than monoids living in endofunctor categories.).
In the past thirty years, categorical algebra has taken the concept of a monoid and said "hmm..., what if we generalize the idea of a "law of composition"?" And that's it. That's where my knowledge stops. I have a pretty good handle on the story of the classical algebraic structures and how they relate to one another, but I don't know the story of these more modern structures.
If any of you can tell me this story, I would appreciate it tremendously. I'm not talking about the history of the subject, I'm talking about the story of the mathematics itself. Bourbaki was able to, with the power of hindsight, tell the story of "classical modern algebra" not only by investigating algebraic structures themselves, but by investigating their structural relationships. This is more than just categories and functors between them. It's a comprehensive approach to algebra, the likes of which I have not been able to find for the structures in question. This is more than just a nice story about operads, clubs, generalized multicategories, or what have you. It should be a general overview of what structures have been developed and how they are related.
I'm interested in hearing as much of the story as you're willing to tell. This would make a great nLab page as well, and I encourage those of you who are interested in telling the story to write up a page. I intend to use this information to guide my studies in the next few weeks, as this story is part of the key to efficiency. Thank you.
I am not sure that the idea of generalisation was the central one in the developments that you mention. It was there, for sure, but there is also an idea of ’observation’ and ’analogy’. Monoidal categories were ’discovered’ invented or whatever to encode observed behaviour and so understand what made that behaviour tick! Once encoded there was then the … I have seen that structure elsewhere… moment and the process extends, repeats, etc.
For me, (but that is for me, not necessarily for anyone else) the key steps were Lawvere’s thesis, Beck’s thesis, May and Stasheff’s work in homotopy theory and then probably Boardman-Vogt. The more purely categorical side of things, which I only got to appreciate later, would involve bits of the La Jolla volume and, of course, the extensive work of Australian category theorists.
The ‘story’ of categorical algebra in the sense you seem to want is perhaps not yet quite clear. I have looked and the themes that motivate classical algebra such as ’structural theorems’ in terms of classification ,(semi-simple thingies etc.) are not that clearly generalisable in a useful way to the categorical algebraic setting.
I suspect that ’willing’ in the last paragraph of you posting is highly optimistic… try ’able’ instead!!!!
I am not sure that the idea of generalisation was the central one in the developments that you mention.
I know that. That was my cartoon version of how it all “went down”, as it were. To elaborate, Peter May woke up one morning and decided he was going to come up with operads, completely without motivation. After he was done, he had to come up with a name, and after one week thinking of nothing else… The rest, they say, is history.
But in all seriousness, the “story” I’m looking for (I was trying to use poetic wording because this is in the “Mathematics, Physics, and Philosophy” forum) is really just something like a view of how these new structures (and others that I haven’t even heard of, most likely) fit into the big picture, something like a page that discusses when a structure is built from other structures, which structures generalize other structures. The “story” is something like a chart of significant algebraic structures, with a discussion of how they differ from one another.
For example, topological spaces were created to generalize metric spaces, and when Weil discovered uniform spaces, he knew exactly where they fit in, snug between topological spaces and metric spaces. Does an operad induce a club structure on something? How exactly do generalized multicategories generalize multicategories? Sometimes more general formulations have simpler definitions or are easier to understand. Sometimes they’re not. Is it easier to define a generalized multicategory, then define a regular one, then define generalized operads as those generalized multicategories with one object and similarly for standard operads? How do clubs fit into this whole picture? Do they generalize generalized multicategories and operads further? Do they only generalize operads? These are the kinds of things one should know when devising a plan of attack to learn this much material. Is it worth learning a convoluted definition first, only to find out that there is a much simpler and elegant way to do it from more generality?
An example: The definition of a scheme as a locally ringed space is technically correct, but we know now that they are really sheaves on the affine etale site satisfying a covering condition by representable functors, and that this definition is the one that generalizes to algebraic spaces and algebraic stacks. The presentation of a scheme as a locally ringed space is a nice technical tool, but the definition of a scheme as this functor of points is the more essential one.
However, thank you for the references. I will definitely check them out.
Edit: I’m wondering, can all of these structures of categorical algebra be described as monads on an appropriate category? I know that this is true for operads and multicategories, but I’m not sure about clubs, or any of the other structures that I have not yet heard of.
Edit 2: In light of my previous edit, is there any place I could get a rundown of the major new algebraic structures beyond operads, multicategories, their generalizations, and clubs? Obviously the algebraic higher categories can be given as the algebras of some (generalized?) operad (and hence as the algebras of some monad), but I’m sure that there are plenty of other structures, since it seems too good to be true (good only in terms of progress) that I would only have to learn about five different (but all closely related) new(ish) algebraic structures.
Categorical algebra is not category theory. It is the study of algebra using categories as a tool.
I do not quite agree. I think that the categorical algebra is the appropriate generalization of algebra to categorical context. Thus it studies those aspects of categorical constructions which are in spirit of algebra. Look at Borceux Handbook 3 vols. and you will see that only small part of it is about the study of algebra using categories.
By the way, I will go now to create an entry categorical algebra.
@Zoran: Yes, sure, I agree. That’s a good enough definition for me. I was only clarifying that I didn’t want to talk about all of category theory, just this generalization of algebra.
Minor point: I’m not sure about any of these proposed “definitions” of “categorical algebra”, if the idea is to capture how people have historically employed that phrase. (I don’t mind redefining the phrase to serve a local purpose however, which is what I took Harry to be doing at the beginning of the discussion.) For example, when Peter Freyd wrote
Perhaps the purpose of categorical algebra is to show that which is trivial is trivially trivial
I think he meant something broader by “categorical algebra” than what is being discussed here. Similarly for the title of the Proceedings of the 1965 La Jolla Conference.
Somehow I feel I intuitively understand what people like Freyd mean, and it’s not just about algebraic structures on or in categories. For example, all those elementary but important lemmas on pullbacks that one learns when first studying category theory: that’s categorical algebra. Any time you make some general observation about adjoint functors: that’s categorical algebra. In other words, if pressed to say what a lot of these people mean by the phrase, my guess is that they approximately mean the body of general “categorical nonsense” calculations in their own right, as just another piece of algebra.
(There’s something healthy about this attitude. It’s healthy to demystify category theory for people, by occasionally reminding them that categories per se are just another species of (essentially) algebraic structures, and as algebra they are no more or less abstract than say Lie algebras or anything else. Too often people allow themselves to be frightened by thinking categorical reasoning is super-abstract, but seen as just another form of algebra, you can be very matter-of-fact about it.)
I suppose one could object and ask “what part of category theory isn’t categorical algebra in this sense?” I’m not sure I can say exactly. But it would be weird for me to consider topos theory, for example, as just categorical algebra: the intuitions and motivations come from so many directions that it’s much, much more than just algebra per se (and as you get more specific, this becomes even plainer: the flavor of realizability toposes is rather different from the flavor of sheaves on a space). As a starting point, you could take the title “Toposes, Theories, and Triples”. Toposes: not just categorical algebra. Theories: rather closer to categorical algebra. Triples: that’s definitely categorical algebra!
@Todd: My idea of categorical algebra is something like “anything that can be realized as a monoid”. This gives us monads (monoids in the endomorphism category), operads, generalized operads, multicategories, generalized multicategories, all types of monoidal categories, etc. While topos theory may incorporate algebraic machinery, it is not pure algebra in this sense. This seems pretty in-line with what you’re saying about “Toposes, Theories, and Triples”, although it may be somewhat less restrictive in some cases and more restrictive in others.
That’s fine, and it’s also an interesting question. The offshoots of the root idea of “monoid” are indeed quite considerable. You had said something else: “It is the study of algebra using categories as a tool,” and Zoran countered with, “categorical algebra is the appropriate generalization of algebra to categorical context.” I was merely pointing out that IMO neither of those quite fits how other people use the phrase. But if that’s understood and you want to redefine it anyway just for the specific purpose of framing an interesting question regarding the scope of the idea of “monoid”, as opposed to say mandating usage in the nLab, then that’s fine.
A lot of the concepts you cite can be seen as instances of monad in a bicategory, and from a logical standpoint it’s often a matter of finding the right bicategorical niche in which the concept naturally lives. That gives monoids, monads, operads, generalized operads, multicategories, and generalized multicategories, depending on how you choose the bicategory. The case of monoidal category I see more as a categorification of “monoid”, and so it’s not described so simply as an instance of a monad in a bicategory, but perhaps more of an instance of pseudomonad in a tricategory or pseudomonoid in a monoidal bicategory or something. But that’s not particularly clarifying; it only becomes more compelling unifying concept only after one develops many disparate notions of such pseudomonads.
Is the theory of higher pseudomonads as mysterious as the theory of weak n-categories and weak omega-categories?
(You just reminded me to add categories themselves to the list of monad-concepts!)
In broad brush strokes, the idea is that the data and axioms of a monoidal n-category should be deducible from the data and axioms of a (n+1)-category (according to the Baez-Dolan periodic table), and that n-dimensional (pseudo)monads also have data and axioms in that vein. God (or the devil) is in the details, but qualitatively speaking that should give some idea of the level of “mysteriousness”.
Weak n-categories and weak infinity-categories are not nearly so terrifying as they were 15 years ago (when they were just being born, really). If you wanted to get them better under your belt, you might try looking at Tom Leinster’s “10 definitions” paper (after some preliminaries, he shows how to compress many of the notions of n-category of the day down to two pages apiece). To get more of the details, you might try his book – it’s very carefully and attractively written. The conceptual machinery behind these definitions, although they take some work to digest, is not so bad – if you can hack modern-day algebraic geometry and algebraic topology, then certainly you can hack this as well, without any question.
Well, what are the major difficulties today in the field of algebraic higher categories? Why haven’t they found as much application as the homotopy-theoretic approach?
In any case, I prefer any noton of categorical algebra which is at least much wider than the one proposed in the first post by Harry.
My guess is that the main difficulty is that there aren’t enough paid professionals currently working on algebraic higher categories for them to be as competitive yet in today’s marketplace of ideas. Also, the homotopy-theoretic approach can draw on decades of accumulated wisdom in homotopy theory, and the experts are quick to see how to fit that wisdom into the newer infinity-category technology. So there’s built-in moral support for this to be the current dominating trend. But it seems to me that people like Batanin are also developing model category connections between say his notion of algebraic higher category and the more “geometric” notions. Also, I feel the algebraic approach could have big pay-off in terms of, say, giving usable and effective algebraic models for homotopy types.
Speaking for myself: I haven’t yet developed enough homotopy-theoretic finesse to use my notion to prove things like the homotopy hypothesis for Trimble -categories, but surely I feel that’s just a matter of technique. (It would also help if I weren’t so isolated and could spend some hours learning from being in the same room as people who could teach me.)
As for mathematical as opposed to sociological difficulties: one example is that it seems to be a lot more straightforward to define the n-category of n-functors in the geometric approaches than in the algebraic approaches.
@Todd:
From IRC:
( @thermo ) you ought to learn some model of (n, 1)-categories so that you know what these intermediate groupoids are
( fpqc ) suggest one to me!
( @thermo ) i found trimble’s model to be the most palatable
( fpqc ) and I should read about it where?
( @thermo ) good question
( @thermo ) i guess it’s discussed in here, ’definition Tr’ http://arxiv.org/abs/math/0107188
( @thermo ) but the exposition is kind of skeletal
So it seems that people are already using algebraic weak n-categories (at least groupoids) like you said.
Well, what are the major difficulties today in the field of algebraic higher categories? Why haven’t they found as much application as the homotopy-theoretic approach?
On top of what Todd already said: algebraic higher categries are simply harder than geometric ones. That directly corresponds to the fact that they extract more explicit data. It is harder to deal with explicitl chocies of composites, than to have a guarantee for composites. At the same time, extracting the specific choices also means that more informaiton is made explicit.
@Harry #14: yes, it’s always nice for me to hear that these approaches aren’t yet historical footnotes! Score one for kids today. :-) Wonder if thermo read any of the notes at the nLab?
Second what Urs said. Ultimately one could say that there are black boxes in the geometric approaches, but that people like Joyal handle them smoothly and with exquisitely good taste. I nevertheless maintain that it would be of interest to develop viable theories that are fully constructive and essentially algebraic, especially as computer-aided algebra continues to be developed.
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