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I haven’t really looked much into doctrinology, as such, even though I think I am at least roughly familiar with all the particular examples that I have seen used.
I know that people (for instance Jim Dolan) like to use “doctrine” as a synonym for just “2-category”, and this is what you follow in your book (p. 15). (By the way, do you maybe want to say there “-doctrine” for -category?)
But don’t we want to put some extra conditions somewhere? It’s a tad disappointing to read a definition which just says: “We define Y to be X. ” We would want something on top of to justify giving the result a different name.
Of course I know thi happens often. For instance “presheaf” is just a synonym for “contravariant functor”. But there it is okay because the next definition will define “sheaf”, which is a contravariant functor with some extra properties.
I would hope the same for doctrine. A definition that just says “A doctrine is a 2-category” should add something else. No?
It’s conceivable to me that some or all historical uses of “doctrine” should really have been saying “2-category”. But if that’s the case, then I think people should learn to say “2-category” when that’s what they mean, rather than defining a new word to mean the same concept.
My current understanding is that a doctrine should be more than a 2-category: it lives on a 2-category like Cat, and what it is is a type of “theory”, together with a corresponding kind of structured object in which we can talk about semantics for theories of that type. E.g. finitely complete categories with finite-limit theories, finite-product categories with Lawvere theories, monoidal categories with operads, etc.
One formal notion which I think makes some good sense of this is “a monad on a proarrow equipment” (such as Prof). Such a monad comes with a notion of algebra (with respect to the functor-like arrows), which are the structured objects, and also a notion of generalized multicategory (defined using the profunctor-like arrows), which are the “theories”, and also (in well-behaved cases) an adjunction between the two which implements the syntax/semantics. This isn’t perfect, though, since it fails to capture some examples like monoidal categories with props, or cartesian closed categories with lambda calculus, but I think that in the situations where it applies, it does a pretty good job of capturing what I’ve gathered a “doctrine” is intuitively supposed to mean.
Defining a doctrine to “be” its 2-category of structured objects means forgetting the distinction between (say) the algebraic theory of groups and the free cartesian category on a group. But the process of building the one from the other, and the resulting relationship between syntax and semantics, is, I think, a significant part of the “study of theories” that shouldn’t be lightly discarded. Remembering the 2-monad that gave rise to the 2-category of structured objects is a little better, but it still doesn’t really include enough information to define a theory separately from the structured object it gives rise to.
Thanks, Mike, that reflects my feeling, though I’d need to think about the “monad on a proarrow equipment”-statement.
Much more naively, it has always seemed to me that there are lots of 2-categories that probably nobody would seriously want to call a doctrine. For instance all 2-categories of the for , for a commutative monoid. Or all 2-groupoids. Even though I realize this can probably be understood as degenerate cases.
Urs, in what I read and listened from Dolan’s video, Dolan meant by doctrine roughly a 2-monad, but was not entirely happy of reducing the notion to it and said that it is meant more than that originally.
My experience is with literature mentioning doctrines is that I am always confused, as it is not clear what someone means. When I hear a definite statement like 2-monad, I am at home, and can read the paper.
Frederic,
thanks for the reply. I understand the pedagogical purpose of the definition in your book, and that’s good. Maybe it would make sense to just briefly add a sentence or some other indication that a doctrine is a bit more than just any random 2-category (or n-category). Because, also on pedagogical grounds, I think it can also be confusing to see a definition that seems to be lacking some clause.
Zoran,
yes, I agree. And also I think if one intends to define doctrine = 2-monad, then again one could ask why there needs to be a new term.
Well, sometimes new terms are good to amplify a new point of view. I guess all we are saying is that whatever it is, it should not be left a mystery to the reader.
It is left mistery because each school takes it for granted, and sometimes cites hard to find papers as background. Somehow this happened a lot with word doctrine in old papers from 1970s which I attempted to partially read.
also on pedagogical grounds, I think it can also be confusing to see a definition that seems to be lacking some clause.
I agree! I fully agree with simplifying things for pedagogical purposes, but I also believe strongly that people should be told when things are being simplified.
That seems like not a bad definition. Although it seems to me that one might also want to consider doctrines that live in “2-doctrines” other than the 2-doctrine of finite 2-limits. (-:
Now you’re back, Todd, you couldn’t look at this thread on hyperdoctrines, could you? I certainly got myself in a huge tangle about Frobenius, Beck-Chevalley and the like.
Welcome back Todd, we miss often your insightful clarifications :)
Mike #13 is just what I was thinking, but I take it more seriously. We don’t have only one notion of “theory”, so why should we have only one notion of “doctrine”?
But, Toby, for theories there are qualifiers what we may think like “Lawvere”, “geometric”, “essentially algebraic”, “single’sorted”, “many-sorted”, “finite-limit” and one uses those occasionally (say at the beginning of the paper). Besides such things make it harder for non-category theorists to read the literature and is one of the reasons why the literature written by orthodox category theorists is not much popular among practical mathematicians. Should we multiply the problem or make it more friendly to spread into central parts of mathematics ?
I find bad the common inclination of category theorists “others should go along with our (non-explicited) intentions as we understand the best”. I suffered a lot from “doctrine” referrals in Australian school papers where those sometimes pop out in the middle of a paper without an excuse and an explanation.
Zoran, I don’t understand your complaint. Are you saying that one should always use the appropriate adjective, rather than beginning with “in this paper, ’theory’ means ’geometric theory’?” That seems like multiplying verbiage unnecessarily.
Zoran is simply saying that it is a pain to follow an account where it is not clear what “doctrine” is supposed to mean.
Right, it is OK to say it couple of times precisely (say in the intro and in the beginning of the technical part, people do not necessarily read fully linearly) and then go on (and give a reference where the exact variant is defined exactly the same way).
Okay, sure. I thought you were disagreeing with Toby somehow, since you started #17 with “But”.
I believe that I also agree with Zoran #17.
Linked to JB’s page on doctrines, and to ’A duality relative to a limit doctrine’.
Linked on doctrine? Yes, it looks like it.
That’s what I meant, yes.
I see it says at doctrine
a doctrine could also reasonably be called a “2-theory.”
We also have an entry 2-Lawvere theory which links to work by Power and Lack. No mention of doctrine in the entry or in those slides. Is that surprising?
Is that surprising?
I think that was part of the point of the above dicussion:
if you mean to talk about algebraic 2-theories , why say doctrine instead? If you mean to talk about 2-monads why say doctrine instead. If you mean to talk about 2-operads (hehe) why say doctrine instead?
If you mean to talk about neither of these but about “doctrines”, then say what you mean.
But irrespective of best terminological practice, shouldn’t doctrine be seen as part of higher algebra and linked to from there? Or would you advocate the more radical solution of removing it?
Why ’hehe’ for 2-operads? There’s no page for them I see.
shouldn’t doctrine be seen as part of higher algebra and linked to from there?
It is linked to from monad and 2-monad. I’ll add a link to higher algebra as soon as I have brought the Lab back (it’s not having a good day today)
Why ’hehe’ for 2-operads?
Because operads are one example of a concept where the higher version is more familiar to people than the 1-categorical version. Most of operad literature is really about -operads.
That accounts for the first “he”. The second is due to the fact that despite of this, I am not aware of any -operad theory at all.
Okay, I have added “2-monad / doctrine” to the item lists at higher algebra.
Thanks. Would have done it myself, but I’m still seeing how things fit together.
Most of operad literature is really about (∞,1)-operads
I don’t agree. Most of operad literature is about 1-operads which are used as a way to present (∞,1)-structures. But the operads themselves are defined in a purely 1-categorical way, internal to a symmetric monoidal 1-category.
Yes, sure. But as opposed to the situation with categories, most of the literature that says “operad” by default means “enriched in spaces” or “enriched in chain complexes”, which are the enriched presentations for -operads.
Hardly ever does one meet somebody talking about operads who considers Set-enriched operads. I think the only people who do are actually pure category theorists.
Yes, true. Just my usual annoying behavior of complaining about phrasing. (-:
32 Yes but there is lots of work on operads in algebraic categories, where¸no spaces are involved. Y ou guys are inclined to read algebraic topology, so you see majority where algbarist does not.
Could you give an example for a citation of the kind you are thinking of, where algebraic operads are used?
I understood Urs’ phrase “enriched in chain complexes” to include the algebraic cases that I’ve encountered. Zoran, are you saying there are algebraic uses of operads that don’t even involve any chain complexes or homological algebra?
Right, most is for dg setting, which is the variant of what you said (btw, not every dg category is about stable infinity as one often works in other characteristics or not pretriangulated). But there are also many papers, though less, where operads in monoidal categories or even in sets are used just to talk about algebraic theories, and no dg or infinity is in the works. For example, Manin has one paper with Borisov on the archive where they give a setting for noncommutative geometry via operads in monoidal categories, ends and coends, nothing is dg or alike. Durov’s algebraic geometry is about finitary algebraic theories in sets, and he goes back between monadic and operadic viewpoint. Livernet has some works in dg setting but also many about operads for modelling various kind of 1-algebras like Leibniz, dual Leibniz, dendriform etc. The work of Fresse on Lie theory for algebras over operads considers also operads in algebraic setup, no dg or alike, everything 1-categorical and it gives very useful generalizations of 1-categorical Lie theory for other pairs of Koszul dual operads. Of course, the Koszul duality involves homological algebra but operads themselves and their algebras are studied in non-dg world (though they have versions/extensions in dg world which are not considered there).
By the way, what is the status of higher operads in the sense of Batanin from the (n,k)-classification viewpoint ?
Among the article on algebraic operads (non-dg) there is also lots of articles related to combinatorics of trees, combinatorics of Feyman diagram, dendriform algebras and so on. See e.g. papers by Ebrahimi-Fard (arxiv) and by Li Guo web).
Cf. also J-L. Loday’s book
dedicated to algebraic operads. His next big book with Bruno Valette covers a lot about dg case however.
Thanks, Zoran, I wasn’t aware of that. (In particular, I didn’t know that Durov used operads in addition to Lawvere theories.)
Are stable -categories always characteristic zero? I thought they could be arbitrary chararcteristic; don’t chain complexes in any characterisic still form a stable model category?
Non-pretriangulated dg-categories, I would argue, are still fundamentally an -categorical notion, even though not captured by the particular context of stable -categories. Stable -categories are automatically enriched over the -category of spectra; but not all spectrally-enriched -categories are going to be stable. It’s like the -version of abelian categories versus Ab-enriched categories. And -categorically, chain complexes are just a restricted class of spectra, right?
Batanin’s higher operads I guess I would consider, in general, as -categorical objects, but presented (as ordinary operads are) using 1-categorical objects.
Durov also has a paper on the arXiv on “generalized operads” where this point of view is a bit elaborated coming from a choice of 2-monad in certain vectoid setup and attached to certain classifying vectoids. That is the paper listed at vectoid. The generalized operads comprise symmetric operads, nonsymmetric operads and some new objects which have in one example similar combinatorics of higher operations.
I agree about chain complexes, of course.
According to some, including Grothendieck, it is not good to attach negative names to mathematical terms (e.g. he was against the name “perverse sheaf” for a mathematical entity). Dogma, hm, not always negative.
I’m still not convinced that n-categories need a name other than “n-category”. I think a proliferation of names for the same concept is a barrier to cross-disciplinary communication.
What was the thinking in putting the ’hyper’ into hyperdoctrines?
What was the thinking in putting the ’hyper’ into hyperdoctrines?
God (or Lawvere) knows. A really offhand guess is that the ’hyper’ has to do with the matrix of adjoint relationships between quantifiers and pullings back.
40 Mike, I also think that the stable (inf,1)-cats can be of arbitrary characteristic, just the concept of pretriangulated A-infinity then does not agree so well with that concept then.
Zoran 47: Why is that? I’m not that familiar with pretriangulated -categories, but I would have thought that you could just work over a ground ring of arbitrary characteristic.
Again, A-infinity also works in other characteristics, but the correspondence between pretringulated A-infinity and stable A-infinity in this case does not work as in characteristic zero. I am not an expert on that though :)
Can you say anything about why the correspondence doesn’t work, despite not being an expert?
Well, there is nothing deep. One first goes from chain complex enrichement to topological enrichement (or by spectra if you prefer) by Dold-Kan. Now the usual thing like if one wants to extend the Dold-Kan from complexes to algebras, say for dg-algebras vs. simplicial algebras one needs characteristic zero assumption. To see that this is relevant consider the case with one object, i.e. a single A-infinity algebra rather than A-infinity category.
Thanks, but you have to back up even further for me; what goes wrong with a positive characteristic -algebra?
Helps motivate the troops, maybe, but estranges them from the natives at the same time. Couldn’t you just say “We’re going to classify mathematical theories in a coordinate-free way; the objects we use to represent these coordinate-free theories are called ’n-categories’.”? Mathematicians ought to be used to learning new words when they learn a new subject. Don’t we ask our undergraduates to go along with it when we say “We’re going to classify geometry and calculus in a coordinate free-way; the objects we use to represent this are called ’manifolds’.”?
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