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nLab > Latest Changes: Higher doctrines

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  1.  
    • CommentRowNumber1.
    • CommentAuthorfpaugam
    • CommentTimeDec 29th 2011
    • (edited Dec 29th 2011)
    A new contribution
    • CommentRowNumber2.
    • CommentAuthorfpaugam
    • CommentTimeDec 29th 2011
    • (edited Dec 29th 2011)
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2011
    • (edited Dec 29th 2011)

    someone erased my contribution to the doctrine part.

    Could you give a link? I don’t see what you are referring to. Nothing has been erased at doctrine, as far as I can see. (?)

    • CommentRowNumber4.
    • CommentAuthorfpaugam
    • CommentTimeDec 29th 2011
    Sorry, i was just in a bad mood.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2011
    • (edited Dec 29th 2011)

    Okay, but I don’t really understand.

    Are you saying nothing has been erased after all?

    If it has, I would like to know what and where. We don’t usually erase each other’s contributions unless for serious reason and after having discussed these reasons.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeDec 29th 2011

    I moved the page to higher doctrine, where it is more likely to receive incoming links (although actually I put in redirects so other links will also work). This means that the link at the top of this page is temporarily a victim of the cache bug.

    I also fixed the headers so that the table of contents will work.

    • CommentRowNumber7.
    • CommentAuthorfpaugam
    • CommentTimeDec 30th 2011
    Yes, i think that nothing has been erased, contrary to what i first thought.

    I just mixed it up with the discussion we had on the nforum some months ago about this section.

    Thanks to Toby for the corrections and sorry for disturbing.
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2011

    No problem. Good to see you contribute!

    • CommentRowNumber9.
    • CommentAuthorfpaugam
    • CommentTimeJan 11th 2012
    I modified the higher doctrine definition by saying that it is a weak n-category described by generators and relations, without giving more details. One of the problem of the notion is to give a precise definition (sketch-like, n-monad-like, etc...) of this notion. I would prefer the sketch-like one, because it is easier for me to describe finiteness conditions (like coherence) in this setting. In lower dimensional cases, at least, we have many examples, and there is also the example of k-monoidal n-categories, treated by induction using a Lawvere tensor product in higher categories.

    So now, a doctrine is not exactly the same as a weak n-category, and i guess it is closer to the categorical logicians' viewpoint
    (a language, a syntax, and we look at the semantics through categorical methods).

    I can keep this funny name in my master course, without looking like a clown.

    ;-)
    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJan 11th 2012

    I don’t understand; can you explain in what sense the doctrine of “categories with finite products”, say, is presented by generators and relations?

    • CommentRowNumber11.
    • CommentAuthorfpaugam
    • CommentTimeJan 18th 2012
    I mean that it is a 2-category of categories with additional structures given by operations (here, the product).

    For example these structures may be described by a 2-monad on the 2-category of categories.

    One may also describe them in a sketch-theoretical viewpoint. This would mean: fix a family of index higher categories \{I\} and \{J\}
    for cones an cocones, and consider the corresponding sketch, given by higher categories equiped with such cones and cocones.
    When cones and cocones are limits and colimits, one gets categories with a given type of fixed additional structures (here finite products).
    This may be a definition of a doctrine of higher categories with additional structures given by generators and relations.
    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2012

    Are you calling the structure such as the product assigning functor C×C→CC\times C\to C a “generator” and the laws that it satisfies “relations”? That’s not how the words “generators and relations” are usually used.

    • CommentRowNumber13.
    • CommentAuthorfpaugam
    • CommentTimeJan 22nd 2012
    About generators and relations, i just mean this in a ``quotient of a free object'' spirit.

    The idea is: take n-categories, and put some additional structures on them (higher sketch), given by a finite procedure and/or finite expressions (this will be called finite coherence) and ask these structures to be respected. I would call this a doctrine: and (n+1)-category given by generators and relations, starting form the (n+1)-category of n-categories.

    One should, in this setting, be able to define a coherent theory (finiteness condition in the presentation, that can be easily checked) such that the axiom of choice implies the existence of a model (like in usual model theory).

    The usual theorem of model theory (logic) is that a finitely coherent theory is coherent (i.e., admits a set-valued model).

    One would like to have a similar things, that says that a finitely coherent higher theory is coherent
    (i.e., admits an n-cat valued model). This is what i called the doctrinal axiom of choice in the nlab page on higher doctrines.
    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJan 22nd 2012

    Okay. In that case, I don’t think that “generators and relations” is a correct phrase to use here. A 2-category (for instance) given by generators and relations should mean that you specify a collection of objects, then a collection of 1-morphisms between those objects, freely generating all composites of those 1-morphisms, then specify a collection of 2-morphisms between those 1-morphisms, freely generating all composites of those 2-morphisms, then quotient by some relations identifying 2-morphisms (and, perhaps, 1-morphisms as well, although in the weak world this is equivalently given by adding invertible 2-morphisms). Giving any object via “generators and relations” means to present it as a (perhaps higher) quotient of free objects — in this case, free (n+1)(n+1)-categories.

    I’ve edited the definition at higher doctrine with a proposed replacement; what do you think?

    • CommentRowNumber15.
    • CommentAuthorfpaugam
    • CommentTimeJan 23rd 2012
    Generators and relations is not completely clear to me, at this point, and i think it is actually what makes unclear the notion of finite coherence for a general theory in a general doctrine... We should be free to use the most intuitive language, and i don't think it is necessary to call a category with generators and relation what you say: the Ehresmann sketch viewpoint also means some kind
    of generator and relation approach for categories (class of cones, cocones, etc...), and it is better adapted to the doctrinal approach, that is based on it, at least from the viewpoint i talk about on higher doctrines.

    The point is to define a theory by a family of generating diagrams, generating cones and cocones (this is the categorical notion of syntax). One then studies semantics using purely higher categorical methods. This is what i mean by generators and relations.

    I prefer to say ``generators and relations'' because it is more intuitive than ``finitary structure'', that is reserved to specialized people.

    I don't understand why you want to put such a complicated name to talk about a simple idea, that is inductive construction of an object from simpler objects (this is what i mean by generators and relations).

    Are you sure that a coherent topos is given by a finitary structure?

    It is the category of sheaves on a Grothendieck site generated by finite coverings, so one may interpret it as the doctrine of models of a theory of limit sketches given by categories with nerves of coverings.

    This is not very algebraic/finitary, but it is coherent in the sense of the topoi doctrine (that is the doctrine of models of another doctrine: limit sketches).

    Constructing doctrines by inductive methods is really as the heart of the notion of finite coherence that i would like to define, to get the doctrinal axiom of choice (combining Goedel's model theory methods of logic with higher categorical theories).

    I think the point is to construct a ``doctrine machine'', like for example Lawvere's tensor products of theories, to define complicated theories from simple ones, by a sketch like method, a la Ehresmann. One may then define finite coherence using the process of construction we used. For example, this is what we do for topoi... One defines a finitary topos (=coherent one) as a topos whose
    topology is generated by finite coverings. How could one generalize this to higher theories? This is the main question i would like to answer.
    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 23rd 2012

    Sorry for sticking my nose in, but since the “definition” was obviously intended loosely or tenuously, why not incorporate the intended sketch-like meaning right into the definition? Something like, “a doctrine D is a higher category with structure described by a higher Ehresmann sketch, presented by ’generators’ (e.g., pasting diagrams, higher notions of cones/cocones) and ’relations’ (e.g., higher limit/colimit assumptions on the cones/cocones).”

    I’m making this up on the spot; obviously other wordings could be given to suit taste. The main thing is to avoid giving a wrong impression; if one says something like “weak nn-category defined by generators and relations”, then someone like Mike might think of presentations by weak nn-computads or the like, which is the wrong impression (and I take it that’s his main objection).

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2012

    Yes Todd, that’s right (about my objection). It is not that I want a complicated name, but that I do not want a wrong name. I do not believe that we should always “feel free to use the most intuitive language” if by “intuitive” one means “intuitive to the person writing it” — we need to also respect standard usage. The term “group” is not perhaps very intuitive, but we shouldn’t feel free to call groups “symmetroids” or something that may seem more intuitive, because then all the mathematicians in the world who know what groups are wouldn’t know what we were talking about. There is a completely standard meaning of what “presented by generators and relations” means for any kind of algebraic gadget, and higher categories are an algebraic gadget; thus we are not free to redefine “presented by generators and relations” to mean something different for them.

    What is true, as Todd suggests, is that a sketch is a presentation of a theory in some doctrine (depending on the kind of sketch) by generators and relations. But the category of models of that sketch is not presented by generators and relations.

    • CommentRowNumber18.
    • CommentAuthorfpaugam
    • CommentTimeJan 29th 2012
    I see... I like to keep a loosely definition until i have the right theorem, that for this theory would be the doctrinal axiom of choice: every finitely coherent theory of a doctrine has a model.

    The approach in this story is to completely forget what you know about set theory, and start directly with a given metalanguage of n-categories. So generators and relations may mean what you want, i guess. I understand that you don't agree, but this is my personal opinion.

    Since i don't have the axiom of choice, i don't want to specialize to a particular notion of ``generators and relations''. I tend to think that describing a doctrine as a sub-(n+1)-category of n-Cat described by an n-monad (finitary or not) is also some kind of generators and relations descriptions. Higher sketches is another possibility. And there is also the Kan extension approach, that may be used. If you are able to prove the doctrinal axiom of choice, you can specify a more precise definition. The loosy definition is enough for all examples i have in mind, and is helpful for pedagogical reasons (otherwise, you close the subject to non specialists, and i am doing applied mathematics, so i can't do that).

    If you accept to forget about set theory and the definition of n-categories, and only work with them as being given as a metalaguage, the notion of generators and relations is whatever you want.

    Also: it is possible and often useful to define inductively complicated doctrines as models for theories of simpler ones (by higher tensor products or other constructions), and this is also a kind of generator and relation construction, in some sense. At this point, i don't see the interest of specializing to a particular notion of generators and relations, since we don't have the doctrinal axiom of choice.

    Also, what is completely standard for you is not really standard for me (i see at least 4 different notions of generator and relation descriptions), but i guess it will be hard to get an agreement on this page of the nlab, and we already wasted too much time on this: the notion is clearly useful to me, from a pedagogical but also mathematical viewpoint (to see various things from above with a uniform way of looking at them), and if you don't like it, i can't force you. I don't want to make the presentation obscure by giving a more precise definition, unless one has a more precise existence result for models.
    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2012

    If you accept to forget about set theory and the definition of n-categories, and only work with them as being given as a metalaguage, the notion of generators and relations is whatever you want.

    No, the meaning of any particular mathematical term is not dependent on the foundational system in which you work. The meaning of “group” doesn’t depend on whether mathematics is formalized in set theory, or type theory, or unformalized. Sure, the precise definition of a group as “a set with X operations” or “a type with X operations” may be written in different words, but the meaning is the same. Your meaning of “generators and relations” does not, as far as I can tell, accord with the standard one.

    • CommentRowNumber20.
    • CommentAuthorfpaugam
    • CommentTimeJan 30th 2012
    Come on Mike, you just don't like this section and my approach. I can understand that, after the too many discussion that we had the two of us on this. I already had similar kinds of ideologic discussions on other subjects with collegues and i know by experience that it leads to nowhere, so i prefer that we stop this, because it is not anymore constructive. I think we are now more discussing for our personal and particular interests, and not for the good of the community.

    If it really piss you off, erase it or make it more obscure than it is, but it would not be very conformal to the positive spirit of sharing ideas that is at the basis of the nlab. Perhaps what i wrote is interesting for someone else than you.
    • CommentRowNumber21.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 30th 2012
    • (edited Jan 30th 2012)

    I hope it’s not taken amiss if I speak up again (which might be warranted because I think my previous comment might have been misunderstood).

    First, I don’t have any hard objections to an nLab article being a little bit loose or tentative, although in such cases I think it would be good to admit the speculative nature right at the top of the page. Something like, “Note: this is at present a speculative article, mostly due to Frédéric Paugam.” Or something similar. I’ve written speculative things myself at the nLab proper, although my more recent tendencies have been to put more speculative material on my personal ncatlab web page, and advertise here at the nForum if I think it’d be of interest. But as I say, either venue (nLab or personal nLab webpage) seems okay to me.

    Second, my reading is that whatever disagreement there is over “generators and relations” is not such a big deal – if I understand Mike correctly, his objection was that the earlier revision sounded like it referred to “generators and relations qua the theory of nn-categories”. It’s clear that you intend more general meanings of “generators and relations”, which might refer e.g. to presentations of higher monads, or of higher sketches, etc., in other words qua general finitary structures on higher categories, taking “generators” to be roughly synonymous with “data”, and relations to be roughly synonymous with “axioms”. And I didn’t read Mike as having any sort of disagreement with that approach (ideological or otherwise) – Mike, please correct me if I’m wrong – I think he just meant the original wording, which is not there in the present revision, sounded slightly off, and that’s all he meant. In particular, I don’t think it had anything to do with the viability of the program or approach itself, or styles of doing research, or anything like that.

    (Actually, the approach and style that I see so far remind me of what James Dolan has been thinking about in the past few years. Are you in contact with him, Frédéric?)

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2012

    Thanks, Todd, you’re exactly right. I am very happy with the idea of higher doctrines being monads presented by generators and relations; in fact that’s exactly the way I like to think about doctrines! The only thing I have a problem with is saying that the category of theories in such a doctrine, i.e. the category of algebras for that monad, is presented by generators and relations, when it’s actually the monad itself that is being so presented.

    • CommentRowNumber23.
    • CommentAuthorfpaugam
    • CommentTimeFeb 2nd 2012
    • (edited Feb 2nd 2012)
    Ok, Mike, i see what you mean. Yes, i agree that a doctrine should just be something like an n-monad (or an n-category of n-sketches), and its algebras are theories. So i think we may change this page by saying that a doctrine is an (n+1)-category ---whose objects are n-categories with additional structures---, given by generators and relations (e.g., algebras for an (n+1)-monad, or n-sketches). I personally tend to prefer n-sketches, because they are easy to define once we have a notion of higher limit.

    The idea of basing mathematics on higher category theory, from the point of view of categorical logic, is not speculative at all, so it is not necessary to say this in the front of the page. It is just a nice swiss knife to understand and define mathematical objects that appear in physics, from my point of view. However, i agree that the problem of defining a good notion of coherence in a given doctrine is a bit speculative,and that we are able to define it properly only for topoi and first order theories. I will change this aspect of the story when i get some time, and clearly separate the speculative aspects from the standard ones.

    The main problem that we had to face in our discussion, Mike, is that you don't agree to call doctrine an (n+1)-category, and theory an object of it. In practice, this definition is enough for applications, since i always define doctrines by describing explicitely the corresponding (n+1)-category, or constructing it as models of a simplified theory in another doctrine, or by higher Lawvere tensor products (see for example the E_k-theory, that gives you k-monoidal n-categories in a sketch like approach, by the recent theorem of Lurie higher algebra).

    It is not clear that all these examples can be described by an n-monad, or by an n-sketch. So by giving a more precise definition, you are reducing the domain of applications: you will need a hard theorem (like Lurie's for k-monoidal n-categories as models of E_k in n-cat) to define your doctrine in your way, even if it is very easy to define in mine.

    This is a definition of mathematical theory, based on the metalanguage of n-category, that is very useful pedagogically, i think, to understand how to define the right higher generalizations of classical objects. The best example is given by symmetric monoidal n-categories: you first define the doctrine of monoidal \infty,n-categories, as models of the Lawvere theory of monoids in the
    n+1-category of categories with products (i put infinity everywhere, here). Then you define the monoidal category Gamma of sets with bijections. Then you define symmetric monoidal categories as models of (Gamma,\otimes) in (nCat,\times), in the doctrine of monoidal categories.
    This gives you, i think (tell me if i'm wrong), the right definition of symmetric monoidal infinity,n-categories.
    This is hard to define by using another method, but i use to think that this presentation of the doctrine of symmetric monoidal higher categories is a kind of generators and relations presentation, even if one has to pass through monoidal categories. How do you treat this example using n-monads or n-sketches? This will not help i think, we prefer to use the definition that can be given in one line using imbricated doctrines, not a definition that takes an article. Remark, we may also use the Baez-Dolan definition, but the use of E_k may be useful in practical applications.

    This help (at least, this helps me) to categorify, if you prefer. It is important to give names that clearly indicate why you are using things. This is a completely personal opinion, and i understand that you don't agree with it.

    I had introduced this doctrinal axiom of choice to indicate that thinking this way may be a useful guide to the use of higher categories in applied mathematics (through categorical logics), that's all. To make this less speculative, one should be able to prove the properties on sequences in compact or complete spaces by these methods (which essentially reduce to the usual axiom of choice, in this case).
    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 2nd 2012

    So i think we may change this page by saying that a doctrine is an (n+1)-category —whose objects are n-categories with additional structures—, given by generators and relations (e.g., algebras for an (n+1)-monad, or n-sketches)

    If you left out the dashes, I would be happy: “a doctrine is an (n+1)-category whose objects are n-categories with additional structures given by generators and relations.” The point being that it is the “additional structures” which are given by generators and relations, not the (n+1)-category (putting the dashes in implies the latter).

    No need to rehash an old argument. I agreed a while ago that defining “doctrine” to mean “(n+1)-category” could be pedagogically useful, until we have a workable precise definition of “structure presented by generators and relations”. I have also agreed since long ago that monads are probably not the right precise definition of doctrine.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 2nd 2012

    The idea of basing mathematics on higher category theory, from the point of view of categorical logic, is not speculative at all

    To a mathematician, I think the phrase “basing mathematics on” sounds like a foundational enterprise, e.g. replacing set theory. If that’s what it means, then it is definitely speculative, but I don’t think that’s actually what you mean.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeFeb 2nd 2012

    but I don’t think that’s actually what you mean.

    I think Frédéric said what he means in the next sentence:

    It is just a nice swiss knife to understand and define mathematical objects

    A while back I would have been inclined to chime into that song on this occasion, which is a nice tune, to my mind.

    • CommentRowNumber27.
    • CommentAuthorfpaugam
    • CommentTimeFeb 2nd 2012
    • (edited Feb 2nd 2012)
    I think Urs has understood my point. I probably tend to mix the notion of foundation with that of swiss knife. That is mostly due to my personal ignorance on foundational stuffs. I thought that ECTS tells you that categories may be a good base for mathematics. Perhaps i misunderstood the point.

    A propos: the use of citations taken out of a complete dialog makes discussion quite strange. It's not so easy to discuss online, because you have to be very careful about what you say, more than in oral discussion...
    • CommentRowNumber28.
    • CommentAuthorfpaugam
    • CommentTimeFeb 2nd 2012
    Mike, about the dashes, you put them wherever you want. We agree in substance.
    • CommentRowNumber29.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2012

    Yes, that’s why I said “I don’t think that’s actually what you mean.” (-: I’m glad we agree.

1 to 29 of 29

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