categorical "varieties of algebras" (fwd)
I am forwarding this to the categories list, where I am sure there will be many answers, but I have never myself delved into these interesting questions. Please be sure to copy your answers to him (although he should probably subscribe to the list). --M ---------- Forwarded message ---------- Date: Sun, 13 Dec 2009 23:15:11 +0200 From: Alexandru Chirvasitu <chirvasitua@gmail.com> To: barr@math.mcgill.ca Subject: categorical "varieties of algebras" Dear Prof. Barr, My name is Alexandru Chirvasitu, and I am a first-year mathematics graduate student at UC Berkeley. I apologize for bothering you wih this, especially since you don't know me, but I was kind of at a loss: I don't really know any people working in the areas I am interested in personally, so I thought I'd give this a go :). I'm quite sure you'll be able to clear this out straight away. Before coming to Berkeley, I was interested in applying category-theoretic methods to study coalgebras, Hopf algebras, and other such creatures: http://arxiv.org/abs/0907.2881 It became clear later on that to get some further insight into the universal constructions useful for these problems (Hopf envelopes of co (or bi) algebras, free Hopf algebra with bijective antipode on a Hopf algebra, etc.), it would be useful to apply some Tannaka reconstruction techniques and move the free constructions "up the categorical ladder": free monoidal category on a category, (left) rigid envelope of a monoidal category, etc. Unfortunately, I couldn't find any results stating clearly (clearly for someone who is perhaps not *too* familiar with the higher categorical machinery) that such free categories always exist. Of course, the few constructions I needed can easily be done by hand, but what I had in mind was some kind of higher categorical analogue of the fact that the forgetful functor from a variety of algebras to another variety of algebras with "fewer operations" has a left adjoint. There's also an issue of how strict things should be. For what I was doing, the following setting is typical: consider the category whose objects are (not necessarily strict, but that's not very important here) monoidal categories with a specified left dual and specified (co)evaluation maps for every object, and whose morphisms are the functors which preserve all of this structure *strictly*. Then I wanted to conclude that the forgetful functor from this to *Cat* has a left adjoint, which should be easy enough. To state my question properly, I'm thinking about a category whose objects are categories *C* endowed with certain "operations", consisting of functors from *C^n x (C opposite)^m* to *C* (example: the specified left dual in the above example is a contravariant functor), appropriately natural transformations between such functors (example: the evaluation maps in a rigid tensor category as above form, together, a dinatural transformation), and equations involving these natural transformations; the morphisms are functors which preserve all the structure *strictly*. Consider the forgetful functor from this to a similar category, but with "fewer operations" (this could easily be made precise). Then, is there a result stating that such a functor always has a left adjoint? I expect it should be easy enough to employ a form of the Adjoint Functor Theorem (working with universes say, to have everything set-theoretically sound) to prove something like this, and I was thinking about writing it up for further reference. My problem was that I can't seem to be able to tell exactly what is well-known and has been written up, what is folklore and trivial, etc. Also, even though a statement as outlined above and to which I could refer would be completely satisfactory, I realize that after destrictification things become much more interesting, and you probably get some neat bicategorical results. I must once again apologize for intruding on your time like this, for the potential silliness of a newbie's question, and for the ramble factor and length of this message :). I hope you do get to reply. Thank you, Alexandru [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Alexandru, I would recommend the paper "Partial Horn logic and cartesian categories" that I wrote with Eric Palmgren (Annals of Pure and Applied Logic 145 (3) (2007), pp. 314 - 353; doi:10.1016/j.apal.2006.10.001). Free categories of various kinds usually do exist, an idea that comes out well in Higgins's 1971 "Notes on categories and groupoids" (now reissued as a TAC reprint). The underlying machinery, again recognized for a long time, relies on the notion of left exact theories, alias finite limit theories, cartesian theories or essentially algebraic theories. In the formulation as an essentially algebraic theory, the operators may be partial but their domains of definition can be expressed in terms of equations using operators defined "earlier" (i.e. there is an ordering on the operators). For categories, the core example is composition, which is partial and defined when an equation holds between the domain of one morphism and the codomain of the other. But the same phenomenon arises, for example, when adding morphisms in an Abelian enriched category or composing 2-cells in a 2-category. The fundamental result for left exact theories is that a forgetful functor, from a category of algebras for one theory to that for another with fewer operators (or fewer equations, for that matter), has a left adjoint. I believe the result is covered in chapter 4 Barr and Wells' "Toposes, Triples and Theories", or at least is inferrable from Kennison's Theorem as stated there. Palmgren and I proved the fundamental freeness result (our Theorem 29, Free Partial Model Theorem) in a way that makes much clearer the connection with the well known result for algebraic theories. (For algebraic theories, with all operators total, you just take a term model and factor out a congruence.) We used a logic, minimally adapted from the standard account of categorical logic as in the Elephant, in which terms are only partially defined. We also described a simple "quasi-equational" mode of theory presentation equivalent to left exact theories. The proof is then very similar to the algebraic case, taking a partial term model and factoring out a partial congruence. It is also more elementary than that in TTT. I hope this will answer your question in most cases. Strictness is an issue, as you mention. Our treatment is very syntactic in nature, and expects strictness of homomorphisms. However, we do have techniques that allow this to be relaxed - see our Theorem 56. We also give various examples from category theory, as well as discussing some of the history of the result. Regards, Steve Vickers. Michael Barr wrote:
From: Alexandru Chirvasitu <chirvasitua@gmail.com>
Dear Prof. Barr,
... Unfortunately, I couldn't find any results stating clearly (clearly for someone who is perhaps not *too* familiar with the higher categorical machinery) that such free categories always exist. Of course, the few constructions I needed can easily be done by hand, but what I had in mind was some kind of higher categorical analogue of the fact that the forgetful functor from a variety of algebras to another variety of algebras with "fewer operations" has a left adjoint.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
i am wondering why is the public image of category theory so important for us. i mean, if category theory is a powerful and useful tool, as it is, then it should be able to take care for itself. bread does not need advertising. i have been with categories for many years. i think in categories, and i used them in each and every one of my research projects, in every piece of software that i designed, in every paper that i wrote. but sometimes it is easier to get to the point without spelling out all definitions in full generality. and without tackling the opprobium. e.g., i worked on networks, and have papers about trust networks, and reputation networks, and recommender systems. a network is a weighted graph, and it composes to some extent, because a friend of a friend is almost like a friend, but a friend of a friend of a friend etc, six hops removed --- is probably not a friend. but you can do with networks a lot of what you can do with categories: make arrow networks, adjoin colimits... anyway, i defined all that, and mentioned categories, but did not really advertise them. was that a mistake? maybe it wasn't such a good work, and i would have done a disservice to category theory by advertising it in a bad paper. i have been using categories in my little crypto modules, and in serious reduction proofs, for more than 5 years (cryptography is a theory of functions after all!), but i first gave a crypto talk using categories last week. and this was not a talk to hard-core cryptographers. i think category theory sometimes suffers from our advertising. even in a good paper, advertising is advertising. there are places for that, and there are places where it is better not to do. i do understand that we need to take care for the public image of our work. funding depends on that, hiring depends on that. but maybe we should clearly state that this is a matter of advocacy and of influence, and not mix it up with Promoting the Truth. i somehow think that the truth can take care for itself. but as always, maybe i am wrong. -- dusko [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andrew, You wrote
Let me make these remarks a little more concrete with a request (or a challenge if you prefer). In my department, the colloquium is called "Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've been writing too many scripts lately!). I'm giving this talk in January. My original plan was to say something nice and differential, with lots of fun pictures of manifolds deforming or knots unknotting, or something like that. However, the discussion here has set me to thinking about saying something instead about category theory. It is a pearl of mathematics, it does have a certain beauty, there's certainly a lot that can be said, even to a fairly applied audience as we tend to have here (it is the Norwegian university of Science and Technology, after all), even without talking about programming (about which I know nothing).
But for such a talk, I need a story. I don't mean a historical one (I'm not much of a mathematical historian anyway), I mean a mathematical one. I want some simple problem that category theory solves in an elegant fashion. It would be nice if there was one that used category theory in a surprising way; beyond the idea that categories are places in which things happen (so perhaps about small categories rather than large ones).
A colloquium is a good place for expressing wild ideas. But they must be related to something everyone can understand and touch. I suggest you talk about "The field with one element" if you think the subject can fit your audience. http://en.wikipedia.org/wiki/Field_with_one_element Many things in this subject are very speculative but there are also a few concrete developpements. One is the algebraic geometry "under SpecZ" of Toen and Vaquié. Another due to Borger is using lambda-rings. What is a lambda-ring? In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)] [Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring" and they call a pre-lambda-ring a "lambda-ring"] The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L) can be defined in a natural way if we use category theory. Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings can be defined to be the theory of natural operations on the functor V. The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] which takes an element x\in M to the power series 1+tx. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andrew, Please disregard my suggestion about a new definition of lambda ring! My memory may have failed me! I am not sure the new definition is right! Best, André -------- Message d'origine-------- De: categories@mta.ca de la part de Joyal, André Date: mar. 15/12/2009 15:14 À: Andrew Stacey; categories@mta.ca Objet : categories: Re: A well kept secret Dear Andrew, You wrote
Let me make these remarks a little more concrete with a request (or a challenge if you prefer). In my department, the colloquium is called "Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've been writing too many scripts lately!). I'm giving this talk in January. My original plan was to say something nice and differential, with lots of fun pictures of manifolds deforming or knots unknotting, or something like that. However, the discussion here has set me to thinking about saying something instead about category theory. It is a pearl of mathematics, it does have a certain beauty, there's certainly a lot that can be said, even to a fairly applied audience as we tend to have here (it is the Norwegian university of Science and Technology, after all), even without talking about programming (about which I know nothing).
But for such a talk, I need a story. I don't mean a historical one (I'm not much of a mathematical historian anyway), I mean a mathematical one. I want some simple problem that category theory solves in an elegant fashion. It would be nice if there was one that used category theory in a surprising way; beyond the idea that categories are places in which things happen (so perhaps about small categories rather than large ones).
A colloquium is a good place for expressing wild ideas. But they must be related to something everyone can understand and touch. I suggest you talk about "The field with one element" if you think the subject can fit your audience. http://en.wikipedia.org/wiki/Field_with_one_element Many things in this subject are very speculative but there are also a few concrete developpements. One is the algebraic geometry "under SpecZ" of Toen and Vaquié. Another due to Borger is using lambda-rings. What is a lambda-ring? In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)] [Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring" and they call a pre-lambda-ring a "lambda-ring"] The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L) can be defined in a natural way if we use category theory. Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings can be defined to be the theory of natural operations on the functor V. The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] which takes an element x\in M to the power series 1+tx. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Andrew The conceptual definition of lamba ring given by Andre', (for which a presentation by useful but complicated identities is theorem rather than a definition) solves an important one of a generic class of problems that I proposed in the article that is bundled with my thesis in TAC Reprints. A simple pedagogically convincing example is given by the following dialogue : I have an example of a general category C and a functor U from it to finite sets; moreover I have a particular object X in C : what information about X can I find using U ? Well, you could count the points of U(X).Yes but that is by no means all. The functor U has a group G of all natural automorphisms, and so U can be lifted across the category of G-sets, thus that number is actually a sum of more refined invariants indexed by the subgroups of G. The generic problem (for the doctrine of algebraic theories rather than for the subdoctrine of permutation representations) considers a specific assignment of an algebraic theory to any morphism of algebraic theories (from Andre's example it should be clear which assignment) and asks for specific calculation (eg a presentation in terms of given presentations, or indeed any information). The construction involves the natural structure of a functor that is not representable in general but hope comes from the fact that this functor preserves filtered colimits and reflexive coequalizers and that some examples are representable or otherwise computable. Bill On Tue 12/15/09 3:14 PM , Joyal, André joyal.andre@uqam.ca sent:
Dear Andrew,
You wrote
Let me make these remarks a little more concrete with a request (or>a challenge if you prefer). In my department, the colloquium is called>"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; I've>been writing too many scripts lately!). I'm giving this talk in January. My>original plan was to say something nice and differential, with lots of fun>pictures of manifolds deforming or knots unknotting, or something like that.>However, the discussion here has set me to thinking about saying something>instead about category theory. It is a pearl of mathematics, it does have>a certain beauty, there's certainly a lot that can be said, even to a fairly>applied audience as we tend to have here (it is the Norwegian university of>Science and Technology, after all), even without talking about programming>(about which I know nothing).
But for such a talk, I need a story. I don't mean a historical one (I'm not>much of a mathematical historian anyway), I mean a mathematical one. I want>some simple problem that category theory solves in an elegant fashion. It>would be nice if there was one that used category theory in a surprising way;>beyond the idea that categories are places in which things happen (so perhaps>about small categories rather than large ones). A colloquium is a good place for expressing wild ideas. But they must be related to something everyone can understand and touch.I suggest you talk about "The field with one element" if you think the subject can fit your audience.
http://en.wikipedia.org/wiki/Field_with_one_element Many things in this subject are very speculative but there are also a few concrete developpements. One is the algebraic geometry "under SpecZ" of Toen and Vaquié.Another due to Borger is using lambda-rings. What is a lambda-ring? In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ringto be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)][Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"and they call a pre-lambda-ring a "lambda-ring"] The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)can be defined in a natural way if we use category theory. Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in theopposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Setwe obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings can be defined to be the theory of natural operations on the functor V. The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] which takes an element x\in M to the power series 1+tx.
Best, André
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Here are some comments on the discussion about category theory: 1. I testify that categories can be fruitfully introduced to undergraduates. In the late sixties, I followed 3 cohorts of students from their entrance at the university (just after the Baccalaureat) up to their graduation. I introduced category theory in the mid of their first year and thoroughly used it later on in my courses on Algebra, Topology, Differential Calculus and functional spaces, and Algebraic Topology. (All these courses have been multigraphed in Amiens.) Since the university in Amiens was only beginning to develop, there were not many students, but most of them seemed to enjoy categories and about 10% of those who completed the cycle went on to do research (generally using categories) and obtained university positions. However I had to stop my experiment because several of my colleagues did not appreciate categories:(t was a very bad time for them in France in the early 70's. 2. Applications of categories begin to be welcomed in the most varied scientific domains. An example is our general model "Memory Evolutive Systems" for 'natural' complex autonomous systems, such as biological or social systems; when we first introduced it in the 90's, people were somewhat skeptical, but now more people accept it, in particular cognitive scientists are taking a real interest in its application to cognitive systems (our model MENS). 3. Thus, though one of the older living "veterans" of the 60's "war", I am not pessimistic (as John Baez seems to imply). I think my generation has lost a battle, partially because of too much precipitation and not always enough diplomacy, but I feel that the youngsters are winning the war and I'ld try to help them as much as possible for I have kept all my enthusiasm. So best wishes to the categories and to all their friends for 2010 Andree [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I would just like to say thank you to all those who responded to my request for some ideas for a story to tell on category theory. I intended to reply to each one but got swamped by grading exams and didn't get round to it before the holiday (I still intend to reply but it will have to wait a week now) so this is a "holding email" just to say that I'm grateful for the ideas and to apologise for not responding (yet) individually. God jul og godt nyttår! Andrew [For admin and other information see: http://www.mta.ca/~cat-dist/ ] Status: RO
Hello, I still like to add some remarks. Category theory is one part of mathematics, and it should be treated not better, but not worse than others. It looks more important to me that categorical thinking becomes popular in other areas in mathematics. Some years ago, a functional analyst needed half an our to prove that homeomorphic Banach spaces have homemorphic duals, a simple consequence of the fact that all functors preserve isomorphisms. Another example from my own experience: People, who worked about orthomodular lattices noticed that they have no tensor product. So they tried to weaken the notions and ended up with effect algebras, but unfortunately they did not admit a tensor product either. Su people looked for other notions. But they had already shown that a tensor product of effect algebras exists if one admits 0=1; i.e. the tensor product my collapse. But because they did not admit this, they had to formulate their result more complicated. Later I saw that tensor products of orthomodular posets exist if one admits 0=1; the easy proof uses the Adjoint Functor Theorem and does not give much insight into the structure. It also seems to work for orthomodular lattices. My preference for orthomodular posets rather than lattices is also inspired by categorical thinking. The idempotents of an arbitrary ring with 1 form an orthomodular poset, and this construction yields a functor. This is the non-commutative analogue to the Boolean algebra of idempotents of an arbitrary ring. But most people were inspired by quantum dynamics and were looking for an abstraction for the set of projections of a Hilbert space. Here joins and meets exist (somehow accidentially) because projections correspond to closed subspaces. But they are not continuous and have no physical meaning in general. I think it is often better to look for functorial notions than to use ad-hoc-abtractions. Greetings Reinhard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (8)
-
Andree Ehresmann -
Andrew Stacey -
Dusko Pavlovic -
F William Lawvere -
Joyal, André -
Michael Barr -
Reinhard Boerger -
Steve Vickers