Hi, whenever I'm teaching basic category theory, students ask me if there is a connection between limits in the categorical sense and limits in the analytical sense, e.g. the limit of a sequence of real numbers. I've never found an answer to this question. So I'd be very grateful for answers to one of the following: - Can the limit of a sequence of real numbers be expressed as a categorical limit (of course it can if the sequence is monotone, but what if it is not)? - Why have people chosen the term "limit" in category theory? (And, by the way, who has defined it first?) Many thanks in advance Tobias -------------------------------------------------------------- Tobias Schröder FB Mathematik und Informatik Philipps-Universität Marburg WWW: http://www.mathematik.uni-marburg.de/~tschroed email: tschroed@mathematik.uni-marburg.de
Tobias Schroeder asks: - Can the limit of a sequence of real numbers be expressed as a categorical limit (of course it can if the sequence is monotone, but what if it is not)? A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories? As for his second question: - Why have people chosen the term "limit" in category theory? (And, by the way, who has defined it first?) In the beginning, the only diagrams that had limits were "nets", that is, diagrams based on directed posets. I believe it was Norman Steenrod in his dissertation who first used the term. Before his dissertation the Cech cohomology of a space was defined only as the numberical invarients that arose as a limit of a directed set of such invariants. It was Steenrod who perceived that Cech cohomology could be defined as an abelian group. For that he needed to invent the notion of a limit of a directed diagram of groups. In the 50s the fact that one didn't need the diagram to be directed was considered startling. At least two of us tried to avoid the word "limit" in this more general setting. Jim Lambek was pushing "inf" and "sup", a suggestion I wish I had heard. Not having heard it, I was pushing "left root" and "right root" (one was, after all, supplying a root to a generalized tree. sort of). All to no avail. So now we have "finite limits" and "finitely continuous".
Tobias Schroeder writes:
- Can the limit of a sequence of real numbers be expressed as a categorical limit (of course it can if the sequence is monotone, but what if it is not)?
I think I have an answer to this question (without cheating). It may be well known or wrong (I haven't carefully checked the details, but I believe that they are correct). Given a metric space X with distance function d, construct a category, also called X, as follows. The objects of the category X are the points of the space X. An element of the hom-set X(x,y) is a triple (r,x,y) with r a real number such that d(x,y)<=r. The composite of the arrows r:x->y and s:y->z is the arrow s+r:x->z. This is well defined by virtue of the triangle inequality d(x,z)<=d(x,y)+d(y,z). By virtue of the condition d(x,x)=0, we have identities. Notice that all arrows are mono. Of course, because the category X is small and it is not a preordered set, it doesn't have all limits. But some limits do exist. CLAIM: Let x_n be a sequence of points of X, and, for each n, let the arrow r_n:x_{n+1}->x_n be d(x_n,x_{n+1}). If the sum of r_k over k>=0 exists, then this omega^op-diagram has a categorical limit. The source of the limiting cone is the metric limit l of the sequence. The projection p_n:l->x_n is the sum of r_k over k>=n. If q_n:m->x_n is another cone, then the mediating map u:l->m exists (and will be automatically unique), because, by definition of cone and of our category, q_n will have to be bigger than r_n, and then u=q_n-r_n does the job. Remarks. (1) For any given Cauchy sequence, one can construct an equivalent Cauchy sequence for which the assumption in the second sentence of the claim fails. Using classical logic, for any given Cauchy sequence, one can construct an equivalent Cauchy sequence for which the assumption holds. (2) In (some flavours of) constructive mathematics, the notion of a Cauchy sequence "with fixed rate of convergence" is taken as basic. This often is taken to mean that d(x_n,x_n+1)<=c^n for a fixed c with 0<c<1. For such sequences, the assumption is satisfied. Recall that a map f:X->Y is called non-expansive if d(fx,fx')<=d(x,x'). If the natural numbers are metrized by d(m,n)=c^min(m,n) for m/=n, to get a space N, then such a Cauchy sequence is just a non-expansive map N->X. It converges if and only if the non-expansive map has a non-expansive extension to N_{infty}, the metric completion of N (which, topologically, is the one-point compactification of N). And non-expansive maps are functors---see (3) below. (3) Recall that a map f:X->Y is called lipschitz if there is a constant c for which d(fx,fx')<=c.d(x,x'). A lipschitz map f:X->Y gives rise to a functor f:X->Y defined by f(r:x->x')=c.r:f(r)->f(x'). That is, the object part is given by the map itself, and the arrow part is given by multiplication with the lipschitz coefficient. (4) We have taken the arrows r_n to be d(x_n,x_{n+1}). But actually any choice of arrows does the job, provided the sum of r_k over k>=0 is finite. (5) Other two conditions for the distance function of a metric space, which were not used in the definition of the category X, are (i) d(x,y)=0 implies x=y, and (ii) d(x,y)=d(y,x). By the first, our category is skeletal. By the second, it is selfdual. Of course, people have considered generalized metric spaces in which these are not assumed to hold. See, for example, Lawvere's paper "Metric spaces, generalized logic, and closed categories", in which he regards a generalized metric space X as an enriched category with X(x,y)=d(x,y) (so he has hom-numbers instead of hom-sets). Here we have hom-sets (of numbers). MHE
Tobias Schroeder writes:
- Can the limit of a sequence of real numbers be expressed as a categorical limit (of course it can if the sequence is monotone, but what if it is not)?
I think I have an answer to this question (without cheating). It may be well known or wrong (I haven't carefully checked the details, but I believe that they are correct).
the view of (quasi)metric spaces as R+-categories with the hom-objects d(x,y) goes back to lawvere's "metric spaces, generalized logic and closed categories" from 1973. the cauchy completeness of a space was identified with what came to be known as the cauchy completeness of the corresponding category (see kelly's book on enriched categories, or francis borceux handbook). and cauchy completeness of a category amounts to the existence of certain absolute (co)limits: eg over Set, Ab, Cat... to splitting idempotents (which can be done by taking (co)equalizers with id). -- dusko
|A good question. I have no answer, only a similar (and ancient) |question: is there a setting in which adjoint operators on Hilbert |spaces can be seen to be examples of adjoint functors between |categories? i may as well state the obvious (not necesarily right) answer to this: no, not quite. rather, what seems to be going on is that the phenomenon of adjoint linear operators is, in yetter's terminology, a sort of decategorification of the phenomenon of adjoint functors. decategorification is generally a somewhat destructive process, destroying the morphisms between objects, and since the morphisms are so intrinsic to the definition of adjoint functor it seems too much to hope for that the decategorified phenomenon of adjoint linear operators could actually qualify as a special case of adjoint functors. there are suggestive indications, though, that all of the really interesting special cases of adjoint linear operators in physics, for example, are decategorifications of interesting pairs of adjoint functors. (for example so-called "creation and annihilation operators on fock space" have categorified analogs that live on a categorified analog of fock space whose objects/vectors are something like joyal's "species of structure".) so roughly: the general phenomenon of adjoint linear operators is technically probably not quite a genuine special case of adjoint functors. the actual interesting special cases of adjoint linear operators, however, are often seen to be mere shadows of more interesting cases of adjoint functors.
i wrote: |phenomenon of adjoint linear operators is, in yetter's terminology, a |sort of decategorification of the phenomenon of adjoint functors. now that i think about i guess it was crane rather than yetter who started using the term "categorification". is it correct that lawvere and schanuel use the term "objectification" (or something like that) to mean pretty much the same thing as what crane meant by "categorification"? i think i might actually prefer "objectification" here but i mostly hang out near sub-communities where "categorification" has caught on to a certain extent.
The category of finite sets and isomorphisms versus the category with objects 1,2,...,n,... and the symmetric groups \Sigma_n as morphisms The category of `modules' over one is equivalent to The category of `modules' over the other Is this somewhere in the literature or just folk lore? Maybe for one n at a time?? thanks jim
I very much agree with James Dolan's response to the question of comparing categorical limits and adjoint functors with their abstract counterparts in analysis. Other words that have been used for "objectification" and "categorification" are "laxification" and "identity breaking". The original questions were a bit like asking: "Is the plus in an abelian group a categorical coproduct?" Lots of abelian groups can arise by taking isomorphism classes and using a categorical coproduct: but then we lose the beautiful universal property. Along the same lines, I enjoy bicategories, with coproducts in their homcategories (preserved by composition), much more than additive categories. Not only is every global coproduct in such a bicategory also a global product, but the projections from the global products are right adjoint to the coprojections into the coproduct. Ross
Peter Freyd wrote:
A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories?
probably not, but the they seem to be instances of the same general structure. (it is simple, pretty old, and i am sure many have noticed it, but since no one mentioned it, here it goes.) let U : Cat ---> CAT be the embedding of small categories in all, and let Y: Cat^op ---> CAT map each small category A to the presheaves Psh(A). now look at the (pseudo)comma category U/Y. each category A is represented in it by the yoneda embedding A-->Psh(A). the morphisms between A-->Psh(A) and B --> Psh(B) are exactly the pairs of adjoint functors between A and B. on the other hand, let I: Vec---> Vec be the identity functor, and let * : Vec^op ---> Vec take a vector space V to its dual V*. look at the comma category I/*. each hilbert space V is represented in it by the obvious linear map V-->V*. the morphisms between V-->V* and W-->W* are exactly the adjoint pairs of operators between V and W. playing around a bit, these two comma categories can be thought of as Chu(CAT,Set) and Chu(Vec,R) respectively. so both sorts of adjunctions are the instances of the chu morphisms. they are the chu morphisms on the "representation" objects, in the form X --> R^X, where R is the dualizing object. -- dusko PS infact, one could start from Chu(SET,Set), and define categories as the profunctors A-->Set^A which form a monoid with respect to the profunctor composition. you'd get only the object part of the adjoint functors as the morphisms of this chu, but the arrow part follows from the adjunction (i think). now can we characterize hilbert spaces in a similar way within Chu(Vec,R)? this seems to be a completely different kind of question. in particular, it is possible to define "profunctors" with respect to R or C, like we did with respect to Set, and we can compose them, but hilbert spaces do not seem to be monoids with respect to this composition, at least the way it occurs to me. if there is no such composition that they are, then hilbert spaces are like R-enriched graphs, rather than categories.
Peter Freyd wrote:
A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories?
I once answered a similar question. In 1974 or 1975, I published a paper "Adjoint functors induced by adjoint linear transformations" in the Proceedings of the AMS. The idea is that a pair of adjoint linear transformations between two linear topological vectorspaces, e.g., a special case is Hilbert spaces, naturally map to an adjoint pair of functors between the categories which are the lattices of closed subspaces, i.e., a galois connection. Cheers, Paul H. Palmquist Paul Palmquist <Paul_Palmquist@compuserve.com>@compuserve.com> on 05/16/2001 08:35:30 AM To: Paul Palmquist <phpalmquist@west.raytheon.com> cc: Subject: categories: Re: Limits -------------Forwarded Message----------------- From: Dusko Pavlovic, INTERNET:dusko@kestrel.edu To: [unknown], INTERNET:categories@mta.ca Date: 5/10/01 4:29 AM RE: categories: Re: Limits Peter Freyd wrote:
A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories?
probably not, but the they seem to be instances of the same general structure. (it is simple, pretty old, and i am sure many have noticed it, but since no one mentioned it, here it goes.) let U : Cat ---> CAT be the embedding of small categories in all, and let Y: Cat^op ---> CAT map each small category A to the presheaves Psh(A). now look at the (pseudo)comma category U/Y. each category A is represented in it by the yoneda embedding A-->Psh(A). the morphisms between A-->Psh(A) and B --> Psh(B) are exactly the pairs of adjoint functors between A and B. on the other hand, let I: Vec---> Vec be the identity functor, and let * : Vec^op ---> Vec take a vector space V to its dual V*. look at the comma category I/*. each hilbert space V is represented in it by the obvious linear map V-->V*. the morphisms between V-->V* and W-->W* are exactly the adjoint pairs of operators between V and W. playing around a bit, these two comma categories can be thought of as Chu(CAT,Set) and Chu(Vec,R) respectively. so both sorts of adjunctions are the instances of the chu morphisms. they are the chu morphisms on the "representation" objects, in the form X --> R^X, where R is the dualizing object. -- dusko PS infact, one could start from Chu(SET,Set), and define categories as the profunctors A-->Set^A which form a monoid with respect to the profunctor composition. you'd get only the object part of the adjoint functors as the morphisms of this chu, but the arrow part follows from the adjunction (i think). now can we characterize hilbert spaces in a similar way within Chu(Vec,R)? this seems to be a completely different kind of question. in particular, it is possible to define "profunctors" with respect to R or C, like we did with respect to Set, and we can compose them, but hilbert spaces do not seem to be monoids with respect to this composition, at least the way it occurs to me. if there is no such composition that they are, then hilbert spaces are like R-enriched graphs, rather than categories. ----------------------- Internet Header -------------------------------- Sender: cat-dist@mta.ca Received: from mailserv.mta.ca (mailserv.mta.ca [138.73.101.5]) by sphmgaae.compuserve.com (8.9.3/8.9.3/SUN-1.9) with ESMTP id HAA02263 for <76600.1050@compuserve.com>; Thu, 10 May 2001 07:29:48 -0400 (EDT) Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f4AAwqm21469 for categories-list; Thu, 10 May 2001 07:58:52 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3AF9FA78.9968DFEE@kestrel.edu> Date: Wed, 09 May 2001 19:18:32 -0700 From: Dusko Pavlovic <dusko@kestrel.edu> X-Mailer: Mozilla 4.72 [en] (X11; U; SunOS 5.5.1 sun4u) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Limits References: <200105032338.f43NcI203820@math-cl-n03.ucr.edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk
participants (8)
-
Dusko Pavlovic -
jdolan＠math.ucr.edu -
jim stasheff -
Martin Escardo -
Paul H Palmquist -
Peter Freyd -
Ross Street -
Tobias Schroeder