Date: Mon, 19 Oct 1998 17:14:40 +0100 From: Manuel Bullejos Subject: categories: Comma categories Does any body know if comma categories have been defined in enriched contexts? I have an idea of how they can be defined in some particular contexts, such as Cat-categories or Simplicial-categories, but I don't know if there is a general definition or even if a definition in the above two contexts can be found in the literature. Thanks Manuel Bullejos Subject: Re: categories: Comma categories Date: Mon, 19 Oct 1998 11:19:50 -0600 From: Vaughan Pratt >Does any body know if comma categories have been defined in >enriched contexts? I'm not sure if it's what you have in mind, but combining comma categories and enrichment is the theme of Casley, R.T., Crew, R.F., Meseguer, J., and Pratt, V.R., ``Temporal Structures'', Mathematical Structures in Computer Science, Volume 1:2, 179-213, July 1991. The abstract is at my web page as http://boole.stanford.edu/chuguide.html#P2 Vaughan Pratt Date: Tue, 20 Oct 1998 10:26:56 +1000 From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: Comma categories >Does any body know if comma categories have been defined in >enriched contexts? Lawvere's La Jolla paper, where general comma categories were introduced, showed how to construct them from pullbacks and a "cylinder" (or "arrow object") construction. John Gray (SLNM p. 254) showed that cylinder is a universal notion which a 2-category may or may not have. I pointed out [Fibrations and Yoneda's lemma in a 2-category, Lecture Notes in Math. 420 (1974) 104-133; MR53#585] that finite completeness for a 2-category should mean that it have pullbacks, a terminal object, and cylinders (a similar idea was in my PhD thesis for differential graded categories which are finitely complete when they admit pullbacks, a zero object and "suspension"). Finite completeness for 2-categories is further analysed in [Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149-181; MR53#5695]. More generally, finite completeness for a V-category A (= a category with homs enriched in V) means that its underlying category has finite ordinary limits, which are preserved by representables A(a,-) into V, and that it admits cotensoring by the "finite" objects of V. There is some choice about what you mean by "finite" object in V however "finitely presentable" is often the right thing. Sometimes, as in the case of V = Cat, the finite objects are generated by a few finite objects - that is why "cylinder" plays the important role in 2-categories (it is the finite generating object, cotensor with which is cylinder). So why am I going on about finite limits in 2-categories? Well, Lawvere's construction shows that comma objects exist in any finitely complete 2-category. Comma objects are particular finite limits just like pullbacks. In particular, there is a 2-category V-Cat of V-categories, V-functors and V-natural transformations. It is certainly complete (as a 2-category) for any decent V. So, indeed, it is well known that comma objects (or comma V-categories) exist. They have their uses but NOT for the wonderful use that Lawvere put them to: Lawvere provided a formula for left (right) Kan extensions of ordinary functors which involves taking a colimit (limit) over a comma category. [Indeed, more is true; see my definition of "pointwise Kan extension" in "Fibrations and Yoneda's lemma in a 2-category".] However, this formula does not work even for additive categories (= categories enriched in the monoidal category of abelian groups). Regards, Ross http://www.mpce.mq.edu.au/~street/ Date: Tue, 20 Oct 1998 17:11:48 -0400 (EDT) From: F W Lawvere Subject: categories: Re: Comma categories A crucial point is whether the recipient of the enriching is cartesian or not. Note that fully internalising always must involve a cartesian aspect since one must diagonalize on the parametrizers of families of objects (at least) in order to explain eg natural transformations, even if the parametrizers for individual homs are not cartesian (eg linear or metric). One can envisage replacing individual "comma" categories by families of categories parametrized by (commutative) coalgebras, which seems just a way of constructing a cartesian category for the purpose, to which it may or may not be adequate. Symmetric monoidal categories in which the unit object is terminal seem to have a special role, but that may be illusory.(After all "any" smc is covered by one with that additional property) . Perhaps the affine modules ( see my paper "Grassmann's dialectics and category theory") constitute a good test case for proposed constuctions Bill Lawvere. ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Mon, 19 Oct 1998, Manuel Bullejos wrote: > > Does any body know if comma categories have been defined in > enriched contexts? > > I have an idea of how they can be defined in some particular > contexts, such as Cat-categories or Simplicial-categories, but I > don't know if there is a general definition or even if a > definition in the above two contexts can be found in the > literature. > > Thanks > > Manuel Bullejos > > > Date: Thu, 22 Oct 1998 10:03:49 +1000 (EST) From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: comma? Dear Carlos > Hi. I'm interested in following the discussion of enriched categories and >``comma objects'' on the categories list, and particularly in what you just >wrote. But...what is a ``comma category'' or more generally ``comma object''? Comma categories are explained, for example, in Mac Lane's book "Categories for the working mathematician" Grad Texts in Math #5 (Springer). In a sense they are "lax pullbacks" defined when given two functors with the same codomain. Hence, just as we can carry over the notion of pullback in Set to any category, we can carry over, by representability, the notion of comma category from Cat to any 2-category (I call them comma objects in SLNM 420). > Also, was your PhD thesis published? I used the notion of differential >graded category in a paper about vector bundles a while ago; then found out >that Kapranov et al had a few earlier papers about it, but I didn't know >that it came from even before that. Exaggerating only slightly, the whole reason enriched category theory got going in the 60s was to deal with the example of differential graded categories: these are categories enriched in chain complexes (Eilenberg-Kelly "Closed categories" LaJolla 1965). As I understand it, that example prompted Sammy and Max to their joint work: they had each used DG-categories before. My thesis [0] was not published in full. For the published papers [2], [5] on the thesis, I took out the stuff on DG-categories. However, [25] uses the DG-category approach very strongly and gives new proofs of more general results (I had some of these at the time of my thesis but hadn't included them). The papers provide universal coefficients (or homotopy classification) theorems for diagrams of chain complexes. 0. PhD Thesis: Homotopy Classification of Filtered Complexes, University of Sydney, August 1968. 2. Projective diagrams of interlocking sequences, Illinois J. Math. 15 (1971) 429-441; MR43#4881. 5. Homotopy classification of filtered complexes, J. Australian Math. Soc. 15 (1973) 298-318; MR49#5135. 25. Homotopy classification by diagrams of interlocking sequences, Math. Colloquium University of Cape Town 13 (1984) 83-120; MR86i:55025. Best regards, Ross www.mpce.mq.edu.au/~street/