Subject: categories: Re: comma?

From: [email-masked] (Ross Street)
Date: Thu, 22 Oct 1998 10:03:49 +1000 (EST)

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/