Re: Comma categories

[Robert L Knighten <RLK@knighten.org>]

Bill Lawvere writes:
 >
 >
 > I recently noticed that in Abstract no. 652-4 in the Notices
 > of the AMS volume 14 (1967) page 937, John Gray advocates a
 > systematic treatment of the calculus of comma categories
 > and lists five operations which should be explicitly accounted
 > for in such a calculus.
 > He also mentions that Jon Beck contributed to that discussion.
 >
 > Probably John Gray's notes, if they still exist, would be
 > a helpful guide to someone planning to write a systematic
 > treatment as suggested recently Uwe Wolters.
 >
 > Bill

As a followup to Bill's note, here is a slightly more recent positing by John
Gray to another mailing list on this very topic.

    * To: types@theory.LCS.MIT.EDU
    * Subject: Re: Cobig, Coproduct, and Comma
    * From: gray@symcom.math.uiuc.edu (John Gray)
    * Date: Mon, 20 Mar 89 17:13:53 EST
    * Sender: meyer@theory.LCS.MIT.EDU

Date: Mon, 20 Mar 89 15:32:11 CST

>Cobig, Coproduct, and Comma  Vaughan Pratt  3/19/89
>Formally a comma category is most slickly described as a lax pullback.
>I've attempted an understandable account of this 2-category concept in
>an appendix below.  I'd appreciate pointers to other accounts.

Comma categories are an ancient tool in category theory.
They were introduced in
	F. W. Lawvere, Functorial Semantics of Algebraic Theories
	Thesis, Columbia University, 1963.
He used them in
	 --, The category of categories as a foundation for
	mathematics, Proceedings of the Conference on Categorical
	Algebra, La Jolla 1965, Springer-Verlag, New York.
I discussed them in several places:
	J. W. Gray,  Fibred and cofibred categories, same proceedings
	as above, 21-83.
I gave a brief calculus of comma categories in:
	--, The categorical comprehension scheme, Category theory,
	Homology theory and their Applications III, Lecture Notes in
	Mathematics 99, Springer-Verlag, New York 1969, 242-312.
They are described as "Cartesian quasi-limits" in the book:
	--, Formal category theory: Adjointness for 2-categories,
	Lecture Notes in Mathematics 391, Springer-Verlag, New York 1974.
which is the first place where the lax description of them can be found.
I don't credit it to anybody there, since I assumed it was general knowledge.
The name was changed to "lax limits"  in:
	G. M. Kelly and R. Street, Review of the elements of 2-categories,
	Category Seminar, Lecture Notes in Mathematics 420, Springer-
	Verlag, New York 1974.
The general theory of the properties of lax limits in 2-categories was
discussed independently by Street and me in various publications.  E. g.,
	J. W. Gray, The existence and construction of lax limits,
	Cahiers Top. et Geom. Diff. 21 (1980), 277-304.
	--, Closed categories, Lax limits and homotopy limits, J. Pure
	Appl. Algebra 19 (1980), 127-158.
	--, The representation of limits, lax limits, and homotopy limits
	as sections, in Mathematical Applications of Category Theory,
	Contemporary Mathematics 30 (1984), AMS, 63-83.
	R. Street, Two constructions on lax functors, Cahiers Top. et
	Geom. Diff. 13, (1972), 217-264.
	--, Limits indexed by category-valued 2-functors, J. Pure and
	Applied Alg. 8 (1976), 149-181.

It is of course very gratifying to see these ideas coming around again as
useful tools in the semantics of programming languages.

	John Gray


-- Bob