Jean B?nabou wrote in part:
If F: A --> C and G: B --> C are functors I denote by F/G the comma category they define, and by F//G their 2-pull-back sometimes called their pseudo pull-back.
Answer to Toby Bartels: (i) You propose to compose the spans by pullbacks. Here is an example of two spans A <-- X --> B and B <-- Y --> C such that the functors A <--X and Y --> C are isos, the functors X --> B and B <-- Y have unique quasi inverses, and if Z is the pullback none of the functors A <-- Z and Z --> B is a weak equivalence: Take A=X=Y=C=1, take for B the coarse category with two objects a and b , for X --> B and B <-- Y the functors "a": 1 --> B and B <-- 1: "b" . The pullback Z is 0.
By "pullback", I meant what you above call "2-pull-back" or "pseudo pull-back". Then Z is again (up to isomorphism) 1.
How stronger a counter example do you need? And using zig-zags will make the situation even worse.
Using zigzags, one would compose zigzags directly and use no pullbacks.
(ii) you define equivalences by spans, not "up to anything". With this definition an equivalence between 1 and 1 is any non empty coarse category. Every non empty set determines up to isomorphism such a category. thus there are at least as many equivalences from 1 to 1 in your sense as there are non empty sets. A bit much don't you think?
Indeed, so one must also define natural isomorphism of equivalences. If you have any difficulty, the answer is in Makkai's anafunctor paper: http://www.math.mcgill.ca/makkai/anafun/
(but hurry, because after this mail there might very well be a quick revision of the article).
Nothing is hidden in the nLab. If it changes, click "History" at the bottom of the page. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]