We actually had a post-seminar reference-hunt on this in Stockholm quite recently, and found that the arrows-only definition goes right back to Mac Lane 1948, “Groups, Categories, and Duality”: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079106/pdf/pnas01707-0037.pdf This cites two earlier papers only along with the definition (Mac Lane 1942 and Eilenberg–Mac Lane 1945 — the first two papers to mention categories, right?), but both of those used the objects-and-arrows formulation. So it seems that the two-sorted formulation was considered right from the start, and the arrows-only version either from the start or very soon afterwards. Of course, the original question has already been well answered, but I guess the extra history may be of interest to others as well. Best, –Peter. On Fri, Mar 6, 2015 at 1:49 AM, Jiri Adamek <adamek@iti.cs.tu-bs.de> wrote:
See Definition 3.8 in Herrlich & Strecker: Category Theory (42 years old...).
Cheers Jiri
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxx
On Thu, 5 Mar 2015, Uwe Egbert Wolter wrote:
Some years ago (around 30?) I read a book where it was mentioned that
one could define categories without (explicit) objects in the sense that objects are mimicked by identity morphisms. Unfortunately, I can not reconstruct what book it was.
I know how this works. I would, however, like to have a reference.
Best
Uwe Wolter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]