Category without objects
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/ ]
[Note from moderator: Many responses to this; will forward the first mentioning each source. Thanks to other posters. ] On 5 March 2015 at 11:49, Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no> wrote:
Some years ago (around 30?)
25
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.
Peter J Freyd and Andre Scedrov, "Categories, Allegories" (Elsevier, 1990) ISBN 0 444 70368 3 is one such. Best wishes, Andy Pitts [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The definition can be found in several well-known textbooks: for example, on page 5 of Freyd's "Abelian Categories" and on page 9 of Mac lane's "Categories for the Working Mathematician". Peter Johnstone 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/ ]
"Categories, Allegories" by Freyd and Scedrov defines categories that way, on p.3. -- Peter 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/ ]
See Definition 3.8 in Herrlich & Strecker: Category Theory (42 years old...). Cheers Jiri xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 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/ ]
Much older, 69-70 years old ! In the original paper: S. Eilenberg y S. Mac Lane, General Theory of Natural Equivalences, Trans. Amer. Math. Soc. 58 (1945), 231?294. They already considered the notion of category without objects, but choose to use objects. They wrote: "It is thus clear that the objects play a secondary role, and could be entirely omitted from the definition of a category. However, the manipulation of the applications would be slightly less convenient were this done. [p?g. 238] On spite of this, Ehresmann was a fan of categories without objects: Erhesman adopted and pushed forward the notion of categories without objects, writing many papers and a book where categories did not have objects. C. Ehresmann, Cat?gories topologiques et cat?gories diff?rentiables, Colloque G?om. Diff. Globale (Bruxelles, 1958), Centre Belgue Rech. Math., Louvain, 1959, 137?150. Eduardo Dubuc On 05/03/15 13:49, Jiri Adamek wrote:
See Definition 3.8 in Herrlich & Strecker: Category Theory (42 years old...).
Cheers Jiri
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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/ ]
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/ ]
In fact this point is observed even in the original paper of Eilenberg Mac Lane ! Furthermore, I recently proposed a notion of autocategory, a kind of category without object ; this is a little different from categories, and is based precisely on the distinction between four things : objects, identities, identifiers, domains or codomains (see in CTGD in 2014), and this allows to see with the same "eye" categories, 2-categories, etc.. Best, René Guitart Le 5 mars 2015 à 12:49, Uwe Egbert Wolter a écrit :
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/ ]
participants (8)
-
Andrew Pitts -
Eduardo J. Dubuc -
Jiri Adamek -
Peter Johnstone -
Peter LeFanu Lumsdaine -
René Guitart -
selinger -
Uwe Egbert Wolter