Re: Category without objects
Many thanks for all the immediate replies and all the interesting information. Finally, I could also reconstruct today where I have seen the arrows-only definition around 30 years ago. There is a four page introduction into categories in the first chapter of P.M. Cohn's "Universal Algebra". He outlines that one could do so and gives a corresponding exercise. Best Uwe On 2015-03-06 00:45, Peter LeFanu Lumsdaine wrote:
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.
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I remember Henry Whitehead said that he was very impressed by the axioms for a category in the Eilenberg-Mac Lane paper. A curiosity about the definition is that groupoids were defined by Brandy in 1926, and this definition was used by the Chicago school of algebra and applied to ring theory. Bill Cockcroft told me that the groupoid notion was an influence. In 1985 I asked Eilenberg about this, and said no, since if it had been, they would have used it as an example! I forgot to ask Mac Lane! Ronnie Brown On 06/03/2015 14:42, Uwe Egbert Wolter wrote:
Many thanks for all the immediate replies and all the interesting information.
Finally, I could also reconstruct today where I have seen the arrows-only definition around 30 years ago. There is a four page introduction into categories in the first chapter of P.M. Cohn's "Universal Algebra". He outlines that one could do so and gives a corresponding exercise.
Best
Uwe
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It seems to me: Eheresmann arrived to the concept of categories as a generalization of groupoids, and he was dealing with small internal categories (the set of arrows (or elements) were differential manifolds etc). This explains why he dismissed objects in his later treatment of abstract categories. Eilemberg and MacLane arrived to the concept of categories as an abstraction of the large concrete categories of sets with structure and functions which were considered to be morphisms for the structure. Objects were essential in this approach. That the insight of E. M. to do not dismiss objects in the abstract setting was wonderful is that to-day we can not conceive groupoids without objects. On 7/3/15 11:36, Ronnie Brown wrote:
I remember Henry Whitehead said that he was very impressed by the axioms for a category in the Eilenberg-Mac Lane paper.
A curiosity about the definition is that groupoids were defined by Brandy in 1926, and this definition was used by the Chicago school of algebra and applied to ring theory. Bill Cockcroft told me that the groupoid notion was an influence. In 1985 I asked Eilenberg about this, and said no, since if it had been, they would have used it as an example! I forgot to ask Mac Lane!
Ronnie Brown
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It is difficult to understand "without objects" without any definition of "object". Remember that , already before the 21st century, modern mathematics had begun to overcome medieval metaphysics. In fact ,in the late 1950s, Alexander Grothendieck had made explicit the definition of "subobject", which seems relevant here, as does his powerful legacy of relativization in several senses. Now we understand that a category C in a category U is a truncated simplicial object C0->...->C3 satisfying certain limit conditions. We are free to call C0 'objects" and C1 "maps" and since C0->C1 is a subobject of C1, we could also say that objects "are" maps,but "mimicked by" seems unnecessary (as well as undefined). (Recall that it is actions of such a C in a topos U that form the topos enveloping, as a full subtopos of sheaves, the typical U-topos E->U). To give a category "with objects" in a serious sense would seem to be giving MORE than just a category, for example an interpretation as structuresC-> B^A, the (functor category also emphasized by Grothendieck)of structures of shape A in background B. (Where perhaps B is equipped with an internal embedding in U itself) The case of no structure and featureless background ( which seems to be the default setting of modern mathematics despite the preference of MacLane'sdear teacher for a vonNeuman-like setting) means in particular that the C0 in a category there consists of "lauter Einsen" in the sense of Cantor. Those featureless elements X of C0 do obtain a structure by virtue of C1,C2 because taking the latter into account we can see the inside of X as the "comma" category C/X involving (not only the subobjects of X and their inclusions, but also singular figures and reparameterizations) as very extensively utilized by Grothendieck . Bill
Date: Sat, 7 Mar 2015 14:36:36 +0000 From: ronnie.profbrown@btinternet.com To: Uwe.Wolter@ii.uib.no; p.l.lumsdaine@gmail.com CC: categories@mta.ca Subject: categories: Re: Category without objects
I remember Henry Whitehead said that he was very impressed by the axioms for a category in the Eilenberg-Mac Lane paper.
A curiosity about the definition is that groupoids were defined by Brandy in 1926, and this definition was used by the Chicago school of algebra and applied to ring theory. Bill Cockcroft told me that the groupoid notion was an influence. In 1985 I asked Eilenberg about this, and said no, since if it had been, they would have used it as an example! I forgot to ask Mac Lane!
Ronnie Brown
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Eduardo J. Dubuc -
F. William Lawvere -
Ronnie Brown -
Uwe Egbert Wolter