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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