Who Introduced the Term “Categorical Group”?
Posted by John Baez
I’m writing a paper in honor of Hoàng Xuân Sính’s 90th birthday, and I’m running into a lot of questions.
The term “categorical group” is often used to mean a group object in Cat; these days we also call such a thing a strict 2-group. Who first introduced the term “categorical group”, and when? Perhaps it appeared in the French literature under some name like “groupe catégorique”?
Here are some things I know, which don’t answer my question, but might provide clues.
In 1973 Hoàng Xuân Sính finished her thesis with Grothendieck on a more general concept, which she called “Gr-catégories”; these are now also called 2-groups. In 1978 she published this paper:
- Hoàng Xuân Sính, Gr-catégories strictes, Acta Mathematica Vietnamica 3 (2) (1978), 47–59.
Strict Gr-categories are the same as categorical groups, but I don’t think she uses a term like “groupe catégorique” anywhere in this paper. In this paper she proves (among other things) that every Gr-category is equivalent to a strict one, and that you can get strict ones from crossed modules.
Earlier, Brown and Spencer wrote a related paper:
- Ronald Brown and C. B. Spencer, -groupoids, crossed modules, and the classifying space of a topological group, Proc. Kon. Akad. v. Wet. 79 (1976), 296-302.
-groupoids are the same as categorical groups, but they don’t use the term “categorical group” anywhere. In this paper they prove (among other things) that you can get -groupoids from crossed modules and vice versa. They write:
This result was, we understand, known to Verdier in 1965; it was then used by Duskin [6] ; it was discovered independently by us in 1972. The work of Verdier and Duskin is unpublished, we have found that Theorem 1 is little known, and so we hope that this account will prove useful.
Reference [6] is to an unpublished paper which I have not located:
- J. Duskin, Preliminary remarks on groups. Unpublished notes, Tulane University, 1969.
The first actual use of the term “categorical group” that I’ve found comes considerably later:
- A. Solian, Coherence in categorical groups, Comm. Algebra 9 (1981), 1039-1057.
However, this paper actually deals with with 2-groups, not the strict 2-groups = strict Gr-categories that are nowadays often meant by the term “categorical group”.
Also in 1981 there is a paper in German that uses the term “Kategorie mit Gruppenstruktur”:
- K.-H. Ulbrich, Kohärenz in Kategorien mit Gruppenstruktur, J. Algebra, 72 (1981), pp. 279-295.
but again this is about 2-groups in general, not the strict ones, and I don’t see the term “categorical group” (or some German equivalent) anywhere in this paper.
Re: Who Introduced the Term “Categorical Group”?
I’ve always assumed the “gr” in “gr-category” was “group”. Is it?