### 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, $\mathcal{G}$-groupoids, crossed modules, and the classifying space of a topological group,
*Proc. Kon. Akad. v. Wet.***79**(1976), 296-302.

$\mathcal{G}$-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 $\mathcal{G}$-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 The work of Verdier and Duskin is unpublished, we have found that Theorem 1 is little known, and so we hope that this account will prove useful.

Reference [6] is to an unpublished paper which I have not located: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 Unpublished notes, Tulane University, 1969.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?