nLab
n-fold category

Definition of strict case

A $0$-fold category is a set. An $n$-fold category for $n>0$ is an internal category in $(n-1)$-fold categories. An $n$-fold category is also known as an $n$-tuple category.

In particular, a $1$-fold category is preciesely a category, and a $2$-fold category is precisely a double category (introduced by Charles Ehresmann in 1963).

Analogously, a $0$-fold groupoid is again a set, and an $n$-fold groupoid is an internal groupoid in $(n-1)$-fold groupoids; in particular, a $1$-fold groupoid is a groupoid.

More generally, an $(n,r)$-fold groupoid is an $r$-fold category in $(n-r)$-fold groupoids; compare $(n,r)$-category.

Remarks

• Notice that an $n$-fold category is a strict version of an n-category in that all $n$ composition operations are strictly unital and associative and strictly commute with each other. Still, $n$-fold groupoids model all homotopy n-types. See homotopy hypothesis.

• Analogous to how a group is a groupoid with a single object, one can consider $(n+1)$-fold groupoids for which all morphisms in one of the $(n+1)$ directions are endomorphisms. These are the cat-n-groups.

• By a theorem by Brown and Higgins, strict omega-categories are equivalent to those $\mathrm{\infty }$-fold categories that satisfy a couple of restrictive properties (something like that all 1-categories of $n$-cells for all $n$ are the same and that all the “thin” identity elements exist, called “connections”).

• There are some moves towards a weak notion of $n$-fold category, particularly the case of double bicategory.

References

• Thomas M. Fiore, Simona Paoli, A Thomason Model Structure on the Category of Small $n$-fold Categories (arXiv)

• Ronnie Brown and P.J. Higgins, The equivalence of $\mathrm{\infty }$-groupoids and crossed complexes, Cah. Top. G'eom. Diff. 22 (1981) 371–386.