category object in an (∞,1)-category, groupoid object
The notion of $n$-fold category is what is obtained by iterating the process of forming internal categories $n$-times, starting with sets: an $0$-fold category is just an object of the ambient category (say a set) and then inductively an $n+1$-fold category is a internal category in the category of $n$-fold categories.
If the ambient category is instead an (∞,1)-category such as ∞Grpd, then an $n$-fold category is an n-fold complete Segal space, see there for more details.
DamienC?: unless I miss a point, the above sentence doesn’t seem to be correct. For instance, for n=2, a 2-fold complete Segal space looks rather like a bicategory than a double category.
A $0$-fold category is a set. A (strict) $n$-fold category for $n \gt 0$ is an internal category in the category of $(n-1)$-fold categories. An $n$-fold category is also known as an $n$-tuple category.
In particular, a $1$-fold category is precisely a category, and a $2$-fold category is precisely a double category (introduced by Charles Ehresmann in 1963).
A key advantage of $n$-fold categories is the ease of expressing multiple compositions, and so the idea of “algebraic inverse to subdivision”. This is important because subdivision is a key tool in many local-to-global problems in mathematics and science, and these themselves are an important class of problems.
Thus subdividing an $n$-cube amounts to dividing the cube into small cubes by hyperplanes parallel to the faces. For a $2$-fold category, we can define a composable array $(a_{ij})$ of $2$-dimensional elements (called squares) to be such that any $a_{ij}$ is composable with its immediate neighbours. In such case, the associative and interchange laws imply that the composition $[a_{ij}]$ is well defined. This process is easily extended to $n$-fold categories, using elements say $a_{(r)}$ where $(r)$ is multi-index, and is applied widely in the JPAA papers by Brown and Higgins listed below. It seems much more difficult to express these ideas in the globular or simplicial contexts.
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.
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.
More generally, an $(n,r)$-fold groupoid is an $r$-fold category in $(n-r)$-fold groupoids; compare $(n,r)$-category.
Note also that a category object in the category of groups is actually a groupoid object.
The category of $n$-fold categories is a cartesian closed category. By induction from the statement at Internal category - Cartesian closure.
Even though 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.
By a theorem by Al-Agl, Brown and Steiner, strict omega-categories are equivalent to those $\infty$-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”): these are the “cubical $\omega$-categories with connections”. Because it is relatively straightforward to define a monoidal closed category in the cubical theory, using the formula $I^m \otimes I^n = I^{m+n}$, this leads to a monoidal closed structure for strict globular $\omega$ categories.
This is a category version of a corresponding groupoid theorem of Brown and Higgins which follows from the two papers listed below.
Jean-Louis Loday introduced in the paper listed below the category of what he called $n$-cat groups, but are now called cat$^n$-groups as they are exactly $n$-fold groupoids internal to the category of groups. He showed that these objects model weak, pointed homotopy $n$-types, see homotopy hypothesis. The paper by Brown and Loday shows that these structures can be used, via a van Kampen type theorem, for explicit computations in homotopy theory, and this is further developed in the paper by Ellis and Steiner. This paper relates relates the theory to that of crossed modules, $n$-ad homotopy groups, and the important $n$-ad connectivity theorem, which is related to results on homotopical excision.
Ronnie Brown and P.J. Higgins, The equivalence of $\infty$-groupoids and crossed complexes, Cah. Top. G'eom. Diff. 22 (1981) 371–386.
Ronnie Brown and P.J. Higgins. On the algebra of cubes, J. Pure Appl. Algebra, 21 (1981) 233-260.
G.J. Ellis, and R.J. Steiner. Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads, J. Pure Appl. Algebra, 46 {1987} 117–136.
J.[L. Loday. Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra, 24 (1982) 179202.]
Ronnie Brown and J.-L. Loday. Van Kampen theorems for diagrams of spaces. Topology 26 (1987) 311–335. With an appendix by M. Zisman.
Ronnie Brown, and J.-L. Loday. Homotopical excision, and Hurewicz theorems for $n$-cubes of spaces. Proc. London Math. Soc. (3) 54 (1987) 176–192.
F.A. Al-Agl, Ronnie Brown and R.J. Steiner,Multiple categories: the equivalence between a globular and cubical approach, Advances in Mathematics, 170 (2002) 71–118.
S. Paoli. Internal categorical structures in homotopical algebra. In Towards higher categories, 85–103, IMA Vol. Math. Appl., 152, Springer, New York, 2010.
T. M. Fiore and S. Paoli. A Thomason model structure on the category of small $n$-fold categories. Algebr. Geom. Topol. 10 (2010) 1933–-2008.