An iterated monoidal category, or -fold monoidal category for varying , is an algebraic analogue of the concept of n-fold loop space. By means of a suitable bar construction, the geometric realization of an -fold monoidal category, or rather its group completion, bears a structure of -fold loop space.
Roughly speaking, the iterative idea is that an -fold monoidal category is a (pseudo-)monoid in the monoidal -category of -fold monoidal categories and (normal lax?) -fold monoidal functors. Were the laxity to be strengthened so that the relevant structure constraints become isomorphisms (strong -fold monoidal functors), we would get braided monoidal categories in the case , and symmetric monoidal categories at and beyond (in other words, the concept stabilizes at ). Without that strengthening, however, we get a new type of structure for each , without stabilization.
Let be a 2-category with 2-product?s. Form a new 2-category with 2-products whose
-cells are (pseudo)monoids in .
-cells are normal? lax homomorphisms of monoids.
-cells are lax transformations between normal lax homomorphisms.
The -product structure on is inherited from the -product structure of .
A -fold monoidal category is an ordinary category; the -category of -fold monoidal categories is Cat. By recursion, the -category of -fold monoidal categories is where is the -category of -fold monoidal categories; objects of are of course called -fold monoidal categories.