nLab
iterated monoidal category

Iterated monoidal categories

Idea

An iterated monoidal category, or nn-fold monoidal category for varying nn, is an algebraic analogue of the concept of n-fold loop space. By means of a suitable bar construction, the geometric realization of an nn-fold monoidal category, or rather its group completion, bears a structure of nn-fold loop space.

Roughly speaking, the iterative idea is that an (n+1)(n+1)-fold monoidal category is a (pseudo-)monoid in the monoidal 22-category of nn-fold monoidal categories and (normal lax?) nn-fold monoidal functors. Were the laxity to be strengthened so that the relevant structure constraints become isomorphisms (strong nn-fold monoidal functors), we would get braided monoidal categories in the case n=2n = 2, and symmetric monoidal categories at n=3n = 3 and beyond (in other words, the concept stabilizes at n=3n = 3). Without that strengthening, however, we get a new type of structure for each nn, without stabilization.

Definition

Let CC be a 2-category with 2-products?. Form a new 2-category with 2-products Mon lax,norm(C)Mon_{lax,norm}(C) whose

The 22-product structure on Mon(C)Mon(C) is inherited from the 22-product structure of CC.

Definition

A 00-fold monoidal category is an ordinary category; the 22-category of 00-fold monoidal categories is Cat. By recursion, the 22-category of nn-fold monoidal categories is Mon norm(C)Mon_{norm}(C) where CC is the 22-category of (n1)(n-1)-fold monoidal categories; objects of Mon norm(C)Mon_{norm}(C) are of course called nn-fold monoidal categories.

References

  • Balteanu, Fiedorowicz, Schwänzl, and Vogt, Iterated monoidal categories, (pdf).

Revised on July 15, 2013 20:15:44 by Mike Shulman (107.201.30.59)