nLab iterated monoidal category

Idea

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

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

Definition

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

• $0$-cells are pseudomonoids in $C$.

• $1$-cells are normal? lax morphisms of pseudomonoids.

• $2$-cells are monoidal transformations between normal lax morphisms of pseudomonoids.

The $2$-product structure on $Mon(C)$ is inherited from the $2$-product structure of $C$.

Definition

A $0$-fold monoidal category is an ordinary category; the $2$-category of $0$-fold monoidal categories is Cat. By recursion, the $2$-category of $n$-fold monoidal categories is $Mon_{norm}(C)$ where $C$ is the $2$-category of $(n-1)$-fold monoidal categories; objects of $Mon_{norm}(C)$ are of course called $n$-fold monoidal categories.

Related pages

• 2-fold monoidal categories are a special sort of duoidal category.

• distributivity for monoidal structures

References

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