[Paper Review] Monoidal categories enriched in braided monoidal categories

Scott Morrison, David Penneys|arXiv (Cornell University)|Jan 3, 2017

Algebraic structures and combinatorial models4 references4 citations

TL;DR

This paper introduces monoidal categories enriched in braided monoidal categories, generalizing de-equivariantization. It establishes a bijective correspondence between rigid V-monoidal categories with adjointable unit functors and braided oplax monoidal functors from V to the Drinfeld center of a rigid monoidal category T, revealing that non-complete enrichments exist beyond strong functors.

ABSTRACT

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category $\mathcal T$. Even the basic theory is interesting; it shares many characteristics with the theory of monoidal categories enriched in a symmetric monoidal category, but lacks some features. Of particular note, there is no cartesian product of braided-enriched categories, and the natural transformations do not form a 2-category, but rather satisfy a braided interchange relation. Strikingly, our classification is slightly more general than what one might have anticipated in terms of strong monoidal functors $\mathcal V o Z(\mathcal T)$. We would like to understand this further; in a future paper we show that the functor is strong if and only if the enriched category is `complete' in a certain sense. Nevertheless it remains to understand what non-complete enriched categories may look like. One should think of our construction as a generalization of de-equivariantization, which takes a strong monoidal functor $\mathsf{Rep}(G) o Z(\mathcal T)$ for some finite group $G$ and a monoidal category $\mathcal T$, and produces a new monoidal category $\mathcal T // G$. In our setting, given any braided oplax monoidal functor $\mathcal V o Z(\mathcal T)$, for any braided $\mathcal V$, we produce $\mathcal T // \mathcal V$: this is not usually an `honest' monoidal category, but is instead $\mathcal V$-enriched. If $\mathcal V$ has a braided lax monoidal functor to $\mathsf{Vec}$, we can use this to reduce the enrichment to $\mathsf{Vec}$, and this recovers de-equivariantization as a special case.

Motivation & Objective

• To formalize the theory of monoidal categories enriched in braided monoidal categories, extending classical enrichment in symmetric or symmetric monoidal categories.
• To address the lack of a cartesian product and 2-categorical structure in braided enrichment, which instead satisfies a braided interchange relation.
• To generalize de-equivariantization by constructing T//V for any braided oplax monoidal functor V → Z(T), not just strong ones.
• To clarify the role of completeness in enrichment, showing that strong functors correspond precisely to complete enriched categories.
• To provide a classification framework for V-monoidal categories using the Drinfeld center construction, enabling new constructions in tensor categories and topological quantum field theory.

Proposed method

• Introduce the notion of a strict V-monoidal category, where composition and tensor product satisfy a braided interchange relation expressed via string diagrams.
• Define V-monoidal functors and establish their naturality via separate conditions on each factor, using the interchange relation and associativity in the enriching category V.
• Use the Drinfeld center Z(T) of a rigid monoidal category T as the target for braided oplax monoidal functors FZ: V → Z(T), with F = FZ ◦ R admitting a right adjoint.
• Leverage adjunctions and mates in the Drinfeld center to derive the mate of the composition map, using Frobenius reciprocity and the zig-zag relation.
• Prove that the correspondence in Theorem 1.1 is bijective by constructing inverse constructions: from a V-monoidal category C, extract CV via V(1 → C(a→b)), and from a functor FZ, construct C via a universal property.
• Use string diagram calculus and coherence theorems to verify the braided interchange and naturality conditions, ensuring consistency in the enriched monoidal structure.

Experimental results

Research questions

• RQ1 How can monoidal categories be naturally enriched in a braided monoidal category rather than a symmetric one, and what structural features emerge?
• RQ2 What is the precise role of the Drinfeld center in classifying V-monoidal categories when V is braided but not symmetric?
• RQ3 Why does the natural transformation structure in V-monoidal categories satisfy a braided interchange relation instead of the usual 2-categorical interchange?
• RQ4 What is the significance of completeness in V-enriched categories, and how does it relate to the strength of the functor FZ: V → Z(T)?
• RQ5 Can de-equivariantization be generalized beyond strong monoidal functors, and what new types of monoidal categories arise from oplax functors?

Key findings

• A bijective correspondence is established between rigid V-monoidal categories with adjointable unit functors and braided oplax monoidal functors FZ: V → Z(T), where F = FZ ◦ R admits a right adjoint.
• The classification is strictly more general than strong functors: non-complete enriched categories exist, and they correspond precisely to oplax functors that are not strong.
• The natural transformations in V-monoidal categories satisfy a braided interchange relation, not the usual 2-categorical one, due to the non-symmetric braiding in the enriching category.
• The construction generalizes de-equivariantization: for any braided oplax FZ: V → Z(T), one obtains a V-monoidal category T//V, which reduces to de-equivariantization when V → Vec is lax.
• The mate of the composition map in the Drinfeld center is computed explicitly as (jaηv) ◦ (−⊗C −), using adjunctions and the interchange relation.
• The theory reveals that the category of V-monoidal categories does not admit a cartesian product and lacks a 2-category structure, due to the braided nature of the enrichment.