2017/01/03 by Scott Morrison, Morrison, Scott, David Penneys +1 · 1 citation
Mathematics · #18D10 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1701.00567
openalex publication_date 2017/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 → 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 Rep(G) → 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 → 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 Vec, we can use this to reduce the enrichment to Vec, and this recovers de-equivariantization as a special case.