Monoidal Categories Enriched in Braided Monoidal Categories

Abstract

A. We introduce the notion of a monoidal category enriched in a braided monoidal category V. We set up the basic theory, and prove a classi cation result in terms of braided oplax monoidal functors to the Drinfeld center of some monoidal category 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 classi cation is slightly more general than what one might have anticipated in terms of strong monoidal functors V → Z (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 (T ) for some nite group G and a monoidal category T , and produces a new monoidal category T //G . In our setting, given any braided oplax monoidal functor V → Z (T ), for any braided V, we produce T //V : this is not usually an 'honest' monoidal category, but is instead V-enriched. If 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.

One dimensional gapped quantum phases and enriched fusion categories

“…This produces a grand unification of all gapped/gapless liquid phases with/without onsite symmetries (including symmetry-breaking phases as we will show in this work). In particular, the results in Theorem 1.1 can be reinterpreted by "rotating" the n+1d bulk in Figure 1 to the time direction and reinterpreting the pair (S, φ) as an Z 1 (S)-enriched fusion n-category Z 1 (S) S determined by the braided equivalence φ : Z 1 (R) → Z 1 (S) [MP19,KZ18]. In this process, the n+1d bulk excitations in Z 1 (S) before the rotation are replaced by the topological sectors of symmetric non-local operators in the n+1D spacetime after the rotation.…”

“…Moreover, φ provides a canonical construction of a B-enriched fusion category B S [MP19]. It is natural to expect that S B S as B-enriched fusion categories 4 .…”

Classification of Super-Modular Categories by Rank

“…A particularly useful technique is to formally "condense the fermion" to obtain a fermionic quotient, which has naive fusion rules. These can be studied using the concept of a sVec-enriched fusion category [45,35], but we will not pursue that here. In this article we make some partial progress towards the classification of rank 8, using a stratification by Galois group and some new techniques.…”

Drinfeld center of enriched monoidal categories

“…Suppose B is rigid. Then B can be canonically promoted to a monoidal category B ♯ enriched over B [MP,Section 2.3]. In fact, one needs to promote ⊗ : B × B → B to a well-defined enriched functor.…”