Module Monoidal Categories as Categorification of Associative Algebras

In [arXiv:1509.02937], the notion of a module tensor category was introduced as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow \mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal category $\mathcal{T}$, which is a monoidal functor $F\colon \mathcal{V}\longrightarrow\mathcal{T}$ together with a braided monoidal lift $F^Z\colon \mathcal{V}\longrightarrow Z(\mathcal{T})$ to the Drinfeld center of $\mathcal{T}$. This is a categorification of a unital associative algebra $A$ over a commutative ring $R$ via a ring homomorphism $f\colon R\longrightarrow Z(A)$ into the center of $A$. In this paper, we want to categorify the characterization of an associative algebra as a (not necessarily unital) ring $A$ together with an $R$-module structure over a commutative ring $R$, such that multiplication in $A$ and action of $R$ on $A$ are compatible. In doing so, we introduce the more general notion of non-unital module monoidal categories and obtain 2-categories of non-unital and unital module monoidal categories, their functors and natural transformations. We will show that in the unital case the latter definition is equivalent to the definition in [arXiv:1509.02937] by explicitly writing down an equivalence of 2-categories.