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

Module Monoidal Categories as Categorification of Associative Algebras