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• CommentRowNumber1.
• CommentAuthorjesuslop
• CommentTimeAug 18th 2018
• (edited Aug 18th 2018)

Hi again,

In Module category, it says:

Let $\mathcal{M}$ be a monoidal category and $B\mathcal{M}$ its delooping as a bicategory. A (left) module category is then simply a 2-functor $B\mathcal{M} \to \mathcal{Cat}$.

Further expanding this definition, we have the following data:

• A category $\mathcal{C}$
• A functor $- \triangleright -\colon \mathcal{C} \times \mathcal{M} \to \mathcal{M}$
• A natural isomorphism $\alpha_{A,B,X}\colon A \triangleright (B \triangleright X) \to (A \otimes B) \triangleright X$ satisfying a pentagon axiom involving the associator of $\mathcal{M}$

Shouldn’t item 2 be “A functor $- \triangleright -\colon \mathcal{M} \times \mathcal{C} \to \mathcal{C}$ “?

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeAug 18th 2018

Yes, fixed now.