nForum - Discussion Feed (module category, little checking) 2022-08-15T14:25:51-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher David_Corfield comments on "module category, little checking" (70858) https://nforum.ncatlab.org/discussion/8860/?Focus=70858#Comment_70858 2018-08-18T07:47:18-04:00 2022-08-15T14:25:51-04:00 David_Corfield https://nforum.ncatlab.org/account/20/ Yes, fixed now.

Yes, fixed now.

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jesuslop comments on "module category, little checking" (70854) https://nforum.ncatlab.org/discussion/8860/?Focus=70854#Comment_70854 2018-08-18T06:45:33-04:00 2022-08-15T14:25:51-04:00 jesuslop https://nforum.ncatlab.org/account/1486/ Hi again, In Module category, it says: Let &Mscr;\mathcal{M} be a monoidal category and B&Mscr;B\mathcal{M} its delooping as a bicategory. A (left) module category is then simply a ...

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 $- \triangleright -\colon \mathcal{M} \times \mathcal{C} \to \mathcal{C}$ “?

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