mathlib documentation

category_theory.bicategory.single_obj

Promoting a monoidal category to a single object bicategory. #

A monoidal category can be thought of as a bicategory with a single object.

The objects of the monoidal category become the 1-morphisms, with composition given by tensor product, and the morphisms of the monoidal category become the 2-morphisms.

We verify that the endomorphisms of that single object recovers the original monoidal category.

One could go much further: the bicategory of monoidal categories (equipped with monoidal functors and monoidal natural transformations) is equivalent to the bicategory consisting of

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Promote a monoidal category to a bicategory with a single object. (The objects of the monoidal category become the 1-morphisms, with composition given by tensor product, and the morphisms of the monoidal category become the 2-morphisms.)

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The unique object in the bicategory obtained by "promoting" a monoidal category.

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The monoidal functor from the endomorphisms of the single object when we promote a monoidal category to a single object bicategory, to the original monoidal category.

We subsequently show this is an equivalence.

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