Monoidal functor

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Monoidal functors: a simple guide

- A monoidal category is a category equipped with a tensor product ⊗ and a unit object I, together with rules that describe how tensoring behaves (associativity and unit laws).

- A monoidal functor F: C → D is a functor that respects this structure. It comes with:
- a way to map F(A) ⊗ F(B) into F(A ⊗ B) for every A, B in C, and
- a way to relate the unit I_D to F(I_C).

These are given by natural maps that must satisfy coherence conditions, ensuring that F preserves the monoidal structure up to these maps.

- Different levels of strictness:
- lax monoidal functor: the structure maps exist as specified (not necessarily invertible).
- strong monoidal functor: the structure maps are invertible (isomorphisms).
- Sometimes the term “monoidal functor” is used to refer to either case.

- If the ambient categories C and D are closed monoidal (they have internal homs), there is another common way to describe the structure, using the adjoint relationships between tensor and internal homs.

- Adjunctions and monoidal structure:
- If F: C → D is left adjoint to G: D → C and G is monoidal, then F gets a compatible comonoidal structure. If this induced structure is strong, the adjunction is a monoidal adjunction, and F is a strong monoidal functor.
- Conversely, the right adjoint of a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is strong monoidal.

This gives the main idea: monoidal functors link the tensor and unit of one category to those of another, with varying levels of strictness and important connections to adjunctions.