Monoidal functor

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In category theory, monoidal functors are the "natural" notion of functor between two monoidal categories.

A monoidal functor $F$ between two monoidal categories $(\mathcal C,\otimes,I_{\mathcal C})$ and $(\mathcal D,\bullet,I_{\mathcal D})$ consists of a functor $F:\mathcal C\to\mathcal D$ together with a natural transformation

$\phi_{A,B}:FA\bullet FB\to F(A\otimes B)$

and a morphism

$\phi:I_D\to FI_C$ ,

called the structure morphisms, which are such that for every three objects $A$ , $B$ and $C$ of $\mathcal C$ the diagrams

and

commute in the category $\mathcal D$ .

Suppose that the monoidal categories $\mathcal C$ and $\mathcal D$ are braided. The monoidal functor $F$ is braided when the diagram

commutes for every objects $A$ and $B$ of $\mathcal C$ .

A braided monoidal functor between symmetric monoidal categories is called a symmetric monoidal functor.

Comonoidal (or opmonoidal) functors are defined similarly, with the direction of the structure maps reversed.

A strong monoidal functor is a monoidal functor whose coherence maps are invertible, and a strict monoidal functor is one whose coherence maps are identities.

An example of a monoidal functor is the underlying functor $U:(\mathbf{Ab},\otimes_\mathbf{Z},\mathbf{Z}) \rightarrow (\mathbf{Set},\times,\{*\})$ from the category of abelian groups to the category of sets.

1 Properties
- 1.1 Monoidal functors and adjunctions
2 See also
3 References

[edit] Properties

[edit] Monoidal functors and adjunctions

Suppose that a functor $F:\mathcal C\to\mathcal D$ is left adjoint to a monoidal $(G,n):(\mathcal D,\bullet,I_{\mathcal D})\to(\mathcal C,\otimes,I_{\mathcal C})$ . Then $F$ has a comonoidal structure $(F, m)$ induced by $(G, n)$ , defined by

$m_{A,B}=\varepsilon_{FA\bullet FB}\circ Fn_{FA,FB}\circ F(\eta_A\otimes \eta_B):F(A\otimes B)\to FA\bullet FB$

and

$m=\varepsilon_{I_{\mathcal D}}\circ Fn:FI_{\mathcal C}\to I_{\mathcal D}$ .

If the induced structure on $F$ is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint of a monoidal adjunction is always a strong monoidal functor.

Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.