Monoidal functor

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

A lax 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 two natural transformations, called coherence maps (or mediating maps),

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

and

$\phi:I\to FI$

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 symmetric. The lax monoidal functor $F$ is symmetric when the diagram

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

Colax (or oplax) monoidal functors are defined similarly, with the direction of the coherence 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.

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

[edit] Properties

[edit] Lax monoidal functors and adjunctions

Suppose that a functor $F:\mathcal C\to\mathcal D$ is left adjoint to a lax monoidal $(G,n):(\mathcal D,\bullet,I_{\mathcal D})\to(\mathcal C,\otimes,I_{\mathcal C})$ . Then $F$ has a colax monoidal 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 this induced colax 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 lax monoidal adjunction; conversely, the left adjoint of a lax monoidal adjunction is always a strong monoidal functor.

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