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In category theory, monoidal functors are functors between monoidal categories that "respect the monoidal structures". There are several different ways to interpret this, depending on whether we want the monoidal structure to be preserved strictly, or up to isomorphism, or up to a not-necessarily-invertible connecting map.

We start with the most relaxed version, in which the monoidal structure is only preserved up to a connecting map. Let $(\mathcal C,\otimes,I_{\mathcal C})$ and $(\mathcal D,\bullet,I_{\mathcal D})$ be monoidal categories. A monoidal functor, or lax monoidal functor, from $\mathcal C$ to $\mathcal D$ 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 coherence maps or structure morphisms, which are such that for every three objects $A$ , $B$ and $C$ of $\mathcal C$ the diagrams

and

commute in the category $\mathcal D$ .

A comonoidal functor, or opmonoidal functor, or colax monoidal functor, or oplax monoidal functor, is defined similarly but with the directions of the coherence maps reversed.

A strong monoidal functor (or in some usage, monoidal functor) is a (lax) monoidal functor whose coherence maps are invertible, and a strict monoidal functor is one whose coherence maps are identities.

An example of a (lax) 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.

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.

The rest of this article uses the terminology given first in the definitions above: monoidal functor, comonoidal functor, strong monoidal functor, etc.

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.

[edit] See also

Monoidal natural transformation

[edit] References

Kelly, G. Max (1974), "Doctrinal adjunction", Lecture Notes in Mathematics, 420, 257–280

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