Monoidal functor

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

A monoidal functor Failed to parse (Missing texvc executable; please see math/README to configure.): F

between two monoidal categories Failed to parse (Missing texvc executable; please see math/README to configure.): (\mathcal C,\otimes,I_{\mathcal C})
and Failed to parse (Missing texvc executable; please see math/README to configure.): (\mathcal D,\bullet,I_{\mathcal D})
consists of a functor Failed to parse (Missing texvc executable; please see math/README to configure.): F:\mathcal C\to\mathcal D
together with a natural transformation

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi_{A,B}:FA\bullet FB\to F(A\otimes B)

and a morphism

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi:I_D\to FI_C

, called the structure morphisms, which are such that for every three objects Failed to parse (Missing texvc executable; please see math/README to configure.): A , Failed to parse (Missing texvc executable; please see math/README to configure.): B

and Failed to parse (Missing texvc executable; please see math/README to configure.): C
of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal C
the diagrams

Image:Lax monoidal funct assoc.png,

Image:Lax monoidal funct right unit.png and Image:Lax monoidal funct left unit.png

commute in the category Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal D .

Suppose that the monoidal categories Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal C

and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal D
are braided. The monoidal functor Failed to parse (Missing texvc executable; please see math/README to configure.): F
is braided when the diagram

Image:Lax monoidal funct sym.png

commutes for every objects Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): B
of Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): U:(\mathbf{Ab},\otimes_\mathbf{Z},\mathbf{Z}) \rightarrow (\mathbf{Set},\times,\{*\})

from the category of abelian groups to the category of sets.

Properties

Monoidal functors and adjunctions

Suppose that a functor Failed to parse (Missing texvc executable; please see math/README to configure.): F:\mathcal C\to\mathcal D

is left adjoint to a monoidal Failed to parse (Missing texvc executable; please see math/README to configure.): (G,n):(\mathcal D,\bullet,I_{\mathcal D})\to(\mathcal C,\otimes,I_{\mathcal C})

. Then Failed to parse (Missing texvc executable; please see math/README to configure.): F

has a comonoidal structure Failed to parse (Missing texvc executable; please see math/README to configure.): (F,m)
induced by Failed to parse (Missing texvc executable; please see math/README to configure.): (G,n)

, defined by

Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): m=\varepsilon_{I_{\mathcal D}}\circ Fn:FI_{\mathcal C}\to I_{\mathcal D}

.

If the induced structure on Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

See also

• Monoidal natural transformation

References

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