Monoidal functor

In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.


Let (\mathcal C,\otimes,I_{\mathcal C}) and (\mathcal D,\bullet,I_{\mathcal D}) be monoidal categories. A monoidal functor from \mathcal C to \mathcal D consists of a functor F:\mathcal C\to\mathcal D together with a natural isomorphism

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

between \mathcal{C}\times\mathcal{C}\to\mathcal{D} functors and a morphism

\phi:I_{\mathcal D}\to FI_{\mathcal 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


commute in the category \mathcal D. Above, the various natural transformations denoted using \alpha, \rho, \lambda are parts of the monoidal structure on \mathcal C and \mathcal D.




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


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.

See also


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