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

• The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible.
• The coherence maps of strong monoidal functors are invertible.
• The coherence maps of strict monoidal functors are identity maps.

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

Definition

Let $(𝒞,\otimes ,I_{𝒞})$ and $(𝒟,\bullet ,I_{𝒟})$ be monoidal categories. A lax monoidal functor from $𝒞$ to $𝒟$ (which may also just be called a monoidal functor) consists of a functor $F:𝒞\to 𝒟$ together with a natural transformation

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

between functors $𝒞\times 𝒞\to 𝒟$ and a morphism

$\varphi :I_{𝒟}\to F{I}_{𝒞}$,

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

,

   and   

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

Variants

• The dual of a monoidal functor is a comonoidal functor; it is a monoidal functor whose coherence maps are reversed. Comonoidal functors may also be called opmonoidal, colax monoidal, or oplax monoidal functors.
• A strong monoidal functor is a monoidal functor whose coherence maps ${\varphi }_{A,B},\varphi$ are invertible.
• A strict monoidal functor is a monoidal functor whose coherence maps are identities.
• A braided monoidal functor is a monoidal functor between braided monoidal categories (with braidings denoted $\gamma$) such that the following diagram commutes for every pair of objects A, B in $𝒞$ :

• A symmetric monoidal functor is a braided monoidal functor whose domain and codomain are symmetric monoidal categories.

Examples

• The underlying functor $U:(𝐀𝐛,{\otimes }_{𝐙},𝐙)\to (𝐒𝐞𝐭,\times ,\{\ast \})$ from the category of abelian groups to the category of sets. In this case, the map ${\varphi }_{A,B}:U(A)\times U(B)\to U(A\otimes B)$ sends (a, b) to $a\otimes b$; the map $\varphi :\{*\}\to \mathbb{Z}$ sends $\ast$ to 1.
• If $R$ is a (commutative) ring, then the free functor $𝖲𝖾𝗍,\to R\mathsf{-}𝗆𝗈𝖽$ extends to a strongly monoidal functor $(𝖲𝖾𝗍,\bigsqcup ,\mathrm{\varnothing })\to (R\mathsf{-}𝗆𝗈𝖽,\oplus ,0)$ (and also $(𝖲𝖾𝗍,\times ,\{\ast \})\to (R\mathsf{-}𝗆𝗈𝖽,\otimes ,R)$ if $R$ is commutative).
• If $R\to S$ is a homomorphism of commutative rings, then the restriction functor $(S\mathsf{-}𝗆𝗈𝖽,{\otimes }_S,S)\to (R\mathsf{-}𝗆𝗈𝖽,{\otimes }_R,R)$ is monoidal and the induction functor $(R\mathsf{-}𝗆𝗈𝖽,{\otimes }_R,R)\to (S\mathsf{-}𝗆𝗈𝖽,{\otimes }_S,S)$ is strongly monoidal.
• An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory. Let ${𝐁𝐨𝐫𝐝}_{⟨n-1,n⟩}$ be the category of cobordisms of n-1,n-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension n is a symmetric monoidal functor $F:({𝐁𝐨𝐫𝐝}_{⟨n-1,n⟩},\bigsqcup ,\mathrm{\varnothing })\to (𝐤𝐕𝐞𝐜𝐭,{\otimes }_k,k).$
• The homology functor is monoidal as $(Ch(R\mathsf{-}𝗆𝗈𝖽),\otimes ,R[0])\to (grR\mathsf{-}𝗆𝗈𝖽,\otimes ,R[0])$ via the map $H_{\ast }(C_1)\otimes H_{\ast }(C_2)\to H_{\ast }(C_1\otimes C_2),[x_1]\otimes [x_2]\mapsto [x_1\otimes x_2]$.

Alternate notions

If $(𝒞,\otimes ,I_{𝒞})$ and $(𝒟,\bullet ,I_{𝒟})$ are closed monoidal categories with internal hom-functors ${\Rightarrow }_{𝒞},{\Rightarrow }_{𝒟}$ (we drop the subscripts for readability), there is an alternative formulation

ψ_AB : F(A ⇒ B) → FA ⇒ FB

of φ_AB commonly used in functional programming. The relation between ψ_AB and φ_AB is illustrated in the following commutative diagrams:

Properties

• If $(M,\mu ,ϵ)$ is a monoid object in $C$, then $(FM,F\mu \circ {\varphi }_{M,M},Fϵ\circ \varphi )$ is a monoid object in $D$.

Monoidal functors and adjunctions

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

$m_{A,B}={\epsilon }_{FA\bullet FB}\circ F{n}_{FA,FB}\circ F({\eta }_A\otimes {\eta }_B):F(A\otimes B)\to FA\bullet FB$

and

$m={\epsilon }_{I_{𝒟}}\circ Fn:F{I}_{𝒞}\to I_{𝒟}$.

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

• Monoidal natural transformation