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

$\phi _{A,B}:FA\bullet FB\to F(A\otimes B)$ from functors ${\mathcal {C))\times {\mathcal {C))\to {\mathcal {D))$ to $F$ 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 , and 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 ${\mathcal {D))$ .

### 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 $\phi _{A,B},\phi$ 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 ${\mathcal {C))$ : ## Examples

• The underlying functor $U\colon (\mathbf {Ab} ,\otimes _{\mathbf {Z} },\mathbf {Z} )\rightarrow (\mathbf {Set} ,\times ,\{\ast \})$ from the category of abelian groups to the category of sets. In this case, the map $\phi _{A,B}\colon U(A)\times U(B)\to U(A\otimes B)$ sends (a, b) to $a\otimes b$ ; the map $\phi \colon \{*\}\to \mathbb {Z}$ sends $\ast$ to 1.
• If $R$ is a (commutative) ring, then the free functor ${\mathsf {Set)),\to R{\mathsf {-mod))$ extends to a strongly monoidal functor $({\mathsf {Set)),\sqcup ,\emptyset )\to (R{\mathsf {-mod)),\oplus ,0)$ (and also $({\mathsf {Set)),\times ,\{\ast \})\to (R{\mathsf {-mod)),\otimes ,R)$ 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, which has been recently developed. Let $\mathbf {Bord} _{\langle n-1,n\rangle )$ 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 $F\colon (\mathbf {Bord} _{\langle n-1,n\rangle },\sqcup ,\emptyset )\rightarrow (\mathbf {kVect} ,\otimes _{k},k).$ • The homology functor is monoidal as $(Ch(R{\mathsf {-mod))),\otimes ,R)\to (grR{\mathsf {-mod)),\otimes ,R)$ 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 $({\mathcal {C)),\otimes ,I_{\mathcal {C)))$ and $({\mathcal {D)),\bullet ,I_{\mathcal {D)))$ are closed monoidal categories with internal hom-functors $\Rightarrow _{\mathcal {C)),\Rightarrow _{\mathcal {D))$ (we drop the subscripts for readability), there is an alternative formulation

ψAB : F(AB) → FAFB

of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:  ## Properties

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

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.