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 $$(\mathcal C,\otimes,I_{\mathcal C})$$ 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
 * $$\phi_{A,B}:FA\bullet FB\to F(A\otimes B)$$

between functors $$\mathcal{C}\times\mathcal{C}\to\mathcal{D}$$ 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
 * [[Image:Lax_monoidal_functor_associative.svg|332px]],


 * [[Image:Lax_monoidal_functor_right_unit.svg|225px]]   and    [[Image:Lax_monoidal_functor_left_unit.svg|225px]]

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 $$\mathcal C$$ :


 * [[Image:Lax_monoidal_functor_braided.svg|225px]]


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

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 $$(Ch(R\mathsf{-mod}),\otimes,R[0]) \to (grR\mathsf{-mod},\otimes,R[0])$$ 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 $$(\mathcal C,\otimes,I_{\mathcal C})$$ 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(A ⇒ B) → FA ⇒ FB

of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:
 * [[Image:Applicative Form of Monoidal Coherence Map 01.svg|Commutative diagram demonstrating how a monoidal coherence map gives rise to its applicative formulation]]
 * [[Image:Monoidal Coherence Map from Applicative Formulation.svg|Commutative diagram demonstrating how a monoidal coherence map can be recovered from its applicative formulation]]

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 $$D$$.

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.