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

,

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 ${\mathcal {D}}$ .^[1]

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 ${\mathcal {C}}$ :

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. 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

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 $(M,\mu ,\epsilon )$ is a monoid object in $C$ , then $(FM,F\mu \circ \phi _{M,M},F\epsilon \circ \phi )$ is a monoid object in $D$ .^[2]