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 ${\mathcal {D}}$.

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, 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

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 $D$.

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.

See also

• Monoidal natural transformation

References

• Kelly, G. Max (1974). "Doctrinal adjunction". Category Seminar. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 257–280. doi:10.1007/BFb0063105. ISBN 978-3-540-37270-7.