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

File:Lax monoidal functor associative.svg,

File:Lax monoidal functor right unit.svg    and    File:Lax monoidal functor left unit.svg

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 ${𝒞}$ :

File:Lax monoidal functor braided.svg

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

Examples

• The underlying functor $U:(\mathbf{A}\mathbf{b},{\otimes }_{\mathbf{Z}},\mathbf{Z})\to (\mathbf{S}\mathbf{e}\mathbf{t},\times ,\{\ast \})$ 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 $\varphi :\{*\}\to \mathbb{\mathbb{Z}}$ sends $\ast$ to 1.
• If $R$ is a (commutative) ring, then the free functor $\mathsf{S}\mathsf{e}\mathsf{t},\to R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d}$ extends to a strongly monoidal functor $(\mathsf{S}\mathsf{e}\mathsf{t},\bigsqcup ,\mathrm{\varnothing })\to (R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},\oplus ,0)$ (and also $(\mathsf{S}\mathsf{e}\mathsf{t},\times ,\{\ast \})\to (R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},\otimes ,R)$ if $R$ is commutative).
• If $R\to S$ is a homomorphism of commutative rings, then the restriction functor $(S\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},{\otimes }_S,S)\to (R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},{\otimes }_R,R)$ is monoidal and the induction functor $(R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},{\otimes }_R,R)\to (S\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},{\otimes }_S,S)$ is strongly monoidal.
• An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory. Let ${\mathbf{B}\mathbf{o}\mathbf{r}\mathbf{d}}_{⟨n-1,n⟩}$ 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 $F:({\mathbf{B}\mathbf{o}\mathbf{r}\mathbf{d}}_{⟨n-1,n⟩},\bigsqcup ,\mathrm{\varnothing })\to (\mathbf{k}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t},{\otimes }_k,k).$
• The homology functor is monoidal as $(Ch(R\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d}),\otimes ,R[0])\to (grR\mathsf{-}\mathsf{m}\mathsf{o}\mathsf{d},\otimes ,R[0])$ 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:

Commutative diagram demonstrating how a monoidal coherence map gives rise to its applicative formulation

Commutative diagram demonstrating how a monoidal coherence map can be recovered from its applicative formulation

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

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.
• Perrone, Paolo (2024). Starting Category Theory. World Scientific. doi:10.1142/9789811286018_0005. ISBN 978-981-12-8600-1.