Monoidal functor

Concept in category theory

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

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

ϕ A , B : F A ∙ F B → F ( A ⊗ B ) {\displaystyle \phi _{A,B}:FA\bullet FB\to F(A\otimes B)}

between functors  C × C → D {\displaystyle {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {D}}} and a morphism

ϕ : I D → F I C {\displaystyle \phi :I_{\mathcal {D}}\to FI_{\mathcal {C}}} ,

called the coherence maps or structure morphisms, which are such that for every three objects  A {\displaystyle A} ,  B {\displaystyle B} and  C {\displaystyle C} of  C {\displaystyle {\mathcal {C}}} the diagrams

and

commute in the category  D {\displaystyle {\mathcal {D}}} . Above, the various natural transformations denoted using  α , ρ , λ {\displaystyle \alpha ,\rho ,\lambda } are parts of the monoidal structure on  C {\displaystyle {\mathcal {C}}} and  D {\displaystyle {\mathcal {D}}} .[1]

### Variants

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* 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  ϕ A , B , ϕ {\displaystyle \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  γ {\displaystyle \gamma } ) such that the following diagram commutes for every pair of objects A, B in  C {\displaystyle {\mathcal {C}}}  :

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

## Examples

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* The underlying functor  U : ( A b , ⊗ Z , Z ) → ( S e t , × , { ∗ } ) {\displaystyle 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  ϕ A , B : U ( A ) × U ( B ) → U ( A ⊗ B ) {\displaystyle \phi _{A,B}\colon U(A)\times U(B)\to U(A\otimes B)} sends (a, b) to  a ⊗ b {\displaystyle a\otimes b} ; the map  ϕ : { ∗ } → Z {\displaystyle \phi \colon \\{*\\}\to \mathbb {Z} } sends  ∗ {\displaystyle \ast } to 1.
* If  R {\displaystyle R} is a (commutative) ring, then the free functor  S e t , → R − m o d {\displaystyle {\mathsf {Set}},\to R{\mathsf {-mod}}} extends to a strongly monoidal functor  ( S e t , ⊔ , ∅ ) → ( R − m o d , ⊕ , 0 ) {\displaystyle ({\mathsf {Set}},\sqcup ,\emptyset )\to (R{\mathsf {-mod}},\oplus ,0)} (and also  ( S e t , × , { ∗ } ) → ( R − m o d , ⊗ , R ) {\displaystyle ({\mathsf {Set}},\times ,\\{\ast \\})\to (R{\mathsf {-mod}},\otimes ,R)} if  R {\displaystyle R} is commutative).
* If  R → S {\displaystyle R\to S} is a homomorphism of commutative rings, then the restriction functor  ( S − m o d , ⊗ S , S ) → ( R − m o d , ⊗ R , R ) {\displaystyle (S{\mathsf {-mod}},\otimes _{S},S)\to (R{\mathsf {-mod}},\otimes _{R},R)} is monoidal and the induction functor  ( R − m o d , ⊗ R , R ) → ( S − m o d , ⊗ S , S ) {\displaystyle (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  B o r d ⟨ n − 1 , n ⟩ {\displaystyle \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 : ( B o r d ⟨ n − 1 , n ⟩ , ⊔ , ∅ ) → ( k V e c t , ⊗ k , k ) . {\displaystyle F\colon (\mathbf {Bord} _{\langle n-1,n\rangle },\sqcup ,\emptyset )\rightarrow (\mathbf {kVect} ,\otimes _{k},k).}
* The homology functor is monoidal as  ( C h ( R − m o d ) , ⊗ , R [ 0 ] ) → ( g r R − m o d , ⊗ , R [ 0 ] ) {\displaystyle (Ch(R{\mathsf {-mod}}),\otimes ,R[0])\to (grR{\mathsf {-mod}},\otimes ,R[0])} via the map  H ∗ ( C 1 ) ⊗ H ∗ ( C 2 ) → H ∗ ( C 1 ⊗ C 2 ) , [ x 1 ] ⊗ [ x 2 ] ↦ [ x 1 ⊗ x 2 ] {\displaystyle 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

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If  ( C , ⊗ , I C ) {\displaystyle ({\mathcal {C}},\otimes ,I_{\mathcal {C}})} and  ( D , ∙ , I D ) {\displaystyle ({\mathcal {D}},\bullet ,I_{\mathcal {D}})} are closed monoidal categories with internal hom-functors  ⇒ C , ⇒ D {\displaystyle \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:

## Properties

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* If  ( M , μ , ϵ ) {\displaystyle (M,\mu ,\epsilon )} is a monoid object in  C {\displaystyle C} , then  ( F M , F μ ∘ ϕ M , M , F ϵ ∘ ϕ ) {\displaystyle (FM,F\mu \circ \phi _{M,M},F\epsilon \circ \phi )} is a monoid object in  D {\displaystyle D} .[2]

## Monoidal functors and adjunctions

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Suppose that a functor  F : C → D {\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}} is left adjoint to a monoidal  ( G , n ) : ( D , ∙ , I D ) → ( C , ⊗ , I C ) {\displaystyle (G,n):({\mathcal {D}},\bullet ,I_{\mathcal {D}})\to ({\mathcal {C}},\otimes ,I_{\mathcal {C}})} . Then  F {\displaystyle F} has a comonoidal structure  ( F , m ) {\displaystyle (F,m)} induced by  ( G , n ) {\displaystyle (G,n)} , defined by

m A , B = ε F A ∙ F B ∘ F n F A , F B ∘ F ( η A ⊗ η B ) : F ( A ⊗ B ) → F A ∙ F B {\displaystyle 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 = ε I D ∘ F n : F I C → I D {\displaystyle m=\varepsilon _{I_{\mathcal {D}}}\circ Fn:FI_{\mathcal {C}}\to I_{\mathcal {D}}} .

If the induced structure on  F {\displaystyle 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

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* Monoidal natural transformation

## Inline citations

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1. ^ Perrone (2024), pp. 360–364
2. ^ Perrone (2024), pp. 367–368

## References

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* 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.

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Functor types

* Additive
* Adjoint
* Conservative
* Derived
* Diagonal
* Enriched
* Essentially surjective
* Exact
* Forgetful
* Full and faithful
* Logical
* Monoidal
* Representable
* Smooth

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