Monoidal functor explained

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

( l C, ⊗ ,I _{lC} ) and ( l D,\bullet,I _{lD} ) be monoidal categories. A lax monoidal functor from l C to l D (which may also just be called a monoidal functor) consists of a functor F: l C\to l D together with a natural transformation\phi _{A,B} :FA\bullet FB\to F(A ⊗ B) between functors l{C} x l{C}\to l{D} and a morphism\phi:I _{lD} \to FI _{lC} ,called the coherence maps or structure morphisms, which are such that for every three objects A , B and C of l C the diagrams

,

   and    commute in the category

l D . Above, the various natural transformations denoted using \alpha, \rho, λ are parts of the monoidal structure on l C and l 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 l C :

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

Examples

• The underlying functor

U\colon( Ab , ⊗ _{Z,Z)} → ( Set , x ,\{\ast\}) from the category of abelian groups to the category of sets. In this case, the map \phi _{A,B} \colon U(A) x U(B)\to U(A ⊗ B) sends (a, b) to a ⊗ b ; the map \phi\colon \{*\}\to Z sends \ast to 1.

• If

R is a (commutative) ring, then the free functor Set ,\to R -mod extends to a strongly monoidal functor ( Set ,\sqcup,\emptyset)\to (R -mod , ⊕ , 0 ) (and also ( Set , x ,\{\ast\})\to (R -mod , ⊗ ,R) if R is commutative).

• If

R\to S is a homomorphism of commutative rings, then the restriction functor (S -mod , ⊗ _{S,S)\to(R-mod, ⊗} _{R,R)} is monoidal and the induction functor (R -mod , ⊗ _{R,R)\to(S-mod, ⊗} _{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

Bord _{\langle} 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( Bord _{\langle} ,\sqcup,\emptyset) → ( kVect , ⊗ _{k,k).}

• The homology functor is monoidal as

(Ch(R -mod ), ⊗ ,R[ 0 ]) \to (grR -mod , ⊗ ,R[ 0 ]) via the map H _{\ast(C} _{1) ⊗} H _{\ast(C} _{2)} \to H _{\ast(C} _{1 ⊗} C _{2),} [x _{1] ⊗ [x} _{2]} \mapsto [x _{1 ⊗} x _{2]} .

Alternate notions

If

( l C, ⊗ ,I _{lC} ) and ( l D,\bullet,I _{lD} ) are closed monoidal categories with internal hom-functors ⇒ _{lC} , ⇒ _{lD} (we drop the subscripts for readability), there is an alternative formulation

ψ_AB : F(A ⇒ B) → FA ⇒ FBof 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: l C\to l D is left adjoint to a monoidal (G,n):( l D,\bullet,I _{lD} )\to( l C, ⊗ ,I _{lC} ) . Then F has a comonoidal structure (F,m) induced by (G,n) , defined bym _{A,B} = \varepsilon _{FA\bullet} \circ Fn _{FA,FB} \circ F( η _{A ⊗} η _{B):F(A ⊗} B)\to FA\bullet FB andm = \varepsilon _{I _{lD}} \circ Fn:FI _{lC} \to I _{lD} .

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

• Book: Kelly, G. Max . Doctrinal adjunction . Category Seminar . Springer . Lecture Notes in Mathematics . 420 . 1974 . 978-3-540-37270-7 . 257–280 . 10.1007/BFb0063105.
• Book: Perrone, Paolo . Starting Category Theory. 2024 . World Scientific. 10.1142/9789811286018_0005 . 978-981-12-8600-1.