Monoidal functor

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. 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 : 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==

• 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 ==