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 $(\mathcal C,\otimes,I_)$ and $(\mathcal D,\bullet,I_)$ 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
:$\phi_:FA\bullet FB\to F(A\otimes B)$
from functors $\mathcal\times\mathcal\to\mathcal$ to $F$ and a morphism
:$\phi:I_\to FI_$,
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 $\phi_, \phi$ 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,\otimes_\mathbf,\mathbf) \rightarrow (\mathbf,\times,\)$ from the category of abelian groups to the category of sets. In this case, the map $\phi_\colon U(A)\times U(B)\to U(A\otimes B)$ 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 $\mathsf,\to R\mathsf$ extends to a strongly monoidal functor $(\mathsf,\sqcup,\emptyset)\to (R\mathsf,\oplus,0)$ (and also $(\mathsf,\times,\)\to (R\mathsf,\otimes,R)$ if $R$ is commutative).
* If $R\to S$ is a homomorphism of commutative rings, then the restriction functor $(S\mathsf,\otimes_S,S)\to(R\mathsf,\otimes_R,R)$ is monoidal and the induction functor $(R\mathsf,\otimes_R,R)\to(S\mathsf,\otimes_S,S)$ 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 $\mathbf_$ 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 $F\colon(\mathbf_,\sqcup,\emptyset)\rightarrow(\mathbf,\otimes_k,k).$
* The homology functor is monoidal as (Ch(R\mathsf),\otimes,R \to (grR\mathsf,\otimes,R via the map H_\ast(C_1)\otimes H_\ast(C_2) \to H_\ast(C_1\otimes C_2), _1otimes _2\mapsto _1\otimes x_2/math>.

Alternate notions

If $(\mathcal C,\otimes,I_)$ and $(\mathcal D,\bullet,I_)$ are closed monoidal categories with internal hom-functors $\Rightarrow_,\Rightarrow_$ (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 $(FM,F\mu\circ\phi_,F\epsilon\circ\phi)$ 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 $(G,n):(\mathcal D,\bullet,I_)\to(\mathcal C,\otimes,I_)$. Then $F$ has a comonoidal structure $(F,m)$ induced by $(G,n)$, defined by
:$m_=\varepsilon_\circ Fn_\circ F(\eta_A\otimes \eta_B):F(A\otimes B)\to FA\bullet FB$
and
:$m=\varepsilon_\circ Fn:FI_\to I_$.

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

*
{{Functors

Monoidal categories