nLab
monoidal functor

Idea

A monoidal functor is a functor between monoidal categories that preserves the monoidal structure: a homomorphism of monoidal categories.

Definition

A functor $F:C\to D$ between strict monoidal categories $(C,\otimes )$ and $(D,\otimes )$ is called lax monoidal if it is equipped with a morphism

$$ϵ:I_D\stackrel{}{\to }F(I_C)$$\epsilon : I_D \stackrel{}{\to} F(I_C)

and a natural transformation

$${\mu }_{x,y}:F(x){\otimes }_{D}F(y)\to F(x{\otimes }_{C}y)$$\mu_{x,y} : F(x) \otimes_D F(y) \to F(x \otimes_C y)

satisfying the following conditions

• associativity For all $x,y,z\in C$ we have a commuting diagram

  $$\begin{array}{ccc}F(x)\otimes F(y)\otimes F(z)& \stackrel{\mathrm{Id}\otimes {\mu }_{y,z}}{\to }& F(x)\otimes F(y\otimes z)\\ {}^{{\mu }_{x,y}\otimes \mathrm{Id}}\downarrow & & {\downarrow }^{{\mu }_{x,y\otimes z}}\\ F(x\otimes y)\otimes F(z)& \stackrel{{\mu }_{x\otimes y,z}}{\to }& F(x\otimes y\otimes z)\end{array}$$\array{ F(x) \otimes F(y) \otimes F(z) &\stackrel{Id \otimes \mu_{y,z}}{\to}& F(x) \otimes F(y \otimes z) \\ {}^{\mathllap{\mu_{x,y} \otimes Id}}\downarrow && \downarrow^{\mathrlap{\mu_{x,y \otimes z}}} \\ F(x \otimes y) \otimes F(z) &\stackrel{\mu_{x \otimes y, z}}{\to}& F(x \otimes y \otimes z) }

• unitality For all $x\in C$ we have

  $$\begin{array}{ccc}{I}_D\otimes F(x)& \stackrel{l_{F(x)}}{\leftarrow }& F(x)\\ {}^{ϵ\otimes \mathrm{Id}}\downarrow & & {\downarrow }^{F(l_x)}\\ F(I_C)\otimes F(x)& \stackrel{{\mu }_{I,x}}{\to }& F(I_C\otimes x)\end{array}$$\array{ I_D \otimes F(x) &\stackrel{l_{F(x)}}{\leftarrow}& F(x) \\ {}^{\mathllap{\epsilon \otimes Id}}\downarrow && \downarrow^{\mathrlap{F(l_x)}} \\ F(I_C) \otimes F(x) &\stackrel{\mu_{I,x}}{\to}& F(I_C \otimes x) }

  and

  $$\begin{array}{ccc}F(x)\otimes I_D& \stackrel{r_{F(x)}}{\leftarrow }& F(x)\\ {}^{\mathrm{Id}\otimes ϵ}\downarrow & & {\downarrow }^{F(r_x)}\\ F(x)\otimes F(I_C)& \stackrel{{\mu }_{x,I}}{\to }& F(x\otimes I_C)\end{array}$$\array{ F(x) \otimes I_D &\stackrel{r_{F(x)}}{\leftarrow}& F(x) \\ {}^{\mathllap{Id \otimes \epsilon }}\downarrow && \downarrow^{\mathrlap{F(r_x)}} \\ F(x) \otimes F(I_C) &\stackrel{\mu_{x,I}}{\to}& F(x \otimes I_C) }

Lax monoidal functors are the lax morphism for an appropriate 2-monad.

If $ϵ$ and ${\mu }_{x,y}$ are isomorphisms then $F$ is called a strong monoidal functor. If they are even identities it is called a strict monoidal functor.

In contrast to this, a strong monoidal functor may also be called a weak monoidal functor. Sometimes the plain term “monoidal functor” is used to mean a strong monoidal functor, in which case the general situation is called a lax monoidal functor.

An oplax monoidal functor (with various alternative names including comonoidal), is a monoidal functor from the opposite categories $C^{\mathrm{op}}$ to $D^{\mathrm{op}}$.

A monoidal transformation between monoidal functors is a natural transformation that respects the extra structure in an obvious way.

If the monoidal categories are not strict one obtains correspondingly more coherence diagrams. One way to summarize these is to note that a monoidal category $C$ is equivalently its pointed delooping 2-category/bicategory $BC$ (with a single object and $C$ as its hom-object), then a monoidal functor $C\to D$ is equivalently a 2-functor/pseudofunctor $BC\to BD$. Using this one can infer the coherence diagrams as special cases from those discussed at pseudofunctor.

Properties

Observation

Lax monoidal functors send monoids to monoids.

If $F:(C,\otimes )\to (D,\otimes )$ is a lax monoidal functor and

$$(A\in C,{\mu }_A:A\otimes A\to A,i_A:I\to A)$$(A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A)

is a monoid in $C$, then the object $F(A)$ is naturally equipped with the structure of a monoid in $D$ by setting

$$i_{F(A)}:I_D\stackrel{}{\to }F(I_C)\stackrel{F(i_A)}{\to }F(A)$$i_{F(A)} : I_D \stackrel{}{\to} F(I_C) \stackrel{F(i_A)}{\to} F(A)

and

$${\mu }_{F(A)}:F(A)\otimes F(A)\stackrel{{\nabla }_{F(A),F(A)}}{\to }F(A\otimes A)\stackrel{F({\mu }_A)}{\to }F(A).$$\mu_{F(A)} : F(A) \otimes F(A) \stackrel{\nabla_{F(A), F(A)}}{\to} F(A \otimes A) \stackrel{F(\mu_A)}{\to} F(A) \,.

This construction defines a functor

$$\mathrm{Mon}(f):\mathrm{Mon}(C)\to \mathrm{Mon}(D)$$Mon(f) : Mon(C) \to Mon(D)

between the categories of monoids.

Similarly, an oplax monoidal functor sends comonoids to comonoids.

Observation

For $(C,\otimes )$ a monoidal category write $BC$ for the correspinding delooping 2-category.

Lax monoidal functor $f:C\to D$ correspond to lax 2-functor

$$BF:BC\to BD.$$\mathbf{B}F : \mathbf{B}C \to \mathbf{B}D \,.

If $F$ is strong monoidal then this is an ordinary 2-functor. If it is strict monoidal, then this is a strict 2-functor.

String diagrams

Just like monoidal categories, monoidal functors have a string diagram calculus; see these slides for some examples.

Related concepts

• monoidal functor

• strong monoidal functor

• cartesian functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

• braided monoidal functor

• symmetric monoidal functor

• monoidal (∞,1)-functor

• monoidal (∞,n)-functor