nLab
monoidal functor

Idea

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

Definition

A functor $F:C\to D$ between 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 (…).

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