nLab monoidal functor

Idea

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

Definition

Definition

Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )$ be two monoidal categories. A lax monoidal functor between them is a functor:

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,$$

together with coherence maps:

1. a morphism

   $$\epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})$$

2. a natural transformation

   $$\mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)$$

   for all $x,y \in \mathcal{C}$

satisfying the following conditions:

1. (associativity) For all objects $x,y,z \in \mathcal{C}$ the following diagram commutes

   $$\array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} F(y \otimes_{\mathcal{C}} z) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) }$$

   where $a^{\mathcal{C}}$ and $a^{\mathcal{D}}$ denote the associators of the monoidal categories;

2. (unitality) For all $x \in \mathcal{C}$ the following diagrams commute

   $$\array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }$$

   and

   $$\array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) }$$

   where $\ell^{\mathcal{C}}$, $\ell^{\mathcal{D}}$, $r^{\mathcal{C}}$, $r^{\mathcal{D}}$ denote the left and right unitors of the two monoidal categories, respectively.

If $\epsilon$ and all $\mu_{x,y}$ are isomorphisms, then $F$ is called a strong monoidal functor. (Note that ‘strong’ is also sometimes applied to ‘monoidal functor’ to indicate possession of a tensorial strength.) If they are even identity morphisms, then $F$ is called a strict monoidal functor.

Remark

In the literature often the term “monoidal functor” refers by default to what in def. is called a strong monoidal functor. With that convention then what def. calls a lax monoidal functor is called a weak monoidal functor.

Remark

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

Definition

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

Definition

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

Properties

Proposition

(Lax monoidal functors send monoids to monoids)

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

$$(A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A)$$

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

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

and

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

This construction defines a functor

$$Mon(f) : Mon(C) \to Mon(D)$$

between the categories of monoids in $C$ and $D$, respectively.

More generally, lax functors send enriched categories to enriched categories, an operation known as change of enriching category. See there for more details.

Similarly:

Proposition

(oplax monoidal functors sends comonoids to comonoids)

For $(C,\otimes)$ a monoidal category write $\mathbf{B}C$ for the corresponding delooping 2-category.

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

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

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

• multifunctor

• module over a monoidal functor

• monoidal adjunction

• indexed monoidal category

• cartesian functor

• lax monoidal functor

  • functor with smash products

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

• braided monoidal functor

• symmetric monoidal functor

• monoidal (∞,1)-functor

• monoidal (∞,n)-functor

References

• Saunders MacLane, §XI.2 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, §2.4 in: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) [ISBN:978-1-4704-3441-0, pdf]

  (discussed what we call strong monoidal functors)

• Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras. (pdf)

Exposition of basics of monoidal categories and categorical algebra:

• geometry of physics – categories and toposes, Section 2: Basic notions of categorical algebra