nLab monoidal functor

1. Idea

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

2. Definition

Definition 2.1. 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 2.2. In the literature often the term “monoidal functor” refers by default to what in def. 2.1 is called a strong monoidal functor. With that convention then what def. 2.1 calls a lax monoidal functor is called a weak monoidal functor.

Remark 2.3. Lax monoidal functors are the lax morphisms for an appropriate 2-monad.

Definition 2.4. An oplax monoidal functor (with various alternative names including comonoidal), is a monoidal functor from the opposite categories $C^{op}$ to $D^{op}$.

Definition 2.5. A monoidal transformation between monoidal functors is a natural transformation that respects the extra structure in an obvious way.

3. Properties

Proposition 3.1. (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.

Relation to multicategories

Remark 3.2. Lax monoidal functors between monoidal categories are in correspondence with morphisms between their underlying (representable) multicategories.

Relation to PROs

Remark 3.3. Strong monoidal functors between monoidal categories are in correspondence with morphisms between their underlying (representable) colored PROs.

Remark 3.4. Strict monoidal functors between monoidal categories are in correspondence with morphisms between their underlying colored PROs that preserve the distinguished isomorphisms $() \xrightarrow{\sim} I$ and $(A, B) \xrightarrow{\sim} (A \otimes B)$ for all $A, B$.

Relationships between categories of monoidal categories

Proposition 3.5. The 1-category of strict monoidal categories and strict monoidal functors is not equivalent to the 1-category of monoidal categories and strong monoidal functors.

Proof. The former has an initial object, whereas the latter does not.  ▮

Proposition 3.6. The inclusion from the 1-category of strict monoidal categories and strong monoidal functors into the 1-category of monoidal categories and strong monoidal functors is not an equivalence.

Proof. As mentioned at monoidal category, not every skeletal monoidal category is monoidally equivalent to a strict skeletal monoidal category. Therefore the inclusion is not essentially surjective.  ▮

Proposition 3.7. The inclusion from the 2-category of strict monoidal categories and strict monoidal functors into the 2-category of monoidal categories and strong monoidal functors is not an equivalence.

Proof. Not every strong monoidal functor between strict monoidal categories is equivalent to a strict one. See for example this MathOverflow question.  ▮

Proposition 3.8. The inclusion of the the 2-category of strict monoidal categories and strong monoidal functors into the 2-category of monoidal categories and strong monoidal functors is an equivalence.

Proof. By the coherence theorem for monoidal categories, every monoidal category is strong monoidally equivalent to a strict one.  ▮

4. String diagrams

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

5. 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

• idempotent monoidal functor

• monoidal (∞,1)-functor

• monoidal (∞,n)-functor

6. References

• Samuel Eilenberg, G. Max Kelly, p. 473 in: Closed Categories, in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) 421-562 [doi:10.1007/978-3-642-99902-4]

• 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