2.1.6 Monoidal Functors
We now introduce the unital analogue of Definition 2.1.4.4.
Definition 2.1.6.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A monoidal structure on $F$ is a nonunital lax monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ (Definition 2.1.4.3) which satisfies the following additional conditions:
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the tensor constraint $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$ is an isomorphism in $\operatorname{\mathcal{D}}$ (that is, $\mu $ is a nonunital monoidal structure on $F$).
There exists an isomorphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \xrightarrow {\sim } F(\mathbf{1}_{\operatorname{\mathcal{C}}})$ which is a unit for $F$ (in the sense of Definition 2.1.5.2).
A monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F$ is a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and $\mu $ is a monoidal structure on $F$.
Warning 2.1.6.3. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital lax monoidal functor. If $F$ is a monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, then it is both a nonunital monoidal functor (that is, the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F( X \otimes Y)$ are isomorphisms) and a lax monoidal functor (that is, it admits a unit $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$). However, the converse is false: to qualify as a monoidal functor, $F$ must satisfy the additional condition that $\epsilon $ is an isomorphism.
Example 2.1.6.5 (Strict Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be strict monoidal categories (Definition 2.1.2.1). We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strict monoidal if it is a nonunital strict monoidal functor (Definition 2.1.4.1) which carries the strict unit object $\mathbf{1}_{\operatorname{\mathcal{C}}}$ to the strict unit object $\mathbf{1}_{\operatorname{\mathcal{D}}}$.
Every strict monoidal functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be regarded as a monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, by taking each tensor constraint $\mu _{X,Y}$ to be the identity morphisms from $F(X) \otimes F(Y) = F(X \otimes Y)$ to itself. Conversely, if $(F, \mu )$ is a monoidal functor for which the tensor constraints $\mu _{X,Y}$ and the unit morphism $\epsilon : \mathbf{1}_{\operatorname{\mathcal{D}}} \rightarrow F( \mathbf{1}_{\operatorname{\mathcal{C}}} )$ are identity morphisms in $\operatorname{\mathcal{D}}$, then $F$ is a strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.
Example 2.1.6.6. Let $M$ and $M'$ be monoids, regarded as monoidal categories having only identity morphisms (Example 2.1.2.8). Then lax monoidal functors from $M$ to $M'$ (in the sense of Definition 2.1.5.8) can be identified with monoid homomorphisms from $M$ to $M'$ (in the sense of Definition 1.3.2.3). Moreover, every lax monoidal functor from $M$ to $M'$ is automatically strict monoidal (and therefore monoidal).
Example 2.1.6.7. Let $\operatorname{\mathcal{C}}$ be a monoidal category, and let $\ell : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ be the nonunital monoidal functor of Example 2.1.4.8 (carrying each object $X \in \operatorname{\mathcal{C}}$ to the functor $\ell _{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ given by $\ell _{X}(Y) = X \otimes Y$). Then $\ell $ is a monoidal functor: it admits a unit $\epsilon : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow \ell _{ \mathbf{1} }$ given by the inverse of the left unit constraint of Construction 2.1.2.17. To prove this, it suffices to verify that $\epsilon $ satisfies property $(1)$ of Proposition 2.1.5.13 (Remark 2.1.6.4). Unwinding the definitions, this is equivalent to the assertion that for every object $X \in \operatorname{\mathcal{C}}$, the outer cycle of the diagram
\[ \xymatrix@C =50pt{ X \ar [dd]^{\operatorname{id}_ X} & \mathbf{1} \otimes X \ar [l]_-{ \lambda _ X} \ar [ddr]_{ \operatorname{id}_{ \mathbf{1} \otimes X} } & \mathbf{1} \otimes (\mathbf{1} \otimes X) \ar [l]_-{ \operatorname{id}_{ \mathbf{1} } \otimes \lambda _ X} \ar [d]^{ \alpha _{ \mathbf{1}, \mathbf{1}, X}} \\ & & (\mathbf{1} \otimes \mathbf{1}) \otimes X \ar [d]^{ \upsilon \otimes \operatorname{id}_ X} \\ X & & \mathbf{1} \otimes X \ar [ll]_{ \lambda _ X} } \]
is commutative. In fact, the whole diagram commutes: for the inner cycle on the left this is immediate, and for the inner cycle on the right it follows from the definition of the left unit constraing $\lambda _{X}$ (Construction 2.2.1.11).
Example 2.1.6.8 ($2$-Cochains as Monoidal Structures). Let $G$ be a group and let $\Gamma $ be an abelian group equipped with an action of $G$. Let $\operatorname{\mathcal{C}}$ be the category introduced in Example 2.1.3.3, whose objects are the elements of $G$ and morphisms are given by
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(g,h) = \begin{cases} \Gamma & \text{ if } g =h \\ \emptyset & \text{otherwise.} \end{cases} \]
Then every $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $ can be regarded as the associativity constraint for a monoidal structure $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$. Let us write $\operatorname{\mathcal{C}}(\alpha )$ to indicate the category $\operatorname{\mathcal{C}}$, endowed with the monoidal structure $(\otimes , \alpha )$.
Suppose that we are given a pair of cocycles $\alpha ,\alpha ': G \times G \times G \rightarrow \Gamma $. Unwinding the definitions, we see that monoidal structures on the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}(\alpha ) \rightarrow \operatorname{\mathcal{C}}(\alpha ')$ are given by functions
\[ \mu : G \times G \rightarrow \Gamma \quad \quad (x,y) \mapsto \mu _{x,y} \]
which satisfy the identity
\[ \alpha _{x,y,z} + \mu _{x,yz} + x( \mu _{y,z} ) = \mu _{xy,z} + \mu _{x,y} + \alpha '_{x,y,z} \]
for $x,y,z \in G$. We can rewrite this identity more compactly as an equation $\alpha + d \mu = \alpha '$, where
\[ d: \{ \text{$2$-Cochains $G \times G \rightarrow \Gamma $} \} \rightarrow \{ \text{$3$-Cochains $G \times G \times G \rightarrow \Gamma $} \} \]
is defined by the formula $(d\mu )_{x,y,z} = x(\mu _{y,z}) - \mu _{xy,z} + \mu _{x,yz} - \mu _{x,y}$.
In particular, the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ can be promoted to a monoidal functor from $\operatorname{\mathcal{C}}(\alpha )$ to $\operatorname{\mathcal{C}}(\alpha ')$ if and only if the cocycles $\alpha $ and $\alpha '$ are cohomologous: that is, they represent the same element of the cohomology group $\mathrm{H}^{3}( G; \Gamma )$.
Notation 2.1.6.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be monoidal categories, and let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be monoidal functors. We say that a natural transformation $\gamma : F \rightarrow F'$ is monoidal if it is monoidal when viewed as a natural transformation of lax monoidal functors (Definition 2.1.5.18). We let $\operatorname{Fun}^{\otimes }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ and whose morphisms are monoidal natural transformations. We regard $\operatorname{Fun}^{\otimes }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ as a full subcategory of the category $\operatorname{Fun}^{\operatorname{lax}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of Definition 2.1.5.18 (or as a non-full subcategory of the category $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ of nonunital monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$).
Warning 2.1.6.10. We will not be consistent in our usage of Notation 2.1.6.9. For example, if $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are symmetric monoidal categories (
), then we will sometimes write $\operatorname{Fun}^{\otimes }(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ to denote the category of symmetric monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ (which is a full subcategory of the category of monoidal functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ defined in Notation 2.1.6.9).
Example 2.1.6.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite products, endowed with the cartesian monoidal structure described in Example 2.1.3.2. For any functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, we can regard the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ as endowed with the lax monoidal structure described in Example 2.1.5.17. This lax monoidal structure is a monoidal structure if and only if the functor $F$ preserves finite products. If this condition is satisfied, then the original functor $F$ inherits a monoidal structure (Remark 2.1.6.12).
Example 2.1.6.15 ($1$-Cochains as Natural Transformations). Let $G$ be a group, let $\Gamma $ be an abelian group equipped with an action of $G$, and choose a pair of $3$-cocycles
\[ \alpha , \alpha ': G \times G \times G \rightarrow \Gamma , \]
which we can regard as associativity constraints for monoidal categories $\operatorname{\mathcal{C}}(\alpha )$ and $\operatorname{\mathcal{C}}(\alpha ')$ having the same underlying category $\operatorname{\mathcal{C}}$ (Example 2.1.6.8). Suppose we are given a pair of monoidal structures $\mu $ and $\mu '$ on the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$, which we can identify with $2$-cochains $\mu , \mu ': G \times G \rightarrow \Gamma $ satisfying
\[ \alpha + d \mu = \alpha ' \quad \quad \alpha + d \mu ' = \alpha '. \]
Then the difference $\nu = \mu - \mu '$ is a $2$-cocycle: that is, it satisfies the identity
\[ x \nu _{y,z} - \nu _{xy,z} + \nu _{x,yz} - \nu _{x,y} = 0 \]
for every triple of elements $x,y,z \in G$.
Note that a natural transformation from the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ to itself can be identified with a function
\[ \gamma : G \rightarrow \Gamma \quad \quad x \mapsto \gamma _{x}; \]
that is, with a $1$-cochain on $G$ taking values in the group $\Gamma $. Unwinding the definitions, we see that the natural transformation $\gamma $ is monoidal (with respect to the monoidal structures supplied by $\mu $ and $\mu '$, respectively) if and only if it satisfies the identity
\[ \mu '_{x,y} + x \gamma _{y} + \gamma _{x} = \mu _{x,y} + \gamma _{xy} \]
for every pair of elements $x,y \in G$. We can rewrite this identity more conceptually as $\mu ' + d \gamma = \mu $, where
\[ d: \{ \text{$1$-Cochains $G \rightarrow \Gamma $} \} \rightarrow \{ \text{$2$-Cochains $G \times G \rightarrow \Gamma $} \} \]
is defined by the formula $(d\gamma )_{x,y} = x(\gamma _{y}) - \gamma _{xy} + \gamma _{x}$. In particular, the monoidal functors $(\operatorname{id}_{\operatorname{\mathcal{C}}}, \mu )$ to $( \operatorname{id}_{\operatorname{\mathcal{C}}}, \mu ' )$ are isomorphic if and only if the $2$-cocycle $\nu = \mu - \mu '$ is a coboundary: that is, it has vanishing image in the cohomology group $\mathrm{H}^{2}(G; \Gamma )$.