# Kerodon

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### 2.1.4 Nonunital Monoidal Functors

We now study functors between (nonunital) monoidal categories.

Definition 2.1.4.1 (Nonunital Strict Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories (Definition 2.1.1.5). A nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the following properties:

• The diagram of functors

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]^{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } & \operatorname{\mathcal{D}}}$

is strictly commutative. In particular, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have an equality $F(X) \otimes F(Y) = F(X \otimes Y)$ of objects of $\operatorname{\mathcal{D}}$.

• For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the functor $F$ carries the associativity constraint $\alpha _{X,Y,Z}: X \otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z$ (for the monoidal structure on $\operatorname{\mathcal{C}}$) to the associativity constraint $\alpha _{F(X), F(Y), F(Z) }: F(X) \otimes (F(Y) \otimes F(Z) ) \simeq (F(X) \otimes F(Y) ) \otimes F(Z)$ (for the monoidal structure on $\operatorname{\mathcal{D}}$).

Example 2.1.4.2. Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category. Then the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ is a nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to itself.

For many applications, Definition 2.1.4.1 is too restrictive. In practice, the definition of a (nonunital) monoidal structure $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ on a category $\operatorname{\mathcal{C}}$ often involves constructions which are only well-defined up to isomorphism (see Examples 2.1.3.1 and 2.1.3.2). In such cases, it is unreasonable to require that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ has the property that $F(X) \otimes F(Y)$ and $F(X \otimes Y)$ are the same object of $\operatorname{\mathcal{D}}$. Instead, we should ask for any isomorphism $\mu _{X,Y}: F(X) \otimes F(Y) \xrightarrow {\sim } F(X \otimes Y)$. To get a well-behaved theory, we should further demand that the isomorphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$, and are suitably compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$. We begin by considering a slightly more general situation, where the morphisms $\mu _{X,Y}$ are not required to be invertible.

Definition 2.1.4.3 (Nonunital Lax Monoidal Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. A nonunital lax monoidal structure on $F$ is a collection of morphisms $\mu = \{ \mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ which satisfy the following pair of conditions:

$(a)$

The morphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$: that is, for every pair of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$, the diagram

$\xymatrix@R =50pt@C=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ F(f) \otimes F(g) } & F( X \otimes Y) \ar [d]^{ F( f \otimes g) } \\ F(X') \otimes F(Y' ) \ar [r]^-{ \mu _{X',Y'} } & F( X' \otimes Y' ) }$

commutes (in the category $\operatorname{\mathcal{D}}$). In other words, we can regard $\mu$ as a natural transformation of functors as indicated in the diagram

$\xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\ar [r]^-{\otimes } \ar [d]_{F \times F} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}\times \operatorname{\mathcal{D}}\ar [r]^-{ \otimes } \ar@ {=>}[]+<20pt,20pt>;+<60pt,50pt>_-{\mu } & \operatorname{\mathcal{D}}. }$
$(b)$

The morphisms $\mu _{X,Y}$ are compatible with the associativity constraints on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ in the following sense: for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@R =40pt@C=70pt{ F(X) \otimes (F(Y) \otimes F(Z) ) \ar [r]^-{ \alpha _{F(X), F(Y), F(Z) } } \ar [d]^{ \operatorname{id}_{F(X)} \otimes \mu _{Y,Z} } & (F(X) \otimes F(Y) ) \otimes F(Z) \ar [d]^{ \mu _{X,Y} \otimes \operatorname{id}_{F(Z)}} \\ F(X) \otimes F(Y \otimes Z) \ar [d]^{ \mu _{X,Y \otimes Z} } & F(X \otimes Y) \otimes F(Z) \ar [d]^{ \mu _{X \otimes Y, Z}} \\ F( X \otimes (Y \otimes Z) ) \ar [r]^-{ F( \alpha _{X,Y,Z } )} & F( (X \otimes Y) \otimes Z) }$

commutes (in the category $\operatorname{\mathcal{D}}$).

A nonunital lax monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on $F$. In this case, we will refer to the morphisms $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ as the tensor constraints of $F$.

Definition 2.1.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. A nonunital monoidal structure on $F$ is a lax nonunital monoidal structure $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ on $F$ with the property that each of the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$ is an isomorphism.

A nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a pair $(F, \mu )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $\mu$ is a nonunital monoidal structure on $F$.

Example 2.1.4.5. Let $k$ be a field and let $\operatorname{Vect}_ k$ denote the category of vector spaces over $k$, endowed with the monoidal structure of Example 2.1.3.1. The construction of this monoidal structure involved certain choices: for every pair of vector spaces $U,V \in \operatorname{Vect}_{k}$, we selected a universal $k$-bilinear map $b_{U,V}: U \times V \rightarrow U \otimes _{k} V$. The collection of functions $b = \{ b_{U,V} \} _{U,V \in \operatorname{Vect}_{k} }$ is then a nonunital lax monoidal structure on the forgetful functor $\operatorname{Vect}_{k} \rightarrow \operatorname{Set}$ (where we equip $\operatorname{Set}$ with the monoidal structure given by cartesian products; see Example 2.1.3.2). Note that the tensor product functor $\otimes _{k}: \operatorname{Vect}_ k \times \operatorname{Vect}_ k \rightarrow \operatorname{Vect}_ k$ is characterized by the requirement that it is given on objects by $(U,V) \mapsto U \otimes _{k} V$ and satisfies condition $(a)$ of Definition 2.1.4.3, and the associativity constraint on $\operatorname{Vect}_ k$ is characterized by the requirement that it satisfies condition $(b)$ of Definition 2.1.4.3. Note that $b$ is not a nonunital monoidal structure: the bilinear maps $b_{U,V}: U \times V \rightarrow U \otimes _{k} V$ are never bijective, except in the trivial case where $U \simeq 0 \simeq V$.

Example 2.1.4.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a nonunital strict monoidal functor. Then $F$ admits a nonunital monoidal structure $\{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$, where we take each $\mu _{X,Y}$ to be the identity morphism from $F(X) \otimes F(Y) = F(X \otimes Y)$ to itself.

Conversely, if $(F, \mu )$ is a nonunital monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ with the property that the tensor constraints $\mu _{X,Y}$ is an identity morphism in $\operatorname{\mathcal{D}}$, then $F$ is a nonunital strict monoidal functor.

Example 2.1.4.7. Let $M$ and $M'$ be nonunital monoids, regarded as nonunital monoidal categories having only identity morphisms (Example 2.1.1.3). Then nonunital lax monoidal functors from $M$ to $M'$ (in the sense of Definition 2.1.4.3) can be identified with nonunital monoid homomorphisms from $M$ to $M'$ (in the sense of Variant 1.3.2.8). Moreover, every nonunital lax monoidal functor from $M$ to $M'$ is automatically strict.

Example 2.1.4.8 (The Left Regular Representation). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $\operatorname{End}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ be the category of functors from $\operatorname{\mathcal{C}}$ to itself, endowed with the strict monoidal structure of Example 2.1.1.4. For each object $X \in \operatorname{\mathcal{C}}$, let $\ell _{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the functor given on objects by the formula $\ell _{X}(Y) = X \otimes Y$. The construction $X \mapsto \ell _{X}$ then determines a functor $\ell : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there is a natural isomorphism $\mu _{X,Y}: \ell _{X} \circ \ell _{Y} \xrightarrow {\sim } \ell _{X \otimes Y}$, whose value on an object $Z \in \operatorname{\mathcal{C}}$ is given by the associativity constraint

$(\ell _{X} \circ \ell _{Y})(Z) = X \otimes (Y \otimes Z) \xrightarrow { \alpha _{X,Y,Z} } (X \otimes Y) \otimes Z = \ell _{X \otimes Y}(Z).$

Then $\mu = \{ \mu _{X,Y} \} _{X,Y}$ is a nonunital monoidal structure on the functor $X \mapsto \ell _{X}$: property $(a)$ of Definition 2.1.4.3 follows from the naturality of the associativity constraint on $\operatorname{\mathcal{C}}$, and property $(b)$ is a reformulation of the pentagon identity.

Warning 2.1.4.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. A nonunital strict monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ possessing certain properties. However, a nonunital (lax) monoidal functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ together with additional structure, given by the tensor constraints $\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F(X \otimes Y)$. We will often abuse terminology by identifying a nonunital (lax) monoidal functor $(F, \mu )$ with the underlying functor $F$; in this case, we implicitly assume that the tensor constraints $\mu _{X,Y}$ have been specified.

Definition 2.1.4.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. Let $F,F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors equipped with nonunital lax monoidal structures $\mu$ and $\mu '$, respectively. We say that a natural transformation of functors $\gamma : F \rightarrow F'$ is nonunital monoidal if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@C =50pt@R=50pt{ F(X) \otimes F(Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ \gamma (X) \otimes \gamma (Y) } & F(X \otimes Y) \ar [d]^{\gamma (X \otimes Y)} \\ F'(X) \otimes F'(Y) \ar [r]^-{ \mu '_{X,Y}} & F'(X \otimes Y) }$

is commutative.

We let $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the category whose objects are nonunital lax monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$, and whose morphisms are nonunital monoidal natural transformations, and we let $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the nonunital monoidal functors $(F,\mu )$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$.

Example 2.1.4.11 (Nonunital Algebras). Let $\operatorname{\mathcal{C}}$ be a nonunital monoidal category and let $A$ be an object of $\operatorname{\mathcal{C}}$. A nonunital algebra structure on $A$ is a map $m: A \otimes A \rightarrow A$ for which the diagram

$\xymatrix@R =50pt@C=50pt{ & A \otimes (A \otimes A) \ar [rr]_-{\alpha _{A,A,A}} \ar [dl]_{ \operatorname{id}\otimes m} & & (A \otimes A) \otimes A \ar [dr]^{m \otimes \operatorname{id}} & \\ A \otimes A \ar [drr]^{m} & & & & A \otimes A \ar [dll]_{m} \\ & & A & & }$

is commutative. A nonunital algebra object of $\operatorname{\mathcal{C}}$ is a pair $(A,m)$, where $A$ is an object of $\operatorname{\mathcal{C}}$ and $m$ is a nonunital algebra structure on $A$. If $(A,m)$ and $(A', m')$ are nonunital algebra objects of $\operatorname{\mathcal{C}}$, then we say that a morphism $f: A \rightarrow A'$ is a nonunital algebra homomorphism if the diagram

$\xymatrix@R =50pt@C=50pt{ A \otimes A \ar [r]^-{m} \ar [d]^{ f \otimes f} & A \ar [d]^{f} \\ A' \otimes A' \ar [r]^-{m'} & A' }$

is commutative. We let $\operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ denote the category whose objects are nonunital algebra objects of $\operatorname{\mathcal{C}}$ and whose morphisms are nonunital algebra homomorphisms.

Let $\{ e\}$ denote the trivial monoid, regarded as a (strict) monoidal category having only identity morphisms (Example 2.1.1.3). Then we can identify objects $A \in \operatorname{\mathcal{C}}$ with functors $F: \{ e\} \rightarrow \operatorname{\mathcal{C}}$ (by means of the formula $A = F(e)$). Unwinding the definitions, we see that nonunital lax monoidal structures on the functor $F$ (in the sense of Definition 2.1.4.3) can be identified with nonunital algebra structures on the object $A = F(e)$. Under this identification, nonunital monoidal natural transformations correspond to homomorphisms of nonunital algebras. We therefore have an isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \{ e \} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$.

Example 2.1.4.12. Let $\operatorname{Set}$ denote the category of sets, endowed with the monoidal structure given by cartesian product of sets (Example 2.1.3.2). For each set $S$, we can identify nonunital algebra structures on $S$ (in the sense of Example 2.1.4.11) with nonunital monoid structures on $S$ (in the sense of Variant 1.3.2.8). This observation supplies an isomorphism of categories $\operatorname{Alg}^{\operatorname{nu}}( \operatorname{Set}) \simeq \operatorname{Mon}^{\operatorname{nu}}$, where $\operatorname{Mon}^{\operatorname{nu}}$ is the category of nounital monoids.

Example 2.1.4.13. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ and $\operatorname{\mathcal{D}}^{\operatorname{rev}}$ denote the same categories with the reversed nonunital monoidal structure (Example 2.1.3.5). Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ can be also regarded as a functor from $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ to $\operatorname{\mathcal{D}}^{\operatorname{rev}}$, which we will denote by $F^{\operatorname{rev}}$. There is a canonical bijection

$\xymatrix@R =50pt@C=50pt{ \{ \text{Nonunital lax monoidal structures on F} \} \ar [d]^{\sim } \\ \{ \text{Nonunital lax monoidal structures on F^{\operatorname{rev}}} \} , }$

which carries a nonunital lax monoidal structure $\mu$ to the nonunital lax monoidal structure $\mu ^{\operatorname{rev}}$ given by the formula $\mu ^{\operatorname{rev}}_{X,Y} = \mu _{Y,X}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$, which restricts to an isomorphism $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{rev}}, \operatorname{\mathcal{D}}^{\operatorname{rev}} )$.

Example 2.1.4.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and regard the opposite categories $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ as equipped with the nonunital monoidal structures of Example 2.1.3.4. Then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ determines a functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. There is a canonical bijection

$\{ \text{Nonunital monoidal structures on F} \} \simeq \{ \text{Nonunital monoidal structures on F^{\operatorname{op}}} \} ,$

which carries a nonunital monoidal structure $\mu$ on $F$ to a nonunital monoidal structure $\mu '$ on $F^{\operatorname{op}}$, given concretely by $\mu '_{X,Y} = \mu _{X,Y}^{-1}$. Using these bijections, we obtain a canonical isomorphism of categories $\operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\operatorname{op}} \simeq \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.

Warning 2.1.4.15. The analogue of Example 2.1.4.14 for nonunital lax monoidal functors is false. The notion of nonunital lax monoidal functor is not self-opposite: in general, there is no simple relationship between the categories $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ and $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{D}}^{\operatorname{op}} )$.

Motivated by Warning 2.1.4.15, we introduce the following:

Variant 2.1.4.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. A nonunital colax monoidal structure on $F$ is a nonunital lax monoidal structure on the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ (Definition 2.1.4.3). In other words, a colax monoidal structure on $F$ is a collection of morphisms $\mu = \{ \mu _{X,Y}: F(X \otimes Y) \rightarrow F(X) \otimes F(Y) \} _{X,Y \in \operatorname{\mathcal{C}}}$ which satisfy the following pair of conditions:

$(a)$

The morphisms $\mu _{X,Y}$ depend functorially on $X$ and $Y$: that is, for every pair of morphisms $f: X \rightarrow X'$, $g: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$, the diagram

$\xymatrix@R =50pt@C=50pt{ F(X \otimes Y) \ar [r]^-{ \mu _{X,Y} } \ar [d]^{ F(f \otimes g) } & F( X ) \otimes F(Y) \ar [d]^{ F( f) \otimes F(g) } \\ F(X' \otimes Y' ) \ar [r]^-{ \mu _{X',Y'} } & F( X') \otimes F(Y' ) }$

commutes (in the category $\operatorname{\mathcal{D}}$).

$(b)$

For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram

$\xymatrix@R =40pt@C=70pt{ F(X \otimes (Y \otimes Z) ) \ar [r]^-{ F(\alpha _{X,Y,Z}) } \ar [d]^{ \mu _{X, Y \otimes Z} } & F( (X \otimes Y) \otimes Z) \ar [d]^{ \mu _{X \otimes Y,Z }} \\ F(X) \otimes F(Y \otimes Z) \ar [d]^{ \operatorname{id}\otimes \mu _{Y,Z} } & F(X \otimes Y) \otimes F(Z) \ar [d]^{ \mu _{X,Y} \otimes \operatorname{id}} \\ F( X) \otimes (F(Y) \otimes F(Z)) \ar [r]^-{ \alpha _{F(X),F(Y),F(Z) } } & (F(X) \otimes F(Y)) \otimes F(Z)}$

commutes.

Construction 2.1.4.17 (Composition of Nonunital Monoidal Functors). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be nonunital monoidal categories, and suppose we are given a pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. If $\mu = \{ \mu _{X,Y} \} _{X,Y \in \operatorname{\mathcal{C}}}$ is a nonunital lax monoidal structure on the functor $F$ and $\nu = \{ \nu _{U,V} \} _{U, V \in \operatorname{\mathcal{D}}}$ is a nonunital lax monoidal structure on $G$, then the composite functor $G \circ F$ inherits a nonunital lax monoidal structure, which associates to each pair of objects $X,Y \in \operatorname{\mathcal{C}}$ the composite map

$(G \circ F)(X) \otimes (G \circ F)(Y) \xrightarrow { \nu _{ F(X), F(Y)} } G( F(X) \otimes F(Y) ) \xrightarrow { G( \mu _{X,Y} ) } (G \circ F)(X \otimes Y).$

This construction determines a composition law

$\circ : \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}).$

Remark 2.1.4.18. In the situation of Construction 2.1.4.17, suppose that $\mu$ and $\nu$ are nonunital monoidal structures on $F$ and $G$, respectively: that is, assume that all of the tensor constraints

$\mu _{X,Y}: F(X) \otimes F(Y) \rightarrow F( X \otimes Y) \quad \quad \nu _{U,V}: G(U) \otimes G(V) \rightarrow G( U \otimes V)$

are isomorphisms. Then Construction 2.1.4.17 supplies a nonunital monoidal structure on the composite functor $G \circ F$. We therefore obtain a composition law

$\circ : \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) \times \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\otimes }_{\operatorname{nu}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}).$

We close this section by describing an alternative perspective on nonunital lax monoidal functors. First, we need to review a bit of terminology.

Notation 2.1.4.19 (Oriented Fiber Products). Let $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$ be categories, and suppose we are given a pair of functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$. We let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ denote the iterated pullback $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{E}}) } \operatorname{Fun}( [1], \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{E}}) } \operatorname{\mathcal{D}}$. We will refer to $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ as the oriented fiber product of $\operatorname{\mathcal{C}}$ with $\operatorname{\mathcal{D}}$ over $\operatorname{\mathcal{E}}$. More concretely:

• An object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is a triple $(C, D, \eta )$ where $C$ is an object of the category $\operatorname{\mathcal{C}}$, $D$ is an object of the category $\operatorname{\mathcal{D}}$, and $\eta : F(C) \rightarrow G(D)$ is a morphism in the category $\operatorname{\mathcal{E}}$.

• If $(C,D,\eta )$ and $(C', D', \eta ')$ are objects of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$, then a morphism from $(C,D,\eta )$ to $(C', D', \eta ')$ is a pair $(u,v)$, where $u: C \rightarrow C'$ is a morphism in the category $\operatorname{\mathcal{C}}$, $v: D \rightarrow D'$ is a morphism in the category $\operatorname{\mathcal{D}}$, and the diagram

$\xymatrix@R =50pt@C=50pt{ F(C) \ar [r]^-{\eta } \ar [d]^{ F(u) } & G(D) \ar [d]^{G(v)} \\ F(C') \ar [r]^-{\eta '} & G(D') }$

commutes in the category $\operatorname{\mathcal{E}}$.

Remark 2.1.4.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors. The oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is often referred to in the literature as the comma construction on the functors $F$ and $G$, and is commonly denoted by $F \downarrow G$.

Proposition 2.1.4.21. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories, let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ denote the oriented fiber product of Notation 2.1.4.19. Then:

• Let $\mu = \{ \mu _{D,D'} \} _{D,D' \in \operatorname{\mathcal{D}}}$ be a nonunital lax monoidal structure on the functor $G$. Then there is a unique nonunital monoidal structure $\otimes _{\mu }$ on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ with the following properties:

$(1)$

The forgetful functor

$U: \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\quad \quad (C,D,\eta ) \mapsto (C,D)$

is a strict nonunital monoidal functor.

$(2)$

On objects, the tensor product $\otimes _{\mu }$ is given by the formula

$(C, D, \eta ) \otimes _{\mu } (C', D', \eta ') = (C \otimes C', D \otimes D', t(\eta ,\eta ') ),$

where $t(\eta , \eta ')$ is the composition $C \otimes C' \xrightarrow { \eta \otimes \eta '} G(D) \otimes G(D') \xrightarrow { \mu _{D,D'} } G(D \otimes D' )$.

• The construction $\mu \mapsto \otimes _{\mu }$ induces a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Nonunital lax monoidal structures on G} \} \ar [d] \\ \{ \textnormal{Nonunital monoidal structures on \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}} satisfying (1)} \} . }$

Remark 2.1.4.22. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories. We can summarize Proposition 2.1.4.21 more informally as follows: for any functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, choosing a nonunital lax monoidal structure on $G$ is equivalent to choosing a nonunital monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ which is compatible with the existing nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively.

Proof of Proposition 2.1.4.21. Unwinding the definitions, we see that to describe nonunital monoidal structure on the category $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$ satisfying condition $(1)$, one must give the following data:

• For every pair of objects $(C, D, \eta )$ and $(C', D', \eta ')$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply a tensor product $(C, D, \eta ) \otimes (C', D', \eta ' )$. By virtue of the assumption that $U$ is nonunital strict monoidal, this tensor product must be given as a triple $(C \otimes C', D \otimes D', t(\eta , \eta ') )$, for some morphism $t(\eta ,\eta '): C \otimes C' \rightarrow G(D \otimes D')$ in the category $\operatorname{\mathcal{D}}$.

• For every pair of morphisms $(u,v): (C,D, \eta ) \rightarrow (\overline{C}, \overline{D}, \overline{\eta } )$ and $(u',v'): (C', D', \eta ') \rightarrow (\overline{C}', \overline{D}', \overline{\eta }')$ in the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply a tensor product morphism $( C \otimes C', D \otimes D', t(\eta ,\eta ') ) \rightarrow ( \overline{C} \otimes \overline{C}', \overline{D} \otimes \overline{D}', t( \overline{\eta }, \overline{\eta }' ) )$. Note that this morphism is uniquely determined: for $U$ to be a nonunital strict monoidal functor, it must be the pair $(u \otimes u', v \otimes v')$. However, the existence of this morphism imposes the following condition:

$(i)$

If the diagrams

$\xymatrix@R =50pt@C=50pt{ C \ar [r]^-{\eta } \ar [d]^{u} & G(D) \ar [d]^{ G(v) } & C' \ar [r]^-{\eta '} \ar [d]^{u'} & G(D') \ar [d]^{ G(v') } \\ \overline{C} \ar [r]^-{ \overline{\eta } } & G( \overline{D} ) & \overline{C}' \ar [r]^-{ \overline{\eta }' } & G( \overline{D}' ) }$

commute (in the category $\operatorname{\mathcal{C}}$), then the diagram

$\xymatrix@C =50pt@R=50pt{ C \otimes C' \ar [r]^-{ t(\eta ,\eta ') } \ar [d]^{u \otimes u'} & G(D \otimes D') \ar [d]^{ G( v \otimes v') } \\ \overline{C} \otimes \overline{C}' \ar [r]^-{ t( \overline{\eta },\overline{\eta }') } & G( \overline{D} \otimes \overline{D}' ) }$

also commutes.

• For every triple of objects $(C, D, \eta )$, $(C', D', \eta ')$, and $(C'', D'', \eta '' )$ of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, we must supply an associativity constraint

$(C,D, \eta ) \otimes ( ( C', D', \eta ') \otimes (C'', D'', \eta '') ) \simeq ( (C,D, \eta ) \otimes (C', D', \eta ') ) \otimes (C'', D'', \eta '' )$

in $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. By virtue of our assumption that $U$ is nonunital strict monoidal, this associativity constraint is uniquely determined: it must be the pair $(\alpha _{C,C',C''}, \alpha _{D,D',D''} )$ given by the associativity constraints for the nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively. However, the existence of this morphism imposes the following condition:

$(ii)$

For every triple of morphisms $\eta : C \rightarrow G(D)$, $\eta ': C' \rightarrow G(D')$, and $\eta '': C'' \rightarrow G(D'')$, the diagram

$\xymatrix@C =50pt@R=50pt{ C \otimes (C' \otimes C'') \ar [r]^-{ \alpha _{C,C',C''} } \ar [d]^-{ t(\eta , t(\eta ',\eta '') )} & (C \otimes C') \otimes C'' \ar [d]^-{ t( t(\eta , \eta '), \eta '' )} \\ G(D \otimes (D' \otimes D'') ) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'' ) }$

commutes (in the category $\operatorname{\mathcal{C}}$).

If this condition is satisfied, then the associativity constraints are automatically functorial and satisfy the pentagon identity (since the analogous conditions hold in the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, respectively).

Given a collection of morphisms $t(\eta , \eta ')$ satisfying these conditions, we define $\mu = \{ \mu _{D,D'} \} _{D,D' \in \operatorname{\mathcal{D}}}$ by the formula $\mu _{D,D'} = t( \operatorname{id}_{G(D)}, \operatorname{id}_{ G(D') })$. Note that, if $(C,D, \eta )$ and $(C', D', \eta ')$ are arbitrary objects of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$, then we have canonical maps

$(\eta , \operatorname{id}_ D): (C,D,\eta ) \rightarrow (G(D), D, \operatorname{id}_{G(D)}) \quad \quad (\eta ', \operatorname{id}_{D'} ): (C',D', \eta ') \rightarrow (G(D'), D', \operatorname{id}_{ G(D') } ).$

Applying condition $(i)$, we see that the morphism $t(\eta ,\eta ')$ can then be recovered as the composition

$C \otimes C' \xrightarrow {\eta \otimes \eta '} G(D) \otimes G(D') \xrightarrow { \mu _{D,D'} } G(D \otimes D').$

To complete the proof, it will suffice to show that if we are given any system of morphisms $\mu = \{ \mu _{D,D'}: G(D) \otimes G(D') \rightarrow G(D \otimes D') \} _{D,D' \in \operatorname{\mathcal{D}}}$ and we define $t(\eta ,\eta ')$ as above, then $\mu$ is a nonunital lax monoidal structure on $G$ if and only if conditions $(i)$ and $(ii)$ are satisfied.

Using the formula for $t(\eta , \eta ')$ in terms of $\mu$, we can rewrite condition $(i)$ as follows:

$(i')$

If the diagrams

$\xymatrix@R =50pt@C=50pt{ C \ar [r]^-{\eta } \ar [d]^{u} & G(D) \ar [d]^{ G(v) } & C' \ar [r]^-{\eta '} \ar [d]^{u'} & G(D') \ar [d]^{ G(v') } \\ \overline{C} \ar [r]^-{ \overline{\eta } } & G( \overline{D} ) & \overline{C}' \ar [r]^-{ \overline{\eta }' } & G( \overline{D}' ) }$

commute (in the category $\operatorname{\mathcal{C}}$), then the outer rectangle in the diagram

$\xymatrix@C =50pt@R=50pt{ C \otimes C' \ar [r]^-{\eta \otimes \eta '} \ar [d]^{u \otimes u'} & G(D) \otimes G(D') \ar [r]^-{ \mu _{D,D'} } \ar [d]^{ G(v) \otimes G(v')} & G(D \otimes D') \ar [d]^{G(v \otimes v')} \\ \overline{C} \otimes \overline{C}' \ar [r]^-{\overline{\eta } \otimes \overline{\eta }'} & G( \overline{D} ) \otimes G( \overline{D}' ) \ar [r]^-{ \mu _{ \overline{D}, \overline{D}' } } & G( \overline{D} \otimes \overline{D}' ) }$

commutes.

Note that the left square appearing in this diagram is automatically commutative. Assertion $(i')$ is therefore a consequence of the following:

$(a)$

For every pair of morphisms $v: D \rightarrow \overline{D}$ and $v': D' \rightarrow \overline{D}'$ in the category $\operatorname{\mathcal{D}}$, the diagram

$\xymatrix@C =50pt@R=50pt{ G(D) \otimes G(D') \ar [r]^-{ \mu _{D,D'} } \ar [d]^{ G(v) \otimes G(v') } & G(D \otimes D') \ar [d]^{G( v \otimes v')} \\ G( \overline{D} ) \otimes G( \overline{D}' ) \ar [r]^-{ \mu _{ \overline{D}, \overline{D}' } } & G( \overline{D} \otimes \overline{D}' ) }$

commutes (in the category $\operatorname{\mathcal{C}}$).

Conversely, if $(i')$ is satisfied, then $(a)$ can be deduced by specializing to the case $\eta = \operatorname{id}_{G(D)}$, $\eta ' = \operatorname{id}_{G(D')}$, $\overline{\eta } = \operatorname{id}_{ G(\overline{D})}$, and $\overline{\eta }' = \operatorname{id}_{ G( \overline{D}')}$. It follows that $(i)$ is satisfied if and only if $(a)$ is satisfied: that is, if and only if $\mu = \{ \mu _{D,D'} \} _{D,D' \in \operatorname{\mathcal{D}}}$ is a natural transformation.

We can reformulate condition $(ii)$ as follows:

$(ii')$

For every triple of morphisms $\eta : C \rightarrow G(D)$, $\eta ': C' \rightarrow G(D')$, and $\eta '': C'' \rightarrow G(D'')$, the outer rectangle in the diagram

$\xymatrix@C =80pt@R=50pt{ C \otimes (C' \otimes C'') \ar [r]^-{ \alpha _{C,C',C''} } \ar [d]^{\eta \otimes (\eta ' \otimes \eta '') } & (C \otimes C') \otimes C'' \ar [d]^{( \eta \otimes \eta ') \otimes \eta ''} \\ G(D) \otimes (G(D') \otimes G(D'') ) \ar [r]^-{ \alpha _{ G(D), G(D'), G(D'')}} \ar [d]^{\operatorname{id}_{G(D)} \otimes \mu _{D,D'} } & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{\mu _{D,D'} \otimes \operatorname{id}_{G(D'')} } \\ G(D) \otimes G(D' \otimes D'') \ar [d]^{\mu _{D,D'\otimes D''}} & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{ \mu _{ D \otimes D', D''} } \\ G(D \otimes (D' \otimes D'')) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'') }$

commutes (in the category $\operatorname{\mathcal{C}}$).

Since the upper square in this diagram automatically commutes (by the naturality of the associativity constraints on $\operatorname{\mathcal{C}}$), assertion $(ii')$ is a consequence of the following simpler assertion:

$(b)$

For every triple of objects $D,D',D'' \in \operatorname{\mathcal{D}}$, the diagram

$\xymatrix@C =80pt{ G(D) \otimes (G(D') \otimes G(D'') ) \ar [r]^-{ \alpha _{ G(D), G(D'), G(D'')}} \ar [d]^{\operatorname{id}_{G(D)} \otimes \mu _{D,D'} } & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{\mu _{D,D'} \otimes \operatorname{id}_{G(D'')} } \\ G(D) \otimes G(D' \otimes D'') \ar [d]^{\mu _{D,D'\otimes D''}} & (G(D) \otimes G(D') ) \otimes G(D'') \ar [d]^{ \mu _{ D \otimes D', D''} } \\ G(D \otimes (D' \otimes D'')) \ar [r]^-{ G( \alpha _{D,D',D''} ) } & G( (D \otimes D') \otimes D'') }$

commutes (in the category $\operatorname{\mathcal{C}}$).

Conversely, if $(ii')$ is satisfied, then $(b)$ can be deduced by specializing to the case $\eta = \operatorname{id}_{G(D)}$, $\eta ' = \operatorname{id}_{ G(D')}$, and $\eta '' = \operatorname{id}_{ G(D'')}$. We conclude by observing that conditions $(a)$ and $(b)$ assert precisely that $\mu$ is a nonunital lax monoidal structure (Definition 2.1.4.3). $\square$

Remark 2.1.4.23 (Adjoint Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories and suppose we are given a pair of adjoint functors $\xymatrix@1{\operatorname{\mathcal{C}} \ar@ <.4ex>[r]^-{F} & \operatorname{\mathcal{D}} \ar@ <.4ex>[l]^-{G}}$, so that we have an isomorphism of oriented fiber products $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (see Notation 2.1.4.19). Applying Proposition 2.1.4.21 (and the dual characterization of nonunital colax monoidal functors), we see that the following are equivalent:

• The datum of a nonunital lax monoidal structure on the functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

• The datum of a nonunital colax monoidal structure on the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

• The datum of a nonunital monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ which is compatible with the nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (meaning that the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is a nonunital strict monoidal functor).