Kerodon

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2.1.3 Examples of Monoidal Categories

We now illustrate Definition 2.1.2.10 with some examples.

Example 2.1.3.1. Let $k$ be a field and let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$ (where morphisms are $k$-linear maps). For every pair of vector spaces $V,W \in \operatorname{Vect}_{k}$, let us choose a vector space $V \otimes _{k} W$ and a bilinear map

\[ V \times W \rightarrow V \otimes _{k} W \quad \quad (v,w) \mapsto v \otimes w \]

which exhibits $V \otimes _{k} W$ as a tensor product of $V$ and $W$ (see Example 2.1.0.2). The construction $(V, W) \mapsto V \otimes _{k} W$ determines a functor

\[ \otimes _{k}: \operatorname{Vect}_{k} \times \operatorname{Vect}_{k} \rightarrow \operatorname{Vect}_{k}, \]

whose value on a pair of $k$-linear maps $\varphi : V \rightarrow V'$, $\psi : W \rightarrow W'$ is characterized by the identity

\[ (\varphi \otimes _{k} \psi )(v \otimes w) = (\varphi (v) \otimes \psi (w)). \]

For every triple of vector spaces $U, V, W \in \operatorname{Vect}_{k}$, there is a canonical isomorphism

\[ \alpha _{U,V,W}: U \otimes _ k (V \otimes _{k} W) \xrightarrow {\sim } (U \otimes _{k} V) \otimes _{k} W, \]

characterized by the identity $\alpha _{U,V,W}( u \otimes (v \otimes w) ) = (u \otimes v) \otimes w$ for $u \in U$, $v \in V$, and $w \in W$. The pair $( \otimes _{k}, \alpha ) = ( \otimes _{k}, \{ \alpha _{U,V,W} \} _{U,V,W \in \operatorname{Vect}_{k} } )$ is then a nonunital monoidal structure on the category $\operatorname{Vect}_{k}$, in the sense of Definition 2.1.1.5. We can upgrade this to a monoidal structure by taking the unit object $\mathbf{1}$ to be the field $k$ (regarded as a vector space over itself), and the unit constraint $\upsilon : \mathbf{1} \otimes _{k} \mathbf{1} \simeq \mathbf{1}$ to be the linear map corresponding to the multiplication on $k$ (so that $\upsilon (a \otimes b) = ab$).

Example 2.1.3.2. Let $\operatorname{\mathcal{C}}$ be a category. Assume that every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ admits a product in $\operatorname{\mathcal{C}}$. This product is not unique: it is only unique up to (canonical) isomorphism. However, let us choose an object $X \times Y$ together with a pair of morphisms

\[ X \xleftarrow { \pi _{X,Y} } X \times Y \xrightarrow { \pi '_{X,Y} } Y \]

which exhibit $X \times Y$ as a product of $X$ and $Y$ in the category $\operatorname{\mathcal{C}}$. Then the construction $(X,Y) \mapsto X \times Y$ determines a functor $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$, given on morphisms by the construction

\[ ( (f: X \rightarrow X'), (g:Y \rightarrow Y') ) \mapsto ( (f \times g): (X \times Y) \rightarrow (X' \times Y' ) ), \]

where $f \times g$ is the unique morphism for which the diagram

\[ \xymatrix@R =40pt@C=40pt{ X \ar [d]^{f} & X \times Y \ar [l]_-{ \pi _{X,Y} } \ar [d]^{f \times g} \ar [r]^-{ \pi '_{X,Y} } & Y \ar [d]^{g} \\ X' & X' \times Y' \ar [l]_-{ \pi _{X',Y'} } \ar [r]^-{ \pi '_{X',Y'}} & Y' } \]

is commutative.

For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, there is a canonical isomorphism $\alpha _{X,Y,Z}: X \times (Y \times Z) \xrightarrow {\sim } (X \times Y) \times Z$, which is characterized by the commutativity of the diagram

\[ \xymatrix@R =40pt@C=40pt{ & X \times (Y \times Z) \ar [rr]^{ \alpha _{X,Y,Z} }_{\sim } \ar [ddl]_{\pi _{X,Y \times Z}} \ar [drr]|\hole & & (X \times Y) \times Z \ar [dll] \ar [ddr]^{ \pi '_{X \times Y,Z}} & \\ & X \times Y \ar [dl]^-{ \pi _{X,Y} } \ar [dr]_-{ \pi '_{X,Y} } & & Y \times Z \ar [dl]^-{ \pi _{Y,Z} } \ar [dr]_-{ \pi '_{Y,Z}} & \\ X & & Y & & Z. } \]

The category $\operatorname{\mathcal{C}}$ admits a nonunital monoidal structure, with tensor product given by the functor $(X,Y) \mapsto X \times Y$, and associativity constraints given by $(X,Y,Z) \mapsto \alpha _{X,Y,Z}$.

If we assume also that the category $\operatorname{\mathcal{C}}$ has a final object $\mathbf{1}$ (so that $\operatorname{\mathcal{C}}$ admits all finite products), then we can upgrade the nonunital monoidal structure above to a monoidal structure, where the unit object of $\operatorname{\mathcal{C}}$ is $\mathbf{1}$ and the unit constraint $\upsilon $ is the unique morphism from $\mathbf{1} \times \mathbf{1}$ to $\mathbf{1}$ in $\operatorname{\mathcal{C}}$. We refer to this monoidal structure as the Cartesian monoidal structure on $\operatorname{\mathcal{C}}$.

Example 2.1.3.3 (Group Cocycles). Let $G$ be a group with identity element $1 \in G$, and let $\Gamma $ be an abelian group on which $G$ acts by automorphisms; we denote the action of an element $g \in G$ by $(\gamma \in \Gamma ) \mapsto g(\gamma ) \in \Gamma $. A $3$-cocycle on $G$ with values in $\Gamma $ is a map of sets

\[ \alpha : G \times G \times G \rightarrow \Gamma \quad \quad (x,y,z) \mapsto \alpha _{x,y,z}. \]

which satisfies the equations

2.1
\begin{eqnarray} \label{equation:cocycle-identity} w( \alpha _{x,y,z} ) - \alpha _{wx,y,z} + \alpha _{w,xy,z} - \alpha _{w,x,yz} + \alpha _{w,x,y} = 0 \end{eqnarray}

for every quadruple of elements $w,x,y,z \in G$.

Let $\operatorname{\mathcal{C}}$ denote the category whose objects are the elements of $G$, and whose morphisms are given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(g,h) = \begin{cases} \Gamma & \text{ if } g = h \\ \emptyset & \text{ otherwise. } \end{cases} \]

Using the action of $G$ on $\Gamma $, we can construct a functor

\[ \otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}, \]

given on objects by $(g,h) \mapsto gh$ and on morphisms by

\[ ( (\gamma : g \rightarrow g), (\delta : h \rightarrow h) ) \mapsto (\gamma + g(\delta ): gh \rightarrow gh). \]

Unwinding the definitions, one sees that upgrading the functor $\otimes $ to a nonunital monoidal structure on the category $(\otimes , \alpha )$ on $\operatorname{\mathcal{C}}$ is equivalent to choosing a $3$-cocycle $\alpha : G \times G \times G \rightarrow \Gamma $. More precisely, any map $\alpha : G \times G \times G \rightarrow \Gamma $ can be regarded as a natural transformation of functors

\[ \bullet \otimes ( \bullet \otimes \bullet ) \rightarrow (\bullet \otimes \bullet ) \otimes \bullet , \]

and pentagon identity $(P)$ of Definition 2.1.1.5 translates to the cocycle condition (2.1) above.

For any choice of cocycle $\alpha : G \times G \times G \rightarrow \Gamma $, we can upgrade the associated nonunital monoidal structure $(\otimes , \alpha )$ to a monoidal structure on the category $\operatorname{\mathcal{C}}$, by taking the unit object of $\operatorname{\mathcal{C}}$ to be the identity element $1 \in G$ and the unit constraint $\upsilon : 1 \otimes 1 \simeq 1$ to be the element $0 \in \Gamma $.

Example 2.1.3.4 (The Opposite of a Monoidal Category). Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $( \otimes , \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$. Then the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ inherits a nonunital monoidal structure, which can be described concretely as follows:

  • The tensor product on $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is obtained from the tensor product functor $\otimes : \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ by passing to opposite categories.

  • Let $X$, $Y$, and $Z$ be objects of $\operatorname{\mathcal{C}}$, and let us write $X^{\operatorname{op}}$, $Y^{\operatorname{op}}$, and $Z^{\operatorname{op}}$ for the corresponding objects of $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then the associativity constraint $\alpha _{ X^{\operatorname{op}}, Y^{\operatorname{op}}, Z^{\operatorname{op}}}$ for $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is the inverse of the associativity constraint $\alpha _{X,Y,Z}$ for $\operatorname{\mathcal{C}}$.

If the nonunital monoidal category $\operatorname{\mathcal{C}}$ is equipped with a unit structure $( \mathbf{1}, \upsilon )$, then we can regard $( \mathbf{1}^{\operatorname{op}}, \upsilon ^{-1} )$ as a unit structure for the nonunital monoidal category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In particular, every monoidal structure on a category $\operatorname{\mathcal{C}}$ determines a monoidal structure on the opposite category $\operatorname{\mathcal{C}}^{\operatorname{op}}$.

Example 2.1.3.5 (The Reverse of a Monoidal Structure). Let $\operatorname{\mathcal{C}}$ be a category equipped with a nonunital monoidal structure $( \otimes , \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$. Then we can equip $\operatorname{\mathcal{C}}$ with another nonunital monoidal structure $(\otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$, defined as follows:

  • The tensor product functor $\otimes ^{\operatorname{rev}}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is given on objects by the formula $X \otimes ^{\operatorname{rev}} Y = Y \otimes X$ (and similarly on morphisms).

  • The associativity constraint on $\otimes ^{\operatorname{rev}}$ is given by the formula $\alpha ^{\operatorname{rev}}_{X,Y,Z} = \alpha _{Z,Y,Z}^{-1}$.

We will refer to the nonunital monoidal structure $( \otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$ as the reverse of the nonunital monoidal structure $(\otimes , \{ \alpha _{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$. In this case, we will write $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ to denote the nonunital monoidal category whose underlying category is $\operatorname{\mathcal{C}}$, equipped with the nonunital monoidal structure $(\otimes ^{\operatorname{rev}}, \{ \alpha ^{\operatorname{rev}}_{X,Y,Z} \} _{X,Y,Z \in \operatorname{\mathcal{C}}} )$.

If the nonunital monoidal category $\operatorname{\mathcal{C}}$ is equipped with a unit structure $( \mathbf{1}, \upsilon )$, then we can also regard $( \mathbf{1}, \upsilon )$ as a unit structure for the nonunital monoidal category $\operatorname{\mathcal{C}}^{\operatorname{rev}}$. In other words, if $\operatorname{\mathcal{C}}$ is a monoidal category, then we can regard $\operatorname{\mathcal{C}}^{\operatorname{rev}}$ as a monoidal category (having the same underlying category and unit object, but “reversed” tensor product).