Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.1.3.2 (Cartesian Products). 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 $\operatorname{\mathcal{C}}$.