---- > [!definition] Definition. ([[categorical product]]) > We say a [[category]] $\mathsf{C}$ **has (finite) products** if for all objects $A,B$ in $\mathsf{C}$, the [[two-object slice category]] $\mathsf{C}_{A,B}$ has [[terminal object|final objects]]. Such a final object would consist of the data of an object of $\mathsf{C}$, usually denoted $A \times B$, and of two morphisms $A \times B \to A$, $A \times B \to B$; $A \times B$ is called the **product** of $A$ and $B$. ^definition > [!justification] Motivation. > Among objects in $\mathsf{Set}$ there is the notion of a [[cartesian product]] $A \times B=\{ (a,b) : a \in A \text{ and } b \in B \}$. This definition is put in terms of elements, but not all objects of a [[category]] will have elements. But the [[universal property of product sets]] allows us to characterize $A \times B$ together with the natural projections $\pi_{A}:A \times B \to A$ and $\pi_{B}:A \times B \to B$ as [[terminal objects are unique up to a unique isomorphism|'the']] [[terminal object|final]] object in $\mathsf{Set}_{A,B}$ with no mention of set elements at all. ^justification > [!basicexample] > > [!basicnonexample] Warning. > 'Categorical multiplication' need not agree with other notions of multiplication that already exist between objects in $\mathsf{C}$. For example, letting $\text{Obj}(\mathsf{C})=\mathbb{Z}$ and $\leq$ the morphism-determining [[relation]] in [[category#^basic-example-2|this category example]], we find that $k_{1} \times k_{2}=\min(k_{1},k_{2})$! ^basic-example ^warning ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01