---- > [!definition] Definition. ([[product category]]) > For $\mathsf{C}$ and $\mathsf{D}$ two [[category|categories]], their **product category** $\mathsf{C} \times \mathsf{D}$ is the [[category]] whose > - objects are ordered pairs $(C,D)$ where $C$ and $D$ and object of $\mathsf{D}$; > - morphisms are $\text{Hom}_{\mathsf{C } \times \mathsf{D}}\big( (C, D), (C', D') \big)=\text{Hom}_{\mathsf{C}}(C, C') \times \text{Hom}_{\mathsf{D}}(D, D')$. > Composition of morphisms is defined componentwise. ---- #### ---- #### 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 > ```