Categorical product: Difference between revisions

Latest revision as of 01:24, 9 December 2008

Definition

Suppose ${\mathcal {C}}$ is a category and $A_{1},A_{2}\in \operatorname {Ob} {\mathcal {C}}$ . A categorical product, or simply product, of $A_{1}$ and $A_{2}$ is an object $C\in \operatorname {Ob} {\mathcal {C}}$ along with morphisms (called projection maps) $\pi _{1}:C\to A_{1},\pi _{2}:C\to A_{2}$ , such that the following holds:

For any object $D\in {\mathcal {C}}$ and morphisms $f_{i}:D\to A_{i}$ , there is a unique morphism $g:D\to C$ such that $\pi _{1}\circ g=f_{1}$ and $\pi _{2}\circ g=f_{2}$ .

If a categorical product exists for two objects, then the categorical product is unique upto canonical isomorphism: for any two categorical products, there is a unique isomorphism between them that commutes with the projection maps.

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 ==Definition==
-Suppose <math>\mathcal{C}</math> is a category and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''categorical product''', or simply '''product''', of <math>A_1</math> and <math>A_2</math> is an object <math>C \in \operatorname{Ob} \mathcal{C}</math> along with morphisms <math>\pi_1:C \to A_1, \pi_2:C \to A_2</math>, such that the following holds:
+Suppose <math>\mathcal{C}</math> is a category and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''categorical product''', or simply '''product''', of <math>A_1</math> and <math>A_2</math> is an object <math>C \in \operatorname{Ob} \mathcal{C}</math> along with morphisms (called ''projection maps'') <math>\pi_1:C \to A_1, \pi_2:C \to A_2</math>, such that the following holds:
 For any object <math>D \in \mathcal{C}</math> and morphisms <math>f_i:D \to A_i</math>, there is a ''unique'' morphism <math>g:D \to C</math> such that <math>\pi_1 \circ g = f_1</math> and <math>\pi_2 \circ g = f_2</math>.
+If a categorical product exists for two objects, then the categorical product is unique upto canonical isomorphism: for any two categorical products, there is a unique isomorphism between them that commutes with the projection maps.