Vipul at 01:24, 9 December 2008

2008-12-09T01:24:03Z

← Older revision		Revision as of 01:24, 9 December 2008
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	==Definition==		==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>.		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.

Vipul: New page: ==Definition== Suppose $\mathcal{C}$ is a category and $A_1,A_2 \in \operatorname{Ob}\mathcal{C}$ . A '''categorical product''', or simply '''product''', of $A_1...$

2008-12-09T01:21:15Z

New page: ==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...

New page

==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:

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>.

Categorical product - Revision history

Vipul at 01:24, 9 December 2008

Vipul: New page: ==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...

Vipul: New page: ==Definition== Suppose $\mathcal{C}$ is a category and $A_1,A_2 \in \operatorname{Ob}\mathcal{C}$ . A '''categorical product''', or simply '''product''', of $A_1...$