Vipul: /* For two objects */

2008-12-25T12:10:30Z

For two objects

← Older revision		Revision as of 12:10, 25 December 2008
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	# <math>\pi_1 \circ i_1</math> and <math>\pi_2 \circ i_2</math> are the identity maps on <math>A_1</math> and <math>A_2</math> respectively.		# <math>\pi_1 \circ i_1</math> and <math>\pi_2 \circ i_2</math> are the identity maps on <math>A_1</math> and <math>A_2</math> respectively.
	# <math>C</math>, along with the maps <math>\pi_1, \pi_2</math>, is a [[product]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:D \to A_1, f_2:D \to A_2</math>, there exists a unique map <math>g:D \to C</math> such that <math>f_1 = \pi_1 \circ g</math> and <math>f_2 = \pi_2 \circ g</math>.		# <math>C</math>, along with the maps <math>\pi_1, \pi_2</math>, is a [[product]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:D \to A_1, f_2:D \to A_2</math>, there exists a unique map <math>g:D \to C</math> such that <math>f_1 = \pi_1 \circ g</math> and <math>f_2 = \pi_2 \circ g</math>.
	# <math>C</math>, along with the maps <math>i_1, i_2</math>, is a [[coproduct]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:A_1 \to D, f_2: A_2 \to D</math>, there exists <math>g:C \to D</math> such that <math>f_1 = g \circ i_1, ~~f_@~~ = g \circ i_2</math>.		# <math>C</math>, along with the maps <math>i_1, i_2</math>, is a [[coproduct]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:A_1 \to D, f_2: A_2 \to D</math>, there exists <math>g:C \to D</math> such that <math>f_1 = g \circ i_1, f_2 = g \circ i_2</math>.

	===For a finite collection of objects===		===For a finite collection of objects===

	A [[preadditive category]] that admits biproducts for finite collections of objects is termed an [[additive category]].		A [[preadditive category]] that admits biproducts for finite collections of objects is termed an [[additive category]].

Vipul: New page: ==Definition== ===For two objects=== Suppose $\mathcal{C}$ is a category and $A_1,A_2 \in \operatorname{Ob}\mathcal{C}$ . A '''biproduct''' of $A_1$ a...

2008-12-25T12:10:13Z

New page: ==Definition== ===For two objects=== Suppose <math>\mathcal{C}</math> is a category and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''biproduct''' of <math>A_1</math> a...

New page

==Definition==

===For two objects===

Suppose <math>\mathcal{C}</math> is a [[category]] and <math>A_1,A_2 \in \operatorname{Ob}\mathcal{C}</math>. A '''biproduct''' of <math>A_1</math> and <math>A_2</math> is an object that serves the role of both a [[product]] and a [[coproduct]]. More explicitly, it is an object <math>C</math> along with maps <math>i_1:A_1 \to C, i_2:A_2 \to C</math> and maps <math>\pi_1:A_1 \to C, \pi_2:A_2 \to C</math> such that:

# <math>\pi_1 \circ i_1</math> and <math>\pi_2 \circ i_2</math> are the identity maps on <math>A_1</math> and <math>A_2</math> respectively.
# <math>C</math>, along with the maps <math>\pi_1, \pi_2</math>, is a [[product]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:D \to A_1, f_2:D \to A_2</math>, there exists a unique map <math>g:D \to C</math> such that <math>f_1 = \pi_1 \circ g</math> and <math>f_2 = \pi_2 \circ g</math>.
# <math>C</math>, along with the maps <math>i_1, i_2</math>, is a [[coproduct]] of <math>A_1</math> and <math>A_2</math>. In other words, for any object <math>D</math> with maps <math>f_1:A_1 \to D, f_2: A_2 \to D</math>, there exists <math>g:C \to D</math> such that <math>f_1 = g \circ i_1, f_@ = g \circ i_2</math>.

===For a finite collection of objects===

A [[preadditive category]] that admits biproducts for finite collections of objects is termed an [[additive category]].

Biproduct - Revision history

Vipul: /* For two objects */

Vipul: New page: ==Definition== ===For two objects=== Suppose \mathcal{C} is a category and A_1,A_2 \in \operatorname{Ob}\mathcal{C}. A '''biproduct''' of A_1 a...

Vipul: New page: ==Definition== ===For two objects=== Suppose $\mathcal{C}$ is a category and $A_1,A_2 \in \operatorname{Ob}\mathcal{C}$ . A '''biproduct''' of $A_1$ a...