Biproduct: Difference between revisions

Latest revision as of 12:10, 25 December 2008

Definition

For two objects

Suppose $C$ is a category and $A_{1}, A_{2} \in O b C$ . A biproduct of $A_{1}$ and $A_{2}$ is an object that serves the role of both a product and a coproduct. More explicitly, it is an object $C$ along with maps $i_{1} : A_{1} \to C, i_{2} : A_{2} \to C$ and maps $π_{1} : A_{1} \to C, π_{2} : A_{2} \to C$ such that:

$π_{1} \circ i_{1}$ and $π_{2} \circ i_{2}$ are the identity maps on $A_{1}$ and $A_{2}$ respectively.
$C$ , along with the maps $π_{1}, π_{2}$ , is a product of $A_{1}$ and $A_{2}$ . In other words, for any object $D$ with maps $f_{1} : D \to A_{1}, f_{2} : D \to A_{2}$ , there exists a unique map $g : D \to C$ such that $f_{1} = π_{1} \circ g$ and $f_{2} = π_{2} \circ g$ .
$C$ , along with the maps $i_{1}, i_{2}$ , is a coproduct of $A_{1}$ and $A_{2}$ . In other words, for any object $D$ with maps $f_{1} : A_{1} \to D, f_{2} : A_{2} \to D$ , there exists $g : C \to D$ such that $f_{1} = g \circ i_{1}, f_{2} = g \circ i_{2}$ .

For a finite collection of objects

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

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