View source for Homological Algebra/Abelian Category

{{definition|<math>Ab</math>-enriched category|An '''<math>Ab</math>-enriched category''' is a category <math>\mathcal{C}</math> such that:
# <math>\forall a,b \in \operatorname{Obj}(\mathcal{C})</math>, <math>\mathcal{C}(a, b)</math> is an abelian group.
# <math>\forall a, b, c \in \operatorname{Obj}(\mathcal{C})</math>, <math>\circ_{a,b,c} : \mathcal{C}(b,c) \times \mathcal{C}(a,b) \to \mathcal{C}(a,c)</math> is bilinear.}}

{{definition|zero object|A '''zero object''' is an object in an <math>Ab</math>-enriched category that is both initial and terminal. We usually denote it by <math>\mathbf{0}</math>.}}

{{definition|biproduct|Given an <math>Ab</math>-enriched category <math>\mathcal{C}</math>, a '''biproduct''' of <math>a,b \in \operatorname{Obj}(\mathcal{C})</math> is a tuple <math>(c, i_1:a \to c, p_1: c \to a, i_2: b \to c, p_2: c \to b)</math> such that:
# <math>p_1 \circ i_1 = 1_a.</math>
# <math>p_2 \circ i_1 = 0_{a,b}.</math>
# <math>p_1 \circ i_2 = 0_{b,a}.</math>
# <math>p_2 \circ i_2 = 1_b.</math>
# <math>i_1 \circ p_1 + i_2 \circ p_2 = 1_c.</math>
We usually denote <math>c</math> by <math>a \oplus b</math>.}}

{{definition|additive category|An '''additive category''' is an <math>Ab</math>-enriched category <math>\mathcal{C}</math> such that:
# There is a zero product in <math>\mathcal{C}</math>.
# Every <math>a,b \in \operatorname{Obj}(\mathcal{C})</math> has a biproduct.}}

{{definition|(co-)kernel|Given <math>f : a \to b</math> in an <math>Ab</math>-enriched category. A '''(co-)kernel''' of <math>f</math> is a (co-)equalizer of <math>f</math> and <math>0_{a,b}</math>.}}

{{definition|abelian category|An '''abelian category''' is an additive category where:
# Every morphism has a kernel and cokernel.
# Every monomorphism is a kernel and every epimorphism is a cokernel.}}

{{example|The category of all left <math>R</math>-modules of a ring <math>R</math> is an abelian category.}}

== Exercises ==
# Given <math>f : a \to b</math> in an <math>Ab</math>-enriched category with zero object. Prove that <math>f = 0_{a,b}</math> iff <math>f</math> factors through <math>\mathbf{0}</math>.

# Given a biproduct <math>(a \oplus b, i_1, p_1, i_2, p_2)</math> of <math>a</math> and <math>b</math>. Prove that <math>(a \oplus b, i_1, i_2)</math> is a coproduct of <math>a</math> and <math>b</math> and <math>(a \oplus b, p_1, p_2)</math> is a product of <math>a</math> and <math>b</math>.

# In an <math>Ab</math>-enriched category with zero object, a kernel of <math>f: a \to b</math> can be equivalently be characterized as a pullback of <math>\mathbf{0} \to b</math> along <math>f</math>.

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