Basic properties of abelian categories | Stable homotopy theory and higher algebra

Proposition 6.1.5. Let \(\Aa \) be an abelian category.

(1): Every monomorphism \(i\colon A \hookrightarrow B\) is the kernel of its cokernel, and every epimorphism \(p\colon B \twoheadrightarrow C\) is the cokernel of its kernel.
(2): A morphism \(f\) is an isomorphism if and only if it is both a monomorphism and an epimorphism.
(3): The category \(\Aa \) admits all pullbacks and pushouts.
(4): Monomorphisms are closed under pushouts, and any pushout square along a monomorphism is also a pullback square.
(5): Epimorphisms are closed under pullbacks and any pullback square along an epimorphism is also a pushout square.
(6): Every morphism \(f\colon A \to B\) factors uniquely¹ as an epimorphism followed by a monomorphism.

Proof. These are the standard exactness properties of an abelian category; see [Weibel (1994), Section 1.2]. We only recall that the image factorization is obtained by setting \[ \im (f):=\ker (B\twoheadrightarrow \coker (f)), \] and that a pushout of \(f\colon A \to B\) along \(g\colon A \to C\) is computed as the cokernel of \((f,-g)\colon A \to B \oplus C\). These descriptions give the remaining assertions and their duals. □

Notes

¹This is classically formulated as ‘unique up to unique isomorphism’.

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