category theory

# Contents

## Idea

A projective presentation of an object is a realization of that object as a suitable quotient of a projective object.

In homological algebra projective presentations can sometimes be used in place of genuine projective resolutions in the computation of derived functors. See for instance at Ext-functor for examples.

The dual notion is that of injective presentation.

## Definition

### In abelian categories

Let $\mathcal{A}$ be an abelian category. For $X \in \mathcal{A}$ any object, a projective presentation of $X$ is a short exact sequence of the form

$0 \to N \stackrel{i}{\hookrightarrow} P \stackrel{p}{\to} X \to 0 \,,$

hence exhibiting $X$ as the cokernel

$X \simeq coker(N \hookrightarrow P)$

such that $P$ is a projective object.

Revised on September 4, 2012 20:29:40 by Urs Schreiber (131.174.190.104)