# nLab delta-functor

Contents

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

### Theorems

#### Additive and abelian categories

additive and abelian categories

# Contents

## Definition

###### Definition

Let $\mathcal{A}$, $\mathcal{B}$ be two abelian categories.

A homological $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$ is for each $n \in \mathbb{N}$ a functor

$T_n : \mathcal{A} \to \mathcal{B}$

equipped for each short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with a natural transformation

$\delta_n : T_n(C) \to T_{n-1}(A)$

such that for each such short exact sequence there is, naturally a long exact sequence

$\cdots T_{n+1}(C) \stackrel{\delta}{\to} T_n(A) \to T_n(B) \to T_n(C) \stackrel{\delta }{\to} T_{n-1}(A) \to \cdots \,.$

## Examples

The archetypical example is the chain homology functor

$H_\bullet(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}$

from the category of chain complexes of some abelian category (for $\mathbb{N}$-graded complexes).

The universal example are (non-total) right derived functors.

The notion is due to

A textbook account is for instance section 2.1 of