nLab
delta-functor

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Additive and abelian categories

Contents

Definition

Definition

Let π’œ\mathcal{A}, ℬ\mathcal{B} be two abelian categories.

A homological Ξ΄\delta-functor from π’œ\mathcal{A} to ℬ\mathcal{B} is for each nβˆˆβ„•n \in \mathbb{N} a functor

T n:π’œβ†’β„¬ T_n : \mathcal{A} \to \mathcal{B}

equipped for each short exact sequence 0β†’Aβ†’Bβ†’Cβ†’00 \to A \to B \to C \to 0 in π’œ\mathcal{A} with a natural transformation

Ξ΄ n:T n(C)β†’T nβˆ’1(A) \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

β‹―T n+1(C)β†’Ξ΄T n(A)β†’T n(B)β†’T n(C)β†’Ξ΄T nβˆ’1(A)β†’β‹―. \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 β€’(βˆ’):Ch β€’(π’œ)β†’π’œ H_\bullet(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A}

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

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

References

The notion is due to

A textbook account is for instance section 2.1 of

Revised on February 19, 2013 10:55:07 by Urs Schreiber (82.113.121.13)