nLab
hyper-derived functor

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

In the context of homological algebra derived functors are traditionally considered on a model structure on chain complexes and often they are evaluated only on chain complexes that are concentrated in a single degree. If instead they are evaluated on general chain complexes, one sometimes speaks of hyper-derived functors for emphasis.

For more see at derived functor in homological algebra.

Examples

If abelian sheaf cohomology is considered in terms of the derived functor of the global section functor, then the corresponding hyper-derived functor is hypercohomology. This, too, is really just the basic definition of (abelian) cohomology, but not restricted to Eilenberg-MacLane objects concentrated in a single degree.

Properties

There is a certain spectral sequence that can help to compute values of hyper-derived functors. See the section Spectral sequences for hyper-derived functors.

Revised on August 26, 2012 19:19:23 by Urs Schreiber (89.204.137.239)