nLab hyper-derived functor

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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.

Last revised on August 26, 2012 at 19:19:23. See the history of this page for a list of all contributions to it.