homological algebra

(also nonabelian homological algebra)

Introduction

additive and abelian categories

Ab-enriched category

pre-additive category

additive category

pre-abelian category

abelian category

Grothendieck category

abelian sheaves

semi-abelian category

kernel, cokernel

complex

differential

homology

category of chain complexes

chain complex

chain map

chain homotopy

chain homology and cohomology

quasi-isomorphism

homological resolution

simplicial homology

generalized homology

exact sequence,

injective object, projective object

injective resolution, projective resolution

flat resolution

derived category

triangulated category, enhanced triangulated category

stable (∞,1)-category

stable model category

pretriangulated dg-category

A-∞-category

(∞,1)-category of chain complexes

derived functor, derived functor in homological algebra

Tor, Ext

homotopy limit, homotopy colimit

abelian sheaf cohomology

double complex

Koszul-Tate resolution, BRST-BV complex

spectral sequence

spectral sequence of a filtered complex

spectral sequence of a double complex

Grothendieck spectral sequence

Leray spectral sequence

Serre spectral sequence

Hochschild-Serre spectral sequence

diagram chasing

3x3 lemma

four lemma, five lemma

snake lemma, connecting homomorphism

horseshoe lemma

Baer's criterion

singular homology

cyclic homology

Dold-Kan correspondence / monoidal, operadic

Eilenberg-Zilber theorem

universal coefficient theorem

Künneth theorem

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stable homotopy theory

loop space object

suspension object

looping and delooping

stabilization

stable derivator

triangulated category

stable (∞,1)-category of spectra

spectrum

stable homotopy category

smash product of spectra

symmetric smash product of spectra

Spanier-Whitehead duality

A-∞ ring

E-∞ ring

A triangulated functor $F : T \to T'$ between two triangulated categories $T$ and $T'$ is an additive functor that commutes with translation and preserves the distinguished triangles.

(Neeman 01, def. 2.1.1)

Triangulated functors are also called exact functors (not to be confused with the other meaning of exact functor).

tensor triangulated category

Fourier-Mukai transform

Last revised on June 29, 2016 at 15:16:15. See the history of this page for a list of all contributions to it.