homological algebra

(also nonabelian homological algebra)

Introduction

Context

• additive and abelian categories

  • Ab-enriched category

  • pre-additive category

  • additive category

  • pre-abelian category

  • abelian category

  • Grothendieck category

• abelian sheaves

• semi-abelian category

Basic definitions

• 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,

  • short exact sequence, long exact sequence, split exact sequence

• injective object, projective object

  • injective resolution, projective resolution

  • flat resolution

Stable homotopy theory notions

• 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

  • lim^1 and Milnor sequences

  • fr-code

• abelian sheaf cohomology

Constructions

• 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

Lemmas

diagram chasing

• 3x3 lemma

• four lemma, five lemma

• snake lemma, connecting homomorphism

• horseshoe lemma

• Baer's criterion

Schanuel's lemma

Homology theories

• singular homology

• cyclic homology

Theorems

• Dold-Kan correspondence / monoidal, operadic

  • Moore complex, Alexander-Whitney map, Eilenberg-Zilber map

• Eilenberg-Zilber theorem

  • cochain on a simplicial set

• universal coefficient theorem

• Künneth theorem