homological algebra
and
nonabelian homological algebra
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, quasi-isomorphism
chain homotopy
chain homology and cohomology
exact sequence,
injective object, projective object
injective resolution, projective resolution
flat resolution
derived functor
Tor, Ext
homotopy limit, homotopy colimit
abelian sheaf cohomology
derived category
triangulated category, enhanced triangulated category
stable (∞,1)-category
stable model category
pretriangulated dg-category
A-∞-category
(∞,1)-category of chain complexes
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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A chain complex A • in an abelian category 𝒜 is called perfect if
it is bounded;
all objects are projective objects.