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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idempotent,

completion

idempotent complete category, Cauchy complete category, idempotent complete (∞,1)-category

Karoubi envelope, Cauchy completion

projector

idempotent monad, idempotent adjunction

In linear algebra a projector is a linear map $e \colon V \to V$ that “squares to itself” in that its composition with itself is again itself: $e \circ e = e$.

A projector $e$ leads to a decomposition of the vector space $V$ that it acts on into a direct sum of its kernel and its image:

The notion of projector is the special case of that of idempotent morphism.