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Homological delta functor

Definition

Let $A$ and $B$ be abelian categories. A homological delta functor from $A$ to $B$ is a family of additive functors $T_{n} : A \to B, n \geq 0,$ together with, for every short exact sequence $0 \to A \to B \to C \to 0$ in the sense of Exact sequence and short exact sequence in an abelian category, a family of morphisms $\partial_{n} : T_{n} (C) \to T_{n - 1} (A), n > 0,$ such that:

the sequence $\dots \to T_{n} (A) \to T_{n} (B) \to T_{n} (C) \overset{\partial_{n}}{\to} T_{n - 1} (A) \to T_{n - 1} (B) \to T_{n - 1} (C) \to \dots \to T_{0} (C) \to 0$ is exact, and
for every morphism between short exact sequences, the connecting squares $\begin{array}{ccc} T_{n} (C) & \overset{\partial_{n}}{\to} & T_{n - 1} (A) \\ ↓ & ↓ \\ T_{n} (C^{'}) & \overset{\partial_{n}^{'}}{\to} & T_{n - 1} (A^{'}) \end{array}$ commute.

Thus the structure consists of both the long exact sequence and the naturality of its connecting maps.

Depends on

Used by

A nonnatural choice of connecting maps does not form a delta functor Counterexample
Effaceable homological delta functor in positive degrees Definition
Morphism of homological delta functors Definition
Homology as a homological delta functor Example
FALSE: a degree-zero natural transformation between delta functors always extends uniquely False statement
FALSE: any sequence of functors with long exact sequences is a delta functor False statement
Dimension shift for a homological delta functor effaced in the middle Lemma
Left derived functors form a homological delta functor Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)
Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)