DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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Morphism of homological delta functors

Definition

Let $S = (S_{n}, \partial^{S})$ and $T = (T_{n}, \partial^{T})$ be homological delta functors. A morphism of homological delta functors $u : S \to T$ is a family of natural transformations $u_{n} : S_{n} \Rightarrow T_{n}, n \geq 0,$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ and every $n > 0$ , the square $\begin{array}{ccc} S_{n} (C) & \overset{\partial_{n}^{S}}{\to} & S_{n - 1} (A) \\ ↓^{u_{n} (C)} & ↓^{u_{n - 1} (A)} \\ T_{n} (C) & \overset{\partial_{n}^{T}}{\to} & T_{n - 1} (A) \end{array}$ commutes.

Equivalently, the degreewise natural transformations assemble into a morphism of the long exact sequences attached to every short exact sequence.

Depends on

Used by

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)