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

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Universal delta functor

Definition

Let $T$ be a delta functor.

If $T = (T_{n}, \partial^{T})$ is homological, then $T$ is universal when for every homological delta functor $S = (S_{n}, \partial^{S})$ and every natural transformation $u_{0} : S_{0} \Rightarrow T_{0},$ there exists a unique morphism of homological delta functors $u : S \to T$ whose degree-zero component is $u_{0}$ .

If $T = (T^{n}, \partial_{T})$ is cohomological, then $T$ is universal when for every cohomological delta functor $S = (S^{n}, \partial_{S})$ and every natural transformation $u^{0} : T^{0} \Rightarrow S^{0},$ there exists a unique morphism of cohomological delta functors $u : T \to S$ whose degree-zero component is $u^{0}$ .

So universality says that the higher-degree components are forced by the degree-zero data and the delta-functor axioms.

Depends on

Used by

A morphism between universal delta functors is determined in degree zero Corollary
Universal delta functors extending the same degree-zero functor are uniquely isomorphic Corollary
FALSE: a degree-zero natural transformation between delta functors always extends uniquely False statement
An exact base functor has the trivial universal delta functor Proposition
Satellites give the first derived functor Proposition
Effaceable cohomological delta functors are universal Theorem
Effaceable homological delta functors are universal 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)
The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)