Effaceable Functor¶
An additive functor for which every object embeds by a map whose induced functor morphism is zero.
Core Idea¶
An additive functor \(F\) between abelian categories is effaceable when each source object \(A\) has some monomorphism \(u:A\hookrightarrow B\) whose induced morphism \(F(u)\) is zero. This does not require \(F(B)=0\): a zero arrow may point into a nonzero object. When every positive degree of a cohomological \(\delta\)-functor has this property, its degree-zero natural transformations extend uniquely through all degrees, making the family universal.
Scope of Application¶
The condition is used in homological algebra to compare cohomology constructions. Positive right derived functors are effaceable by embeddings into injective objects under standard enough-injectives assumptions. Čech cohomology on presheaves for suitable covers gives another setting where effaceability supports comparison with derived functors.
Clarity¶
Effaceability is a condition on the induced map, not necessarily the target functor value. For each object and, in the degreewise theorem, each positive degree, a different embedding may be chosen. A single additive functor is not automatically a \(\delta\)-functor, and the universal-extension theorem needs the latter's long-exact-sequence structure.
Manages Complexity¶
Instead of comparing higher cohomology terms one by one, show the local zero-map condition and use exactness to force a unique compatible comparison from degree zero. This compresses an all-degrees identification into a repeatable proof obligation without hiding its hypotheses.
Abstract Reasoning¶
Embed \(A\) into \(B\) with \(F^n(A)\to F^n(B)\) zero. The short exact sequence \(0\to A\to B\to B/A\to0\) supplies a long exact sequence in which the relevant degree-\(n\) term can be described through a connecting map from the previous degree. This is the inductive bridge behind universality. An injective \(B\) with \(F^n(B)=0\) is one easy way to obtain the required zero map, not its definition.
Knowledge Transfer¶
The argument recurs when different cohomology constructions admit suitable effacing embeddings, even though their objects and formulas differ. Live Functor is the strict genus; Delta Functor and Derived Functor are related contexts, not whole-node parents of an arbitrary effaceable additive functor.
Relationships to Other Abstractions¶
Current abstraction Effaceable Functor Domain-specific
Parents (1) — more general patterns this builds on
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Effaceable Functor is a kind of Functor Domain-specific
An effaceable additive functor is a functor constrained by every-object monic effacement.
Hierarchy paths (4) — routes to 4 parentless roots
- Effaceable Functor → Functor → Category → Associativity → Invariance
- Effaceable Functor → Functor → Function (Mapping)
- Effaceable Functor → Functor → Category → Closure
- Effaceable Functor → Functor → Category → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Effaceable Functor sits in a sparse region of the domain-specific corpus (64th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Categories, Sheaves & Homotopy (21 abstractions)
Nearest neighbors
- Auslander–Reiten theory — 0.85
- Functor — 0.85
- Normal Morphism — 0.84
- Simplicial Localization — 0.84
- Categorical Trace — 0.84
Computed from structural-signature embeddings · 2026-10-08