Verdier duality

A derived-sheaf duality that exchanges proper direct image with exceptional inverse image and extends Poincaré duality to singular spaces and maps.

Version

v1 · 2026-09-08 · History

Domain-specific #

7404

Origin domain

algebraic topology

Subdomain

algebraic topology

Core Idea

On suitable locally compact spaces, a dualizing complex defines a contravariant functor D, and for a map f the derived compact-support pushforward has right adjoint f-shriek; global forms relate compactly supported cohomology to dual cohomology. Sheaf localization preserves how data vary over singular strata, while derived Hom into the dualizing complex reverses support and cohomological degree in a functorial six-operations framework. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Verdier duality belongs to algebraic topology and is useful where the analyst can specify the typed algebraic topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the coefficient category, finiteness and constructibility conditions, dualizing complex, support convention, functors, and shifts are explicit. The scope is broad within that domain but bounded by the need for the coefficient category, finiteness and constructibility conditions, dualizing complex, support convention, functors, and shifts are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity

The abstraction clarifies a crowded vocabulary by making the coefficient category, finiteness and constructibility conditions, dualizing complex, support convention, functors, and shifts are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Verdier duality can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Verdier duality. Verdier duality compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed algebraic topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the coefficient category, finiteness and constructibility conditions, dualizing complex, support convention, functors, and shifts are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of algebraic topology because they reuse the typed algebraic topology carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Sheaf localization preserves how data vary over singular strata, while derived Hom into the dualizing complex reverses support and cohomological degree in a functorial six-operations framework., and type the carrier, state every parameter and convention in the definition, test that the coefficient category, finiteness and constructibility conditions, dualizing complex, support convention, functors, and shifts are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Verdier duality Domain-specific

Parents (1) — more general patterns this builds on

• Verdier duality is a kind of Duality Prime

  The proposed strict upward parent is prime:duality.

Hierarchy path (1) — routes to 1 parentless root

• Verdier duality → Duality

Neighborhood in Abstraction Space

Verdier duality sits in a crowded region of the domain-specific corpus (7^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Topology & Homology (37 abstractions)

Nearest neighbors

• Constructible sheaf — 0.94
• Poincaré space — 0.93
• Sheaf of algebras — 0.93
• Cotangent sheaf — 0.93
• Rational homotopy theory — 0.92

Computed from structural-signature embeddings · 2026-09-08