Six operations¶
The six-functor formalism relating pullback, pushforward, extraordinary pullback and pushforward, tensor product and internal Hom across geometric categories.
Core Idea¶
Existence and adjunctions depend on the category and morphism class, derived and underived notation differ and projection base-change and duality formulas require hypotheses. A morphism induces paired covariant and contravariant functors, while tensor and internal Hom operate within each fiber category; adjunction, base-change and duality compatibilities make cohomological calculations transport formally across contexts. 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¶
Six operations belongs to homological algebra and is useful where the analyst can specify the typed homological algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the geometric or categorical context and morphism f, source and target derived categories, f-star f-lower-star f-shriek and f-upper-shriek functors, tensor and internal Hom, adjoint pairs, units counits and projection formula, base-change maps, duality and hypotheses for existence and equivalence are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the geometric or categorical context and morphism f, source and target derived categories, f-star f-lower-star f-shriek and f-upper-shriek functors, tensor and internal Hom, adjoint pairs, units counits and projection formula, base-change maps, duality and hypotheses for existence and equivalence 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.
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 Six operations. Six operations 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¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed homological algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
- Lock the constitutive rule. Express the geometric or categorical context and morphism f, source and target derived categories, f-star f-lower-star f-shriek and f-upper-shriek functors, tensor and internal Hom, adjoint pairs, units counits and projection formula, base-change maps, duality and hypotheses for existence and equivalence are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of homological algebra because they reuse the typed homological algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A morphism induces paired covariant and contravariant functors, while tensor and internal Hom operate within each fiber category; adjunction, base-change and duality compatibilities make cohomological calculations transport formally across contexts., and type the carrier, state every parameter and convention in the definition, test that the geometric or categorical context and morphism f, source and target derived categories, f-star f-lower-star f-shriek and f-upper-shriek functors, tensor and internal Hom, adjoint pairs, units counits and projection formula, base-change maps, duality and hypotheses for existence and equivalence are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Six operations Domain-specific
Parents (1) — outgoing typed relations
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Six operations is a kind of Composition Prime
It remains its own entry because its identity is fixed by the geometric or categorical context and morphism f, source and target derived categories, f-star f-lower-star f-shriek and f-upper-shriek functors, tensor and internal Hom, adjoint pairs, units counits and projection formula, base-change maps, duality and hypotheses for existence and equivalence.
Hierarchy path (1) — routes to 1 parentless root
- Six operations → is a kind of Composition → presupposes Gestalt Principles → is a decomposition of Holism
Neighborhood in Abstraction Space¶
Six operations sits in a crowded region of the domain-specific corpus (4th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Homological Algebra Constructions (19 abstractions)
Nearest neighbors
- Zig-zag lemma — 0.93
- Hall algebra — 0.93
- Bar complex — 0.93
- Traced monoidal category — 0.93
- Chain complex — 0.93
Computed from structural-signature embeddings · 2026-10-25