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Canonical static concept record

Presheaf

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Summary

A Set-valued contravariant functor on a small category.

Record metadata

Canonical name: Presheaf
Concept ID: presheaf
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Category theory
Domains: Category theory, Algebraic geometry, Topology
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

small category $C$

Data

functor $F\colon C^{\mathrm{op}}\to\mathbf{Set}$

Notes

This atlas uses the standard $\mathbf{Set}$ -valued meaning so that the sheaf and sheaf-of-algebraic-objects branches are type-correct. Functors $C^{\mathrm{op}}\to D$ for a general category $D$ are $D$ -valued presheaves.

Concept sources

Stacks Project — Presheaves — Presheaves · reference · source ID stacks-presheaf
Wikipedia — Presheaf — Presheaf · expository · source ID wp-presheaf
nLab — Presheaf — Presheaf · reference · source ID nlab-presheaf
Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Functor Presheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_functor_to_presheaf_framework_specialization
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: contravariant Set-valued functor

Authored explanation

A presheaf is a functor from the opposite of a category to Set.

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Small category Presheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_category_presheaf
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose a Set-valued contravariant functor

Authored explanation

Choose a functor $C^{\mathrm{op}}\to\mathbf{Set}$ on the small category $C$ .

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Stacks Project — Presheaves — Presheaves · reference · source ID stacks-presheaf
Wikipedia — Presheaf — Presheaf · expository · source ID wp-presheaf
nLab — Presheaf — Presheaf · reference · source ID nlab-presheaf
Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Presheaf presheaf + site(construction junction)

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_sheaf__presheaf
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: presheaf data

Authored explanation

Supply the contravariant functor to which the sheaf condition will be applied.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Stacks Project — Presheaves — Presheaves · reference · source ID stacks-presheaf
Stacks Project — Sheaves — Sheaves · reference · source ID stacks-sheaf
Wikipedia — Sheaf (mathematics) — Sheaf (mathematics) · expository · source ID wp-sheaf
nLab — Sheaf — Sheaf · reference · source ID nlab-sheaf
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Presheaf Representable presheaf

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: category_presheaf_to_representable_presheaf
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: represented by an object

Authored explanation

Representable presheaves are the subclass of presheaves whose values and restriction maps are naturally isomorphic to hom-sets into one fixed representing object.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

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Presheaf

Summary

Record metadata

Carrier(s)

Data

Notes

Concept sources

Incoming relations (arrows to this concept)

Functor Presheaf

Authored explanation

How to interpret this relation type

Relation sources

Small category Presheaf

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Presheaf presheaf + site(construction junction)

Authored explanation

How to interpret this relation type

Relation sources

Presheaf Representable presheaf

Authored explanation

How to interpret this relation type

Relation sources