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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Site

Open this concept in the interactive graph Read the Markdown equivalent

Summary

A category equipped with a Grothendieck topology or a pretopology of covering families.

Record metadata

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

Carrier(s)

small category $C$

Data

covering sieves or covering families

Axioms / constraints

identity/maximal coverings
pullback stability
transitivity or local character

Canonically induces

category of sheaves

Notes

This atlas uses sites small relative to the ambient universe. Large-site conventions require explicit universe management.

Concept sources

Stacks Project — Sites and Sheaves — introduction — Sites and Sheaves — introduction · reference · source ID stacks-sites
Stacks Project — Grothendieck topologies — Grothendieck topologies · reference · source ID stacks-groth-topology
Wikipedia — Site (mathematics) — Site (mathematics) · expository · source ID wp-site
nLab — Site — Site · reference · source ID nlab-site
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.

Small category Site

Permalink to relation

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

Relation ID: category_to_site
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + Grothendieck covering structure

Authored explanation

Choose covering sieves or covering families on the small category satisfying the Grothendieck-topology axioms.

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 — Sites and Sheaves — introduction — Sites and Sheaves — introduction · reference · source ID stacks-sites
Stacks Project — Grothendieck topologies — Grothendieck topologies · reference · source ID stacks-groth-topology
Wikipedia — Site (mathematics) — Site (mathematics) · expository · source ID wp-site
nLab — Site — Site · reference · source ID nlab-site
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.

Site Grothendieck topos

Permalink to relation

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

Relation ID: e_site_topos
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form the sheaf topos

Authored explanation

Form $\operatorname{Sh}(C,J)$ , the category of sheaves of sets on the site. This is a canonical construction from a site, not an equivalence between the site and the resulting topos.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Stacks Project — Topoi — Topoi · reference · source ID stacks-topos
Wikipedia — Grothendieck topos — Grothendieck topos · expository · source ID wp-groth-topos
nLab — Topos — Topos · reference · source ID nlab-topos
Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Site presheaf + site(construction junction)

Permalink to relation

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

Relation ID: j_sheaf__site
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: covering structure

Authored explanation

Supply the covers on the presheaf’s domain category.

How to interpret this relation type

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

Relation sources

Stacks Project — Sites and Sheaves — introduction — Sites and Sheaves — introduction · reference · source ID stacks-sites
Stacks Project — Sheaves — Sheaves · reference · source ID stacks-sheaf
Wikipedia — Site (mathematics) — Site (mathematics) · expository · source ID wp-site
nLab — Site — Site · reference · source ID nlab-site
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

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Site

Summary

Record metadata

Carrier(s)

Data

Axioms / constraints

Canonically induces

Notes

Concept sources

Incoming relations (arrows to this concept)

Small category Site

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Site Grothendieck topos

Authored explanation

How to interpret this relation type

Relation sources

Site presheaf + site(construction junction)

Authored explanation

How to interpret this relation type

Relation sources