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Canonical static concept record

Grothendieck topos

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Summary

A category equivalent to the category of sheaves on a site.

Record metadata

Canonical name: Grothendieck topos
Concept ID: grothendieck_topos
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)

category $E$

Axioms / constraints

$E$ is equivalent to $\operatorname{Sh}(C,J)$ for some small site $(C,J)$

Notes

A site presents a Grothendieck topos, but is not itself the carrier of the topos; different sites can present equivalent topoi.

Concept 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

Incoming relations (arrows to this concept)

Each relation below ends 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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.