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A category whose objects and morphisms form sets.
Smallness is relative to the ambient universe or foundational convention.
wp-categorynlab-categorystacks-categoriesriehl-category-contextEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require the collections of objects and morphisms to be sets.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-categorynlab-categorystacks-categoriesriehl-category-contextThis is an authored directed relation from the source endpoint to the target endpoint.
A monoid is precisely a category with one object: monoid elements are endomorphisms and multiplication is composition.
Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.
wp-monoidwp-categorynlab-categoryriehl-category-contextEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Retain objects as vertices, morphisms as arrows, and the source/target maps. Parallel arrows and loops remain.
Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.
wp-categorynlab-categorystacks-categorieswp-directed-graphdiestel-graph-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Choose a functor on the small category .
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.
stacks-presheafwp-presheafnlab-presheafriehl-category-contextThis is an authored directed relation from the source endpoint to the target endpoint.
Fixing an object of a small category canonically gives the contravariant hom-functor .
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.
riehl-category-contextThis is an authored directed relation from the source endpoint to the target endpoint.
Choose covering sieves or covering families on the small category satisfying the Grothendieck-topology axioms.
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.
stacks-sitesstacks-groth-topologywp-sitenlab-siteriehl-category-context