generalized variety
A category is a generalized variety if it has sifted colimits and there is a (small) set of strongly finitely presentable objects such that every object is a sifted colimit of objects from . Generalized varieties are like locally strongly finitely presentable categories but without colimits. The relation is similar as between finitely accessible and locally finitely presentable categories. This notion is defined in [AR01, Def. 3.6].
- Related properties: locally strongly finitely presentable, multi-algebraic, sifted colimits
Relevant implications
- generalized variety implies sifted colimits
- generalized variety implies ℵ₁-accessible
- locally strongly finitely presentable is equivalent to cocomplete andgeneralized variety
- multi-algebraic is equivalent to generalized variety andmulti-cocomplete
Examples
There are 28 categories with this property.
- category of abelian groups
- category of algebras
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- category of groups
- category of left modules over a division ring
- category of left modules over a ring
- category of M-sets
- category of monoids
- category of non-empty sets
- category of pairs of sets
- category of pointed sets
- category of rings
- category of rngs
- category of sets
- category of simplicial sets
- category of vector spaces
- discrete category on two objects
- empty category
- poset of extended natural numbers
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
- walking span
Counterexamples
There are 31 categories without this property.
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite sets and injections
- category of finitely generated abelian groups
- category of free abelian groups
- category of Hausdorff spaces
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with ∞ allowed
- category of pointed topological spaces
- category of posets
- category of prosets
- category of schemes
- category of sets and relations
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of Z-functors
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- dual of the category of sets
- poset [0,1]
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- walking idempotent
Unknown
There are 11 categories for which the database has no information on whether they satisfy this property. Please help us fill in the gaps by contributing to this project.
- category of abelian sheaves
- category of finite sets and bijections
- category of finite sets and surjections
- category of locally ringed spaces
- category of metric spaces with non-expansive maps
- category of pseudo-metric spaces with non-expansive maps
- category of sheaves
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- walking fork
- walking parallel pair