generalized variety
A category is a generalized variety if it has sifted colimits and there is an essentially small collection 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 to that between finitely accessible and locally finitely presentable categories. This notion is defined in Adámek–Rosický, Def. 3.6.
Related finitary algebraic, multi-algebraic, sifted colimits
Relevant implications
Examples
There are 52 categories with this property.
- category of abelian groups
- category of algebras
- category of cochain complexes of abelian groups
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- category of fields of characteristic zero
- category of finite sets and bijections
- category of finite sets and surjections
- category of graded abelian groups
- category of graded modules over a graded ring
- category of groups
- category of Jónsson-Tarski algebras
- category of left modules over a division ring
- category of left modules over a non-semisimple 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 quivers
- category of rings
- category of rngs
- category of semigroups
- category of sequences of abelian groups
- category of sequences of sets
- category of set functions and commutative squares
- category of sets
- category of simplicial sets
- category of vector spaces
- category of Z-sets
- delooping of a group
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- discrete category on two objects
- empty category
- forked commutative square
- indiscrete category of sets
- partially ordered set of extended natural numbers
- trivial category
- walking commutative square
- walking composable pair
- walking coreflexive pair
- walking fork
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting
Counterexamples
There are 80 categories without this property.
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of compact Hausdorff spaces
- category of connected sequences of sets
- category of coproducts of Euclidean spaces
- category of countable groups
- category of countable sets
- category of countable-dimensional vector spaces
- category of empty-or-finite pairs of sets
- category of extended metric spaces
- category of F(I)-sets
- category of filtered vector spaces
- 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 finite sets of cardinality a power of 3
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite Z-sets
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of finitely generated abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated groups
- category of finitely generated projective modules over the ring of dual numbers
- category of free abelian groups
- category of Hausdorff spaces
- category of large families of abelian groups
- category of large families of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of large families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- category of locally ringed spaces
- category of long transfinite sequences of abelian groups
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of partially ordered sets
- category of partially ordered sets without isolated points
- category of pointed topological spaces
- category of preordered sets
- category of quivers with finite components
- category of schemes
- category of sets and bijections
- category of sets and relations
- category of sets equipped with a binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets with a distinguished subset
- category of sets with finite-to-one maps
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of uniform spaces
- category of Z-functors
- category of ω-sets
- cocompletion of a discrete–pair join
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category of sets
- dual of the category of sets
- dual of the category of topological spaces
- partially ordered collection of ordinal numbers
- partially ordered set of natural numbers
- preordered set of integers w.r.t. divisibility
- real interval [0,1]
- simplex category
- walking idempotent
Unknown
There are 3 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.