locally finitely presentable
A category is locally finitely presentable if it is locally essentially small*, cocomplete, and there is a set of finitely presentable objects such that every object is a filtered colimit of objects in . This is the same as being locally -presentable.
*Many authors assume the category to be locally small, but this is inconvenient since then locally finitely presentable categories would not be invariant under equivalences of categories.
- Related properties: locally presentable, locally ℵ₁-presentable
- nLab Link
Relevant implications
locally finitely presentable implies locally presentable
Locally finitely presentable categories are by definition the locally -presentable categories.locally finitely presentable implies exact filtered colimits
Special case of Adamek-Rosicky, Prop. 1.59 with .locally finitely presentable implies locally ℵ₁-presentable
trivialfinitary algebraic implies locally finitely presentable
Adamek-Rosicky, Cor. 3.7
Examples
- category of abelian groups
- category of commutative rings
- category of groups
- category of left R-modules
- category of M-sets
- category of monoids
- category of pointed sets
- category of posets
- category of rings
- category of rngs
- category of sets
- category of simplicial sets
- category of small categories
- category of vector spaces
- partial order of extended natural numbers
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of fields
- category of finite abelian groups
- category of finite orders
- category of finite sets
- category of finite sets and bijections
- category of finite sets and injections
- category of finite sets and surjections
- category of finitely generated abelian groups
- category of free abelian groups
- category of metric spaces with ∞ allowed
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of non-empty sets
- category of schemes
- category of sets and relations
- category of smooth manifolds
- category of topological spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- empty category
- partial order [0,1]
- partial order of natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- walking parallel pair of morphisms
Unknown
For these categories the database has no info if they satisfy this property or not.