CatDat

locally finitely presentable

A category is locally finitely presentable if it is locally essentially small*, cocomplete, and there is a set $S$ of finitely presentable objects such that every object is a filtered colimit of objects in $S$. This is the same as being locally $\aleph_0$-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 $\aleph_0$-presentable categories.
• locally finitely presentable implies   exact filtered colimits

  Special case of Adamek-Rosicky, Prop. 1.59 with $\lambda = \aleph_0$.
• locally finitely presentable implies   locally ℵ₁-presentable

  trivial
• finitary 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.

• category of locally ringed spaces
• category of measurable spaces