CatDat

locally presentable

Let $\kappa$ be a regular cardinal. A category is locally $\kappa$-presentable if it is locally essentially small*, cocomplete, and there is a set of $\kappa$-presentable objects $S$ such that every object is a $\kappa$-filtered colimit of objects in $S$. A category is locally presentable if it is locally $\kappa$-presentable for some regular cardinal $\kappa$.
*Many authors assume the category to be locally small, but this is inconvenient since then locally presentable categories would not be invariant under equivalences of categories.

• Related properties: locally finitely presentable, locally ℵ₁-presentable
• nLab Link

Relevant implications

• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• locally finitely presentable implies   locally presentable

  Locally finitely presentable categories are by definition the locally $\aleph_0$-presentable categories.
• locally presentable and  self-dual implies   thin

  This follows from Adamek-Rosicky, Thm. 1.64.
• locally ℵ₁-presentable implies   locally presentable

  trivial
• Grothendieck topos implies   locally presentable and  cogenerator

  For "locally presentable" see Prop. 3.4.16 in Handbook of Categorical Algebra Vol. 3. For "cogenerator" see the nLab.
• Grothendieck abelian implies   locally presentable

  See Deriving Auslander's formula, Cor. 5.2, or Sheafifiable homotopy model categories, Prop. 3.10.

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of groups
• category of left R-modules
• category of M-sets
• category of metric spaces with ∞ allowed
• 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 [0,1]
• partial order of extended natural numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

• 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 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 of natural numbers
• partial order of ordinal numbers
• 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