CatDat

locally presentable

Let $\kappa$ be a regular cardinal. A category is locally $\kappa$-presentable if it satisfies one of the following equivalent conditions:

1. It is $\kappa$-accessible and cocomplete.
2. It is $\kappa$-accessible and complete.
3. It is equivalent to the category of $\kappa$-limit-preserving functors to $\mathbf{Set}$ from a small category with $\kappa$-limits.
4. It is equivalent to the category of models of a small $\kappa$-limit sketch.

• Dual property: locally copresentable
• Related properties: accessible, cocomplete, locally finitely presentable, locally multi-presentable, locally poly-presentable, locally ℵ₁-presentable
• nLab Link

Relevant implications

• accessible andcomplete implies locally presentable
• Grothendieck abelian implies locally presentable
• Grothendieck topos implies cogenerator andinfinitary extensive andlocally presentable
• locally copresentable andlocally presentable implies essentially small
• locally copresentable andself-dual implies locally presentable
• locally presentable andself-dual implies locally copresentable
• locally presentable is equivalent to accessible andcocomplete
• locally ℵ₁-presentable implies locally presentable

Examples

There are 32 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of groups
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• category of metric spaces with ∞ allowed
• category of monoids
• category of pairs of sets
• category of pointed sets
• category of posets
• category of prosets
• category of rings
• category of rngs
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of vector spaces
• poset [0,1]
• poset of extended natural numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism

Counterexamples

There are 35 categories without this property.

• category of combinatorial species
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• 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 pointed topological spaces
• category of pseudo-metric spaces with non-expansive maps
• 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 countable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• dual of the category of sets
• empty category
• poset of natural numbers
• poset of ordinal numbers
• simplex category
• walking fork
• walking idempotent
• walking parallel pair
• walking span

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.

• category of Hausdorff spaces
• category of locally ringed spaces
• category of measurable spaces