locally strongly finitely presentable
A category is a locally strongly finitely presentable if it is cocomplete and there is a set of strongly finitely presentable objects such that every object is a sifted colimit of objects from . There are several equivalent conditions:
- It is equivalent to the category of models of a many-sorted finitary algebraic theory.
- It is equivalent to the category of finite-product-preserving functors to from a small category with finite products (=Lawvere theory).
- It is equivalent to the Eilenberg–Moore category of a finitary (=filtered-colimit-preserving) monad on for some set .
- It is equivalent to the Eilenberg–Moore category of a sifted-colimit-preserving monad on for some set . (cf. [KR12, Proposition 3.3])
- Related properties: locally finitely presentable
- nLab Link
Relevant implications
- finitary algebraic implies locally strongly finitely presentable
- locally strongly finitely presentable implies locally finitely presentable
- locally strongly finitely presentable implies regular
Examples
There are 23 categories with this property.
- category of abelian groups
- category of algebras
- 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 monoids
- category of pairs of sets
- category of pointed sets
- category of rings
- category of rngs
- category of sets
- category of simplicial sets
- category of vector spaces
- poset of extended natural numbers
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
Counterexamples
There are 37 categories without this property.
- 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 continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of non-empty sets
- category of posets
- category of prosets
- category of schemes
- category of sets and relations
- category of small categories
- 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
- dual of the category of sets
- empty category
- poset [0,1]
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- walking fork
- walking parallel pair of morphisms
- walking span
Unknown
There are 5 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.