CatDat

filtered-colimit-stable monomorphisms

A category has filtered-colimit-stable monomorphisms if it has filtered colimits and for every filtered diagram of monomorphisms $(X_i \to Y_i)$ also their colimit $\colim_i X_i \to \colim_i Y_i$ is a monomorphism.

• Dual property: cofiltered-limit-stable epimorphisms
• Related properties: exact filtered colimits, filtered colimits

Relevant implications

• biproducts andfiltered colimits andfiltered-colimit-stable monomorphisms andproducts implies CIP
• Cauchy complete andessentially finite implies filtered-colimit-stable monomorphisms
• cofiltered-limit-stable epimorphisms andself-dual implies filtered-colimit-stable monomorphisms
• exact filtered colimits implies filtered-colimit-stable monomorphisms
• filtered colimits andleft cancellative implies filtered-colimit-stable monomorphisms
• filtered colimits andsubobject-trivial implies filtered-colimit-stable monomorphisms
• filtered-colimit-stable monomorphisms andself-dual implies cofiltered-limit-stable epimorphisms
• filtered-colimit-stable monomorphisms implies filtered colimits
• locally finitely multi-presentable implies filtered-colimit-stable monomorphisms

Examples

There are 53 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite sets and bijections
• category of finite sets and surjections
• category of groups
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• category of measurable spaces
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of semigroups
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• discrete category on two objects
• empty category
• poset [0,1]
• poset of extended natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 27 categories without this property.

• category of Banach spaces with linear contractions
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• category of finitely generated abelian groups
• category of free abelian groups
• category of Hausdorff spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of schemes
• category of sets and relations
• category of sets with finite-to-one maps
• category of smooth manifolds
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• dual of the category of sets
• dual of the category of topological spaces
• poset of natural numbers
• simplex category
• walking idempotent

Unknown

There is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.

• category of locally ringed spaces