effective congruences
A congruence (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism such that we have a cartesian square A category has effective congruences if every congruence in the category is effective.
- Dual property: effective cocongruences
- Related properties: mono-regular, normal, quotients of congruences
- nLab Link
Relevant implications
- additive andeffective congruences andfinitely complete implies normal
- coextensive andcoregular andmono-regular implies effective congruences andMalcev
- core-thin implies effective congruences andquotients of congruences
- effective cocongruences andself-dual implies effective congruences
- effective congruences andextensive implies mono-regular
- effective congruences andregular implies quotients of congruences
- effective congruences andself-dual implies effective cocongruences
- elementary topos implies disjoint finite coproducts andeffective congruences andepi-regular andfinitely cocomplete
- kernels andnormal andpreadditive implies effective congruences
- left cancellative implies coreflexive equalizers andeffective cocongruences andeffective congruences andreflexive coequalizers
- multi-algebraic implies effective congruences
- right cancellative implies coreflexive equalizers andeffective cocongruences andeffective congruences andreflexive coequalizers
Examples
There are 61 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of combinatorial species
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of countable sets
- 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 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 monoids
- category of non-empty sets
- category of pairs of sets
- category of pointed sets
- category of rings
- category of rngs
- category of semigroups
- category of sets
- category of sets and relations
- category of sets with finite-to-one maps
- category of sheaves
- category of simplicial sets
- category of torsion abelian groups
- category of vector 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 [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- simplex category
- trivial category
- walking commutative square
- walking composable pair
- walking coreflexive pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting
Counterexamples
There are 15 categories without this property.
- category of free abelian groups
- category of Hausdorff spaces
- category of locally ringed spaces
- category of measurable spaces
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of pointed topological spaces
- category of posets
- category of prosets
- category of schemes
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of torsion-free abelian groups
Unknown
There are 2 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.