CatDat

effective congruences

A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square $$\begin{CD} E @> f >> X \\ @V g VV @VV h V \\ X @>> h > Y. \end{CD}$$ A category has effective congruences if every congruence in the category is effective.

• Dual property: effective cocongruences
• Related properties: Barr-exact, mono-regular, normal, quotients of congruences
• nLab Link

Relevant implications

• additive andeffective congruences implies normal
• Barr-exact is equivalent to effective congruences andregular
• coextensive andcoregular andmono-regular implies effective congruences
• core-thin implies effective 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 effective congruences
• kernels andnormal andpreadditive implies effective congruences
• left cancellative implies effective congruences
• multi-algebraic implies effective congruences
• right cancellative implies effective congruences

Examples

There are 68 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• 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 finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• 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 an infinite uncountable 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 17 categories without this property.

• category of filtered vector spaces
• 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
• dual of the category of topological spaces

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 pseudo-metric spaces with non-expansive maps