CatDat

quotients of congruences

A congruence (or internal equivalence relation) on an object $X$ of a category is a parallel pair $p_1, p_2 : E \rightrightarrows X$ which is jointly monomorphic, and such that for every object $T$, the image of $(p_1 \circ {-}, p_2 \circ {-}) : \operatorname{Hom}(T, E) \to \operatorname{Hom}(T, X)^2$ is an equivalence relation. The category has quotients of congruences if for each such congruence, there exists a coequalizer of $p_1$ and $p_2$. Note that in the case of a category with binary powers, the corresponding subobjects of $X \times X$ are also commonly referred to as congruences, or as internal equivalence relations.

• Dual property: coquotients of cocongruences
• Related properties: coequalizers, cokernels, regular
• nLab Link

Relevant implications

• cokernels andkernels andpreadditive implies quotients of congruences
• coquotients of cocongruences andself-dual implies quotients of congruences
• core-thin implies quotients of congruences
• preadditive andquotients of congruences andregular implies cokernels
• quotients of congruences andself-dual implies coquotients of cocongruences
• reflexive coequalizers implies quotients of congruences

Examples

There are 69 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 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 Hausdorff spaces
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• 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 sets
• category of sets and relations
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of topological spaces
• 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 fork
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 4 categories without this property.

• category of free abelian groups
• category of metric spaces with continuous maps
• category of schemes
• category of smooth manifolds

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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