CatDat

coequalizers

A coequalizer of a pair of morphisms $f,g : A \to B$ is an object $C$ with a morphism $c : B \to C$ such that $c \circ f = c \circ g$ and which is universal with respect to this property. This property refers to the existence of coequalizers.

• Dual property: equalizers
• nLab Link

Relevant implications

• abelian is equivalent to   additive and  coequalizers and  epi-regular and  equalizers and  mono-regular
• binary coproducts and  coequalizers implies   pushouts
• binary coproducts and  pushouts implies   coequalizers
• cocomplete is equivalent to   coequalizers and  coproducts
• coequalizers and  countable coproducts implies   sequential colimits
• coequalizers and  groupoid implies   thin
• coequalizers and  left cancellative implies   thin
• coequalizers and  self-dual implies   equalizers
• coequalizers implies   Cauchy complete
• connected colimits is equivalent to   coequalizers and  wide pushouts
• equalizers and  self-dual implies   coequalizers
• finitely cocomplete is equivalent to   coequalizers and  finite coproducts
• thin implies   coequalizers and  right cancellative

Examples

There are 38 categories with this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of sets
• category of simplicial sets
• category of small categories
• category of topological spaces
• category of vector spaces
• category of Z-functors
• discrete category on two objects
• empty category
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

There are 12 categories without this property.

• category of fields
• category of finite sets and bijections
• category of finite sets and injections
• category of free abelian groups
• category of schemes
• category of sets and relations
• category of smooth manifolds
• 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
• walking parallel pair of morphisms

Unknown

There are 1 categories for which the database has no information on whether they satisfy this property.

• category of metric spaces with continuous maps