equalizers
An equalizer of a pair of morphisms is an object with a morphism such that and which is universal with respect to this property. This property refers to the existence of equalizers.
- Dual property: coequalizers
- Related properties: coreflexive equalizers, finitely complete, kernels
- nLab Link
Relevant implications
- binary products andcoreflexive equalizers implies equalizers
- binary products andequalizers implies pullbacks
- binary products andpullbacks implies equalizers
- coequalizers andself-dual implies equalizers
- complete is equivalent to equalizers andproducts
- connected limits is equivalent to equalizers andwide pullbacks
- countable products andequalizers implies sequential limits
- equalizers andfinitely cocomplete andlocally cocartesian coclosed implies coregular
- equalizers andright cancellative implies thin
- equalizers andself-dual implies coequalizers
- equalizers andsemi-strongly connected implies cofiltered
- equalizers andzero morphisms implies kernels
- equalizers implies Cauchy complete
- equalizers implies coreflexive equalizers
- finitely complete is equivalent to equalizers andfinite products
- kernels andpreadditive implies equalizers
- thin implies equalizers andleft cancellative
Examples
There are 59 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 fields
- 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 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 continuous maps
- category of metric spaces with non-expansive maps
- category of metric spaces with ∞ allowed
- category of monoids
- 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 schemes
- category of sets
- 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 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
- trivial category
- walking commutative square
- walking composable pair
- walking fork
- walking isomorphism
- walking morphism
- walking span
Counterexamples
There are 11 categories without this property.
- category of finite sets and bijections
- category of finite sets and surjections
- category of non-empty sets
- category of sets and relations
- category of smooth manifolds
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of the additive monoid of natural numbers
- simplex category
- walking idempotent
- walking parallel pair
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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