mono-regular
A category is mono-regular when every monomorphism is regular, i.e. the equalizer of a pair of morphisms. Notice that this is not standard terminology, apparently the literature has no name for this yet. A preadditive category is mono-regular iff it is normal. The notion of a normal category is reserved for categories with zero morphisms, while mono-regular applies to all categories.
- Dual property: epi-regular
- Related properties: normal
- nLab Link
Relevant implications
- epi-regular andself-dual implies mono-regular
- groupoid implies directed limits andleft cancellative andmono-regular andpullbacks andself-dual andwell-powered
- mono-regular andpreadditive implies normal
- mono-regular andregular subobject classifier implies subobject classifier
- mono-regular andself-dual implies epi-regular
- mono-regular implies balanced
- normal implies mono-regular
- subobject classifier implies finitely complete andmono-regular
Examples
There are 32 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of combinatorial species
- 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 left modules over a division ring
- category of left modules over a ring
- category of M-sets
- category of non-empty sets
- category of pairs of sets
- category of pointed sets
- category of sets
- category of sheaves
- category of simplicial sets
- category of vector spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- discrete category on two objects
- dual of the category of sets
- empty category
- simplex category
- trivial category
- walking idempotent
- walking isomorphism
Counterexamples
There are 37 categories without this property.
- category of algebras
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of fields
- 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 monoids
- 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 small categories
- category of smooth manifolds
- category of topological spaces
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- poset [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- walking commutative square
- walking composable pair
- walking fork
- walking morphism
- walking parallel pair
- walking span
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.