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 every monomorphism is a kernel, and this type of category is commonly known as a normal category. We avoid this terminology here since it only applies to a certain type of categories, but mono-regular applies to all categories.
- Dual property: epi-regular
- nLab Link
Relevant implications
- abelian is equivalent to additive and coequalizers and epi-regular and equalizers and mono-regular
- epi-regular and self-dual implies mono-regular
- groupoid implies filtered limits and left cancellative and mono-regular and pullbacks and self-dual and well-powered
- mono-regular and self-dual implies epi-regular
- mono-regular implies balanced
- subobject classifier implies finitely complete and mono-regular
Examples
There are 24 categories with this property.
- category of abelian groups
- category of combinatorial species
- category of finite abelian groups
- category of finite orders
- 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 R-modules
- category of M-sets
- category of non-empty sets
- category of pointed sets
- category of sets
- category of simplicial sets
- category of vector spaces
- category of Z-functors
- delooping of a non-trivial finite group
- delooping of an infinite group
- discrete category on two objects
- empty category
- trivial category
- walking isomorphism
Counterexamples
There are 24 categories without this property.
- category of Banach spaces with linear contractions
- category of commutative rings
- category of fields
- category of free abelian groups
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of monoids
- category of posets
- category of rings
- category of rngs
- 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
- 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
- walking morphism
- walking parallel pair of morphisms
Unknown
There are 3 categories for which the database has no information on whether they satisfy this property.