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
subobject classifier implies finitely complete and mono-regular
The first part holds by convention, and the second part: any monomorphism is the equalizer of \chi_U,\chi_X : X \to \Omega$.abelian is equivalent to additive and equalizers and coequalizers and mono-regular and epi-regular
by definitiongroupoid implies self-dual and mono-regular and pullbacks and filtered limits and left cancellative and well-powered
easymono-regular implies balanced
Any regular monomorphism that is an epimorphism must be an isomorphism.self-dual and mono-regular implies epi-regular
trivial by self-dualityself-dual and epi-regular implies mono-regular
trivial by self-duality
Examples
- 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
- 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
For these categories the database has no info if they satisfy this property or not.