extremal generator
An object of a category is called an extremal generator if it is a generator and for every morphism , if is a bijection, then is an isomorphism. Equivalently, the functor is faithful and conservative. This property refers to the existence of an extremal generator.In a locally essentially small category with coproducts, it is also equivalent to the condition that the canonical morphism is an extremal epimorphism for every object , explaining the terminology (see Prop. 5.3 at the nLab).By definition, is an extremal generator if and only if is an extremal generating collection.
Relevant implications
Examples
There are 87 categories with this property.
- category of abelian groups
- category of abelian sheaves
- category of algebras
- category of Banach spaces with linear contractions
- category of cochain complexes of abelian groups
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of countable groups
- category of countable sets
- category of countable-dimensional vector spaces
- category of extended metric spaces
- category of filtered vector spaces
- category of finite ordered sets
- category of finite sets
- category of finite sets and injections
- category of finite sets of cardinality a power of 3
- category of finite sets of even cardinality
- category of finite sets of odd cardinality
- category of finite-dimensional vector spaces
- category of finite-dimensional vector spaces [countable field]
- category of finite-dimensional vector spaces [finite field]
- category of finite-dimensional vector spaces [uncountable field]
- category of finitely generated abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated groups
- category of finitely generated projective modules over the ring of dual numbers
- category of free abelian groups
- category of graded abelian groups
- category of graded modules over a graded ring
- category of groups
- category of Jónsson-Tarski algebras
- category of large families of abelian groups
- category of large vector spaces over a large field with a small basis
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of left modules over a ring
- category of long transfinite sequences of abelian groups
- category of M-sets
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of monoids
- category of non-empty sets
- category of partially ordered sets
- category of partially ordered sets without isolated points
- category of pointed sets
- category of preordered sets
- category of pseudo-metric spaces with non-expansive maps
- category of rings
- category of rngs
- category of semigroups
- category of sequences of abelian groups
- category of sets
- category of sets and relations
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of sets with finite-to-one maps
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of vector spaces
- category of Z-sets
- delooping of a group
- delooping of a non-trivial finite group
- delooping of an infinite countable group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category of sets
- discrete category on two objects
- dual of the category of sets
- dual of the category of topological spaces
- indiscrete category of sets
- simplex category
- trivial category
- walking coreflexive pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking splitting
Counterexamples
There are 48 categories without this property.
- category of combinatorial species
- category of connected sequences of sets
- category of empty-or-finite pairs of sets
- category of F(I)-sets
- category of fields
- category of fields of characteristic zero
- category of finite abelian groups
- category of finite groups
- category of finite sets and bijections
- category of finite sets and surjections
- category of finite Z-sets
- category of Hausdorff spaces
- category of large families of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of large families of vector spaces which are mostly zero
- category of locally ringed spaces
- category of measurable spaces
- category of pairs of sets
- category of pointed topological spaces
- category of quivers
- category of quivers with finite components
- category of schemes
- category of sequences of sets
- category of set functions and commutative squares
- category of sets and bijections
- category of sets equipped with a binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets with a distinguished subset
- category of sheaves
- category of topological spaces
- category of uniform spaces
- category of Z-functors
- category of ω-sets
- cocompletion of a discrete–pair join
- empty category
- forked commutative square
- partially ordered collection of ordinal numbers
- partially ordered set of extended natural numbers
- partially ordered set of natural numbers
- preordered set of integers w.r.t. divisibility
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking span
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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