extremal generating collection

This property refers to the existence of an extremal generating collection. A collection of objects $S \subseteq \Ob(\C)$ in a category $\C$ is called an extremal generating collection if it is a generating collection and for every morphism $f : A \to B$, $f$ is an isomorphism if and only if for every object $G \in S$ we have $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is a bijection. Equivalently, the functor $$(\Hom(G,-))_{G \in S} : \C \to (\Set^+)^S$$ is faithful and conservative. If $\C$ is locally essentially small and has coproducts, it is also equivalent to the condition that the canonical morphism $$\textstyle\bigsqcup_{G\in S} \bigsqcup_{f\in\Hom(G,A)} G \to A$$ is an extremal epimorphism for every object $A$, explaining the terminology (see Prop. 5.3 at the nLab). The term "extremal generating set" is more common, but within our foundations, the collection $S$ is not necessarily a set; it is just isomorphic to a set.

Relevant implications

• accessible implies extremal generating collection
• balanced andgenerating collection implies extremal generating collection
• coproducts andextremal generating collection andstrongly connected implies extremal generator
• essentially small implies extremal generating collection
• extremal cogenerating collection andself-dual implies extremal generating collection
• extremal generating collection andleft cancellative andlocally finite andsemi-strongly connected implies essentially small
• extremal generating collection andself-dual implies extremal cogenerating collection
• extremal generating collection implies generating collection
• extremal generator implies extremal generating collection
• generating collection andgroupoid implies extremal generating collection

Examples

There are 104 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 combinatorial species
• 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 directed graphs
• category of fields
• category of fields of characteristic zero
• category of filtered vector spaces
• 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 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 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 metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of 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 set functions and commutative squares
• category of sets
• category of sets and relations
• category of sets with a binary relation
• category of sets with a distinguished subset
• category of sets with finite-to-one maps
• category of sheaves
• 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
• 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
• empty category
• forked commutative square
• 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]
• simplex category
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 13 categories without this property.

• category of Hausdorff spaces
• category of large families of sets
• 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 pointed topological spaces
• category of schemes
• category of topological spaces
• category of uniform spaces
• cocompletion of a discrete–pair join
• partially ordered collection of ordinal numbers

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.

• category of Z-functors