CatDat

extremal cogenerator

An object $Q$ of a category is called an extremal cogenerator if it is a cogenerator and for every morphism $f : A \to B$, if ${-}\circ f : \Hom(B,Q)\to\Hom(A,Q)$ is a bijection, then $f$ is an isomorphism. Equivalently, the functor $\Hom(-,Q) : \C^{\op} \to \Set^+$ is faithful and conservative. This property refers to the existence of an extremal cogenerator.
In a locally essentially small category with small products, it is also equivalent to the condition that the canonical morphism $$\textstyle A \to \prod_{f\in\Hom(A,Q)} Q$$ is an extremal monomorphism for every object $A$, explaining the terminology (see Prop. 5.3 at the nLab).
By definition, $Q$ is an extremal cogenerator if and only if $\{Q\}$ is an extremal cogenerating set.

• Dual property: extremal generator
• Related properties: cogenerator, extremal cogenerating set
• nLab Link

Relevant implications

• balanced andcogenerator implies extremal cogenerator
• cogenerator andgroupoid implies extremal cogenerator
• core-connected implies extremal cogenerator
• extremal cogenerating set andproducts andstrongly connected implies extremal cogenerator
• extremal cogenerator andself-dual implies extremal generator
• extremal cogenerator implies cogenerator andextremal cogenerating set
• extremal generator andself-dual implies extremal cogenerator

Examples

There are 55 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable sets
• category of filtered vector spaces
• category of finite ordered sets
• category of finite sets
• category of finite sets and surjections
• 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 Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of M-sets
• 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 non-empty sets
• category of pairs of sets
• category of partially ordered sets
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of set functions and commutative squares
• category of sets
• category of sets and relations
• category of sets with a distinguished subset
• category of sheaves
• category of simplicial sets
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• 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 on two objects
• dual of the category of sets
• simplex category
• trivial category
• walking coreflexive pair
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 29 categories without this property.

• category of algebras
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of countable groups
• category of fields
• category of finite abelian groups
• category of finite groups
• category of finite sets and bijections
• category of finite sets and injections
• category of finitely generated abelian groups
• category of groups
• category of Hausdorff spaces
• category of monoids
• category of rings
• category of rngs
• category of semigroups
• category of sets with finite-to-one maps
• category of small categories
• dual of the category of topological spaces
• empty category
• 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 fork

Unknown

There are 4 categories for which the database has no information on whether they satisfy this property. Please help us fill in the gaps by contributing to this project.

• category of free abelian groups
• category of locally ringed spaces
• category of schemes
• category of Z-functors