CatDat

cogenerating set

A set of objects $S$ is called a cogenerating set if for every pair of parallel morphisms $f,g : A \rightrightarrows B$, $f = g$ holds if and only if for every morphism $h : B \to Q$ with $Q \in S$ we have $h \circ f = h \circ g$. Equivalently, the functor $(\Hom(-,Q))_{Q \in S} : \C^{\op} \to (\Set^+)^S$ is faithful. This property refers to the existence of a cogenerating set.
In a locally essentially small category with small products, it is also equivalent to the condition that the canonical morphism $$A \to \textstyle\prod_{Q\in S} \prod_{f\in\Hom(A,Q)} Q$$ is a monomorphism for every object $A$.

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

Relevant implications

• balanced andcogenerating set implies extremal cogenerating set
• cogenerating set anddisjoint coproducts implies cogenerator
• cogenerating set andgroupoid implies extremal cogenerating set
• cogenerating set andproducts andstrongly connected implies cogenerator
• cogenerating set andself-dual implies generating set
• cogenerator implies cogenerating set
• extremal cogenerating set implies cogenerating set
• generating set andself-dual implies cogenerating set
• thin implies cogenerating set

Examples

There are 81 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• 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-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 free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of Jónsson-Tarski algebras
• 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 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 sequences of abelian groups
• 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 uniform spaces
• 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 on two objects
• dual of the category of sets
• dual of the category of topological spaces
• 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]
• 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 algebras
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• 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

Unknown

There are 3 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 locally ringed spaces
• category of schemes
• category of Z-functors