cogenerator
An object of a category is called a cogenerator if for every pair of parallel morphisms , holds if for every morphism we have . Equivalently, the functor is faithful. This property refers to the existence of a cogenerator.
Relevant implications
Grothendieck topos implies locally presentable and cogenerator
For "locally presentable" see Prop. 3.4.16 in Handbook of Categorical Algebra Vol. 3. For "cogenerator" see the nLab.Grothendieck abelian implies cogenerator
Kashiwara-Schapira, Thm. 9.6.3thin and inhabited implies cogenerator
[dualized] Any object will be a generator for trivial reasons.cogenerator implies inhabited
[dualized] trivialself-dual and generator implies cogenerator
trivial by self-dualityself-dual and cogenerator implies generator
trivial by self-duality
Examples
- category of abelian groups
- category of Banach spaces with linear contractions
- category of finite orders
- category of finite sets
- category of finite sets and surjections
- category of free abelian groups
- category of left R-modules
- category of M-sets
- 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 non-empty sets
- category of pointed sets
- category of posets
- category of sets
- category of sets and relations
- category of simplicial sets
- category of vector spaces
- delooping of a non-trivial finite group
- delooping of an infinite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- 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
- trivial category
- walking isomorphism
- walking morphism
- walking parallel pair of morphisms
Counterexamples
- category of commutative rings
- category of fields
- category of finite abelian groups
- category of finite sets and bijections
- category of finite sets and injections
- category of finitely generated abelian groups
- category of groups
- category of monoids
- category of rings
- category of rngs
- category of small categories
- category of topological spaces
- empty category
Unknown
For these categories the database has no info if they satisfy this property or not.