CatDat

cogenerator

An object $Q$ of a category is called a cogenerator if for every pair of parallel morphisms $f,g : A \to B$, $f = g$ holds if for every morphism $h : B \to Q$ we have $h \circ f = h \circ g$. Equivalently, the functor $\mathrm{Hom}(-,Q) : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}^+$ is faithful. This property refers to the existence of a cogenerator. By definition, $Q$ is a cogenerator if and only if $\{Q\}$ is a cogenerating set.

• Dual property: generator
• Related properties: cogenerating set
• nLab Link

Relevant implications

• cogenerating set andproducts andzero morphisms implies cogenerator
• cogenerator andself-dual implies generator
• cogenerator implies cogenerating set andinhabited
• generator andself-dual implies cogenerator
• Grothendieck abelian implies cogenerator
• Grothendieck topos implies cogenerator andlocally presentable
• inhabited andthin implies cogenerator

Examples

There are 45 categories with this property.

• category of abelian groups
• category of abelian sheaves
• 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 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 pointed sets
• category of posets
• category of prosets
• category of sets
• category of sets and relations
• category of sheaves
• category of simplicial sets
• category of smooth manifolds
• category of topological spaces
• 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
• dual of the category of sets
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking fork
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms
• walking span

Counterexamples

There are 15 categories without this property.

• category of algebras
• category of commutative algebras
• category of commutative monoids
• 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
• empty category

Unknown

There are 5 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 combinatorial species
• category of Hausdorff spaces
• category of locally ringed spaces
• category of schemes
• category of Z-functors