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.

• Dual property: generator
• nLab Link

Relevant implications

• cogenerator and  self-dual implies   generator
• cogenerator implies   inhabited
• generator and  self-dual implies   cogenerator
• Grothendieck abelian implies   cogenerator
• Grothendieck topos implies   cogenerator and  locally presentable
• inhabited and  thin implies   cogenerator

Examples

There are 33 categories with this property.

• 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

There are 13 categories without this property.

• 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

There are 5 categories for which the database has no information on whether they satisfy this property.

• category of combinatorial species
• category of locally ringed spaces
• category of schemes
• category of smooth manifolds
• category of Z-functors