CatDat

generator

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

• Dual property: cogenerator
• nLab Link

Relevant implications

• cogenerator and  self-dual implies   generator
• generator and  self-dual implies   cogenerator
• generator implies   inhabited
• Grothendieck abelian is equivalent to   abelian and  coproducts and  exact filtered colimits and  generator
• Grothendieck topos is equivalent to   coproducts and  elementary topos and  generator and  locally essentially small
• inhabited and  thin implies   generator
• locally presentable implies   cocomplete and  complete and  generator and  locally essentially small and  well-copowered and  well-powered

Examples

There are 42 categories with this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of commutative rings
• category of finite orders
• category of finite sets
• category of finite sets and injections
• category of finitely generated abelian groups
• category of free abelian groups
• category of 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 monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of sets
• category of sets and relations
• category of simplicial sets
• category of small categories
• 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
• 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 5 categories without this property.

• category of fields
• category of finite abelian groups
• category of finite sets and bijections
• category of finite sets and surjections
• empty category

Unknown

There are 4 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 Z-functors