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. By definition, $G$ is a generator if and only if $\{G\}$ is a generating set.

• Dual property: cogenerator
• Related properties: generating set
• nLab Link

Relevant implications

• cogenerator andself-dual implies generator
• coproducts andgenerating set andzero morphisms implies generator
• finitary algebraic implies generator
• generator andself-dual implies cogenerator
• generator implies generating set andinhabited
• Grothendieck abelian is equivalent to abelian andcoproducts andexact filtered colimits andgenerator
• inhabited andthin implies generator

Examples

There are 54 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• 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 Hausdorff spaces
• 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 monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of prosets
• 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
• 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 7 categories without this property.

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

Unknown

There are 4 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 locally ringed spaces
• category of sheaves
• category of Z-functors