CatDat

generating set

A set of objects $S$ is called a generating set if for every pair of parallel morphisms $f,g : A \rightrightarrows B$, $f = g$ holds if and only if for every morphism $h : G \to A$ with $G \in S$ we have $f \circ h = g \circ h$. Equivalently, the functor $(\hom(G,-))_{G \in S} : \mathcal{C} \to (\mathbf{Set}^+)^S$ is faithful. This property refers to the existence of a generating set.

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

Relevant implications

• cogenerating set andself-dual implies generating set
• coproducts andgenerating set andzero morphisms implies generator
• essentially small implies generating set
• generating set andself-dual implies cogenerating set
• generator implies generating set andinhabited
• Grothendieck topos is equivalent to coproducts andelementary topos andgenerating set andlocally essentially small
• locally presentable implies cocomplete andcomplete andgenerating set andlocally essentially small andwell-copowered andwell-powered

Examples

There are 62 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• 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 pairs of 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 sheaves
• 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
• empty category
• 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 is 1 category without this property.

• category of schemes

Unknown

There are 2 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 locally ringed spaces
• category of Z-functors