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

• thin and  inhabited implies   generator

  Any object will be a generator for trivial reasons.
• locally presentable implies   locally essentially small and  well-powered and  well-copowered and  complete and  cocomplete and  generator

  For the non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.
• Grothendieck topos is equivalent to   elementary topos and  coproducts and  generator and  locally essentially small

  Mac Lane & Moerdijk, Appendix, Prop. 4.4
• Grothendieck abelian is equivalent to   abelian and  coproducts and  generator and  exact filtered colimits

  by definition
• generator implies   inhabited

  trivial
• self-dual and  generator implies   cogenerator

  trivial by self-duality
• self-dual and  cogenerator implies   generator

  trivial by self-duality

Examples

• 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

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

Unknown

For these categories the database has no info if they satisfy this property or not.

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