generating collection

This property refers to the existence of a generating collection. A generating collection of a category $\C$ is a collection of objects $S \subseteq \Ob(\C)$ that is essentially small (i.e., isomorphic to a set) and has the following property: if parallel morphisms $f,g : A \rightrightarrows B$ satisfy $f \circ h = g \circ h$ for every morphism $h : G \to A$ with $G \in S$, then $f = g$. Equivalently, the functor $$(\Hom(G,-))_{G \in S} : \C \to (\Set^+)^S$$ is faithful. If $\C$ is locally essentially small and has coproducts, this is also equivalent to the condition that the canonical morphism $$\textstyle\bigsqcup_{G\in S} \bigsqcup_{f\in\Hom(G,A)} G \to A$$ is an epimorphism for every object $A$.

The condition that $S$ is essentially small is important since, without it, every category would have a generating collection, namely its collection of all objects. Notice that $S$ is not assumed to be a set "on the nose", since its objects do not need to be sets; see our foundations for more background. This is why we have not adopted the more common terminology of a "generating set". We did not choose "essentially small generating collection" either since it is too cumbersome and diverges too much from the literature.

Relevant implications

• balanced andgenerating collection implies extremal generating collection
• cocomplete andgenerating collection andlocally essentially small andwell-copowered implies total
• cogenerating collection andself-dual implies generating collection
• coproducts andgenerating collection andstrongly connected implies generator
• disjoint products andgenerating collection implies generator
• extremal generating collection implies generating collection
• generating collection andgroupoid implies extremal generating collection
• generating collection andlocally essentially small implies concretizable
• generating collection andself-dual implies cogenerating collection
• generator implies generating collection
• Grothendieck topos is equivalent to coproducts andelementary topos andgenerating collection andlocally essentially small
• thin implies generating collection

Examples

There are 110 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of coproducts of Euclidean spaces
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of directed graphs
• category of fields
• category of fields of characteristic zero
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• category of finite-dimensional vector spaces
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of finitely generated free abelian groups
• category of finitely generated free modules over Z x Z
• category of finitely generated projective modules over the ring of dual numbers
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of large families of abelian groups
• category of large vector spaces over a large field with a small basis
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of long transfinite sequences of abelian groups
• 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 partially ordered sets
• category of partially ordered sets without isolated points
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets and relations
• category of sets with a binary relation
• category of sets with a distinguished subset
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of uniform spaces
• category of vector spaces
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category of sets
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• forked commutative square
• partially ordered collection of ordinal numbers
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• simplex category
• trivial category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Counterexamples

There are 7 categories without this property.

• category of large families of sets
• category of large families of sets which are mostly empty
• category of large families of sets which are mostly singletons
• category of large families of vector spaces which are mostly zero
• category of locally ringed spaces
• category of schemes
• cocompletion of a discrete–pair join

Unknown

There is 1 category for which the database has no information on whether it satisfies this property. Please help us fill in the gaps by contributing to this project.

• category of Z-functors