Implication Details
Claim: If a category is essentially small, then it has an extremal generating set and is locally essentially small and is well-copowered and is well-powered.
Proof: All conclusions are trivial except perhaps that the category has an extremal generating set. For that, let be a set with one representative of each isomorphism class of objects of the category. Then it is easy to show using the Yoneda Lemma that is an extremal generating set.
Show 50 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of finitely generated abelian groups
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- simplex category
- category of finite sets and injections
- category of finite sets and surjections
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- 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 countable groups
- category of Hausdorff spaces
- category of locally ringed spaces
- category of smooth manifolds
- category of measurable spaces
- partially ordered set of natural numbers
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of schemes
- category of countable sets
- category of combinatorial species
- category of topological spaces
- category of pointed topological spaces
- category of uniform spaces
- category of Z-functors
- preordered set of integers w.r.t. divisibility
- forked commutative square
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking coreflexive pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting