Implication Details
Claim: If a category is direct, then it is one-way and is skeletal.
Proof: The category is one-way since any non-identity endomorphism yields an infinite sequence of equal non-identity morphisms. The category is skeletal since any non-identity isomorphism yields the infinite sequence .
Show 72 categories using this implication
- 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 compact Hausdorff spaces
- category of countable groups
- category of countable sets
- category of fields
- category of filtered vector spaces
- category of finite abelian groups
- category of finite groups
- category of finite sets
- category of finite sets and bijections
- category of finite sets and injections
- 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 free abelian groups
- category of groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of left modules over a division ring
- category of left modules over a ring
- category of locally ringed spaces
- 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 pairs of sets
- category of partially ordered sets
- 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 schemes
- category of semigroups
- category of set functions and commutative squares
- category of sets
- category of sets and relations
- category of sets with a distinguished subset
- 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 vector spaces
- category of Z-functors
- 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 ordinal numbers
- discrete category on two objects
- empty category
- 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
- trivial category
- walking coreflexive pair
- walking isomorphism
- walking splitting