Implication details
Claim: A category has equalizers and is regular-subobject-trivial if and only if it is thin.
Proof: For the forward direction, if are morphisms, their equalizer is a regular monomorphism , hence an isomorphism. But this means . The reverse implication is trivial.
This implication has a dual.
Show 87 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
- category of Banach spaces with linear contractions
- category of sets equipped with a binary relation
- category of sets equipped with an irreflexive binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative rings
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of F(I)-sets
- 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 fields
- category of fields of characteristic zero
- category of free abelian groups
- category of finitely generated free abelian groups
- category of groups
- category of measurable spaces
- category of metric spaces with continuous maps
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of partially ordered sets without isolated points
- category of quivers
- category of quivers with finite components
- category of sets and relations
- category of rngs
- category of schemes
- category of sequences of sets
- category of connected sequences of sets
- category of pointed sets
- category of set functions and commutative squares
- category of sets and bijections
- category of countable sets
- discrete category of sets
- category of large families of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of sets with finite-to-one maps
- indiscrete category of sets
- category of pairs of sets
- category of empty-or-finite pairs of sets
- category of sheaves
- category of combinatorial species
- category of long transfinite sequences of abelian groups
- category of uniform spaces
- category of countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of Z-functors
- category of finite Z-sets
- preordered set of integers w.r.t. divisibility
- cocompletion of a discrete–pair join
- forked commutative square
- category of ω-sets
- real interval [0,1]
- walking commutative square
- walking composable pair
- walking fork
- walking idempotent
- walking isomorphism
- walking morphism
- walking parallel pair
- walking span
- walking splitting