Implication details
Claim: If a category has binary copowers and is retract-trivial, then it is thin.
Proof: For every object the codiagonal is a split epimorphism, hence an isomorphism. Hence, the two inclusions coincide. Now, if are two morphisms, consider the induced morphism and compute .
This implication has a dual.
Show 53 categories using this implication
- 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 sets equipped with a 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 reflexive binary relation
- 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-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 finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of finitely generated groups
- category of metric spaces with continuous maps
- category of finitely generated projective modules over the ring of dual numbers
- category of quivers
- category of schemes
- category of sequences of sets
- category of set functions and commutative squares
- category of sets and bijections
- category of countable 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 sets with finite-to-one maps
- category of non-empty 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 countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of Z-functors
- category of finite Z-sets
- cocompletion of a discrete–pair join
- forked commutative square
- category of ω-sets
- walking fork
- walking idempotent
- walking parallel pair