Implication details
Claim: If a category has binary powers and is core-thin, then it is thin.
Proof: Let be any object. The swap is an automorphism, hence equal to the identity. It follows that the projections are the same. And this means that every two morphisms are the same.
This implication has a dual.
Show 75 categories using this implication
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- delooping of the bicyclic monoid
- delooping of the double bicyclic monoid
- 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 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 commutative algebras
- category of commutative rings
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- simplex category
- category of F(I)-sets
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- 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 free abelian groups
- category of finitely generated free abelian groups
- category of finitely generated free modules over Z x Z
- category of groups
- category of countable groups
- category of Hausdorff spaces
- category of Jónsson-Tarski algebras
- category of M-sets
- category of finitely generated projective modules over the ring of dual numbers
- category of quivers
- category of left modules over a ring
- category of left modules over a division ring
- category of left modules over a non-semisimple ring
- category of sets and relations
- category of sequences of abelian groups
- category of sequences of sets
- category of connected sequences of sets
- category of sets
- category of pointed sets
- category of set functions and commutative squares
- category of countable sets
- 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 pairs of sets
- category of empty-or-finite pairs of sets
- category of sheaves
- category of abelian sheaves
- category of combinatorial species
- category of torsion abelian groups
- category of long transfinite sequences of abelian groups
- category of vector spaces
- category of countable-dimensional vector spaces
- category of large families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- category of Z-functors
- category of finite Z-sets
- category of Z-sets
- forked commutative square
- category of graded abelian groups
- category of graded modules over a graded ring
- category of ω-sets
- category of simplicial sets
- walking coreflexive pair
- walking idempotent
- walking parallel pair
- walking splitting