Implication details
Claim: If a category has disjoint finite coproducts and is regular-quotient-trivial, then it is trivial.
Proof: For any object , the coequalizer of the two coprojections is the codiagonal . Therefore, these two coprojections are equal. But their equalizer is also the unique morphism . It follows that is an isomorphism.
This implication has a dual.
Show 35 categories using this implication
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- category of Banach spaces with linear contractions
- category of cochain complexes of abelian groups
- category of finite sets and injections
- category of filtered vector spaces
- category of finite abelian groups
- category of finite sets of even cardinality
- category of fields of characteristic zero
- category of free abelian groups
- category of finitely generated free modules over Z x Z
- category of Jónsson-Tarski algebras
- category of M-sets
- category of sets with a distinguished subset
- category of finitely generated projective modules over the ring of dual numbers
- 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 sequences of abelian groups
- category of sets
- category of countable sets
- category of sets with finite-to-one maps
- category of abelian sheaves
- category of torsion abelian groups
- category of torsion-free abelian groups
- category of long transfinite sequences of abelian groups
- category of vector spaces
- category of countable-dimensional vector spaces
- category of large vector spaces over a large field with a small basis
- category of Z-sets
- category of graded abelian groups
- category of graded modules over a graded ring
- category of simplicial sets
- walking fork