Implication details
Claim: If a category has kernels, then it has a multi-initial object and has a multi-terminal object.
Proof: If the category is empty, both properties hold trivially. Otherwise, it has a zero object by this result, and is a multi-terminal object and a multi-initial object.
This implication has a dual.
Show 29 categories using this implication
- empty category
- trivial category
- category of abelian groups
- category of finitely generated abelian groups
- category of cochain complexes of abelian groups
- category of finite sets and surjections
- category of filtered vector spaces
- 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 free abelian groups
- 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
- discrete category of sets
- indiscrete category of sets
- category of non-empty sets
- category of abelian sheaves
- 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 graded abelian groups
- category of graded modules over a graded ring
- walking isomorphism