Implication Details
Claim: If a category is distributive, then it has a strict initial object.
Proof: See the nLab or Prop. 3.4 in Introduction to extensive and distributive categories by Carboni-Lack-Walters.
This implication has a dual.
Show 43 categories using this implication
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- category of algebras
- category of Banach spaces with linear contractions
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- category of cochain complexes of abelian groups
- 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 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 monoids
- 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 sets and relations
- category of rings
- category of rngs
- category of sequences of abelian groups
- category of pointed sets
- category of non-empty sets
- category of abelian sheaves
- category of pointed topological spaces
- 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 families of vector spaces which are mostly zero
- category of large vector spaces over a large field with a small basis
- preordered set of integers w.r.t. divisibility
- category of graded abelian groups
- category of graded modules over a graded ring