Implication details
Claim: If a category is core-connected and has an initial object, then it is trivial.
Proof: Every object is isomorphic to the initial object.
This implication has a dual.
Show 52 categories using this implication
- category of abelian groups
- category of large families of abelian groups
- category of finitely generated abelian groups
- 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 reflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of small categories
- category of cochain complexes of abelian groups
- category of compact Hausdorff spaces
- category of coproducts of Euclidean spaces
- category of filtered vector spaces
- 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 M-sets
- category of smooth manifolds
- category of metric spaces with continuous maps
- category of monoids
- partially ordered collection of ordinal numbers
- category of partially ordered sets
- category of preordered sets
- 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 rngs
- category of sequences of abelian groups
- category of sets
- category of pointed sets
- category of countable sets
- 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 idempotent
- walking morphism
- walking splitting