Implication details
Claim: A category is connected and has a multi-initial object if and only if it has an initial object.
Proof: This follows from the dual implication.
This implication has a dual.
Show 82 categories using this implication
- empty category
- discrete category on two objects
- category of algebras
- category of finite sets and bijections
- delooping of a group
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of an infinite uncountable group
- 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 an irreflexive binary relation
- category of sets equipped with a reflexive binary relation
- category of sets equipped with a symmetric irreflexive binary relation
- category of sets equipped with a symmetric reflexive binary relation
- category of commutative algebras
- category of commutative monoids
- category of commutative rings
- simplex category
- category of coproducts of Euclidean spaces
- category of finite sets and injections
- category of finite sets and surjections
- 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 sets of cardinality a power of 3
- category of fields
- category of fields of characteristic zero
- 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 finitely generated groups
- category of Hausdorff spaces
- category of smooth manifolds
- category of metric spaces with non-expansive maps
- category of metric spaces with continuous maps
- category of monoids
- category of sets with a distinguished subset
- partially ordered set of natural numbers
- partially ordered set of extended natural numbers
- partially ordered collection of ordinal numbers
- category of pseudo-metric spaces with non-expansive maps
- category of partially ordered sets without isolated points
- category of finitely generated projective modules over the ring of dual numbers
- category of quivers with finite components
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of sequences of sets
- category of connected sequences of sets
- category of pointed sets
- category of set functions and commutative squares
- category of sets and bijections
- category of countable sets
- discrete category of sets
- category of large families of sets and bijections
- category of large families of sets which are mostly empty
- category of large families of sets which are mostly singletons
- category of sets with finite-to-one maps
- category of non-empty sets
- category of empty-or-finite pairs of sets
- category of combinatorial species
- category of topological spaces
- category of pointed topological spaces
- category of countable-dimensional vector spaces
- category of finite Z-sets
- cocompletion of a discrete–pair join
- forked commutative square
- category of ω-sets
- real interval [0,1]
- category of simplicial sets
- walking composable pair
- walking coreflexive pair
- walking fork
- walking idempotent
- walking morphism
- walking parallel pair
- walking splitting