CatDat

initial object

An initial object is an object that has a unique morphism to every object in the category. This property refers to the existence of an initial object.

• Dual property: terminal object
• Related properties: finite coproducts, multi-initial object
• nLab Link

Relevant implications

• cartesian closed andinitial object implies strict initial object
• coextensive andfiltered colimits andinitial object implies cartesian filtered colimits
• cofiltered andessentially finite andthin implies initial object
• connected andmulti-initial object is equivalent to initial object
• essentially countable andthin andℵ₁-cofiltered implies initial object
• finite copowers implies initial object
• finite coproducts is equivalent to binary coproducts andinitial object
• initial object andleft cancellative implies strict initial object
• initial object andlocally cocartesian coclosed implies cocartesian coclosed
• initial object andmulti-complete implies complete
• initial object andpushouts implies binary coproducts
• initial object andself-dual implies terminal object
• initial object andstrongly connected andterminal object implies pointed
• initial object andthin implies copowers
• initial object andwide pushouts implies cocomplete
• initial object implies cofiltered
• initial object implies ℵ₁-cofiltered
• pointed is equivalent to initial object andzero morphisms
• self-dual andterminal object implies initial object
• strict initial object implies initial object

Examples

There are 71 categories with this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• 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 finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sets
• category of sets and relations
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• dual of the category of sets
• dual of the category of topological spaces
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• trivial category
• walking commutative square
• walking composable pair
• walking fork
• walking isomorphism
• walking morphism
• walking span
• walking splitting

Counterexamples

There are 15 categories without this property.

• category of fields
• category of finite sets and bijections
• category of finite sets and surjections
• category of non-empty sets
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• empty category
• simplex category
• walking coreflexive pair
• walking idempotent
• walking parallel pair

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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