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
- nLab Link
Relevant implications
- cartesian closed and initial object implies strict initial object
- finite coproducts is equivalent to binary coproducts and initial object
- groupoid and initial object implies trivial
- initial object and left cancellative implies strict initial object
- initial object and pushouts implies binary coproducts
- initial object and right cancellative implies strict initial object
- initial object and self-dual implies terminal object
- initial object and wide pushouts implies cocomplete
- initial object implies connected
- pointed is equivalent to initial object and zero morphisms
- self-dual and terminal object implies initial object
- strict initial object implies initial object
Examples
There are 40 categories with this property.
- category of abelian groups
- category of Banach spaces with linear contractions
- category of combinatorial species
- category of commutative rings
- category of finite abelian groups
- category of finite orders
- category of finite sets
- category of finite sets and injections
- category of finitely generated abelian groups
- category of free abelian groups
- category of groups
- category of left R-modules
- category of locally ringed spaces
- category of M-sets
- category of measurable spaces
- category of metric spaces with ∞ allowed
- category of metric spaces with continuous maps
- category of metric spaces with non-expansive maps
- category of monoids
- category of pointed sets
- category of posets
- category of rings
- category of rngs
- category of schemes
- category of sets
- category of sets and relations
- category of simplicial sets
- category of small categories
- category of smooth manifolds
- category of topological spaces
- category of vector spaces
- category of Z-functors
- partial order [0,1]
- partial order of extended natural numbers
- partial order of natural numbers
- partial order of ordinal numbers
- preorder of integers w.r.t. divisiblity
- trivial category
- walking isomorphism
- walking morphism
Counterexamples
There are 11 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 group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- discrete category on two objects
- empty category
- walking parallel pair of morphisms
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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