CatDat

strict initial object

A strict initial object is an initial object $0$ such that every morphism $A \to 0$ is an isomorphism. This property refers to the existence of a strict initial object.

• Dual property: strict terminal object
• Related properties: initial object
• nLab Link

Relevant implications

• cartesian closed andinitial object implies strict initial object
• disjoint finite products andstrict initial object implies thin
• distributive implies strict initial object
• extensive implies strict initial object
• initial object andleft cancellative implies strict initial object
• initial object andright cancellative implies strict initial object
• pointed andstrict initial object implies trivial
• self-dual andstrict initial object implies strict terminal object
• self-dual andstrict terminal object implies strict initial object
• strict initial object implies initial object

Examples

There are 34 categories with this property.

• category of combinatorial species
• category of finite orders
• category of finite sets
• category of finite sets and injections
• category of Hausdorff spaces
• 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 pairs of sets
• category of posets
• category of prosets
• category of schemes
• category of sets
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of Z-functors
• 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

Counterexamples

There are 31 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite sets and bijections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left modules over a division ring
• category of left modules over a ring
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of rings
• category of rngs
• category of sets and relations
• category of vector spaces
• 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
• dual of the category of sets
• 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.

—