CatDat

strict terminal object

A strict terminal object is a terminal object $1$ such that every morphism $1 \to A$ is an isomorphism. This property refers to the existence of a strict terminal object.

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

Relevant implications

• left cancellative and  terminal object implies   strict terminal object
• pointed and  strict terminal object implies   trivial
• right cancellative and  terminal object implies   strict terminal object
• self-dual and  strict initial object implies   strict terminal object
• self-dual and  strict terminal object implies   strict initial object
• strict terminal object implies   terminal object

Examples

There are 8 categories with this property.

• category of commutative rings
• category of rings
• partial order [0,1]
• partial order of extended natural numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism

Counterexamples

There are 43 categories without this property.

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• 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 non-empty sets
• category of pointed sets
• category of posets
• 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
• 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
• partial order of natural numbers
• partial order of ordinal numbers
• walking parallel pair of morphisms

Unknown

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

—