strict terminal object
A strict terminal object is a terminal object such that every morphism 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
strict terminal object implies terminal object
[dualized] by definitionstrict terminal object and pointed implies trivial
[dualized] If is the zero object, then for every object the unique morphism is an isomorphism by assumption.right cancellative and terminal object implies strict terminal object
[dualized] It suffices to prove that in general any monomorphism into an initial object is an isomorphism. If is the unique morphism, then since is initial. But then is a split epimorphism and a monomorphism, hence an isomorphism.left cancellative and terminal object implies strict terminal object
[dualized] Let be a morphism. Let be the unique morphism. It is an epimorphism by assumption. Also, since is initial. But then is a split monomorphism and an epimorphism, hence an isomorphism.self-dual and strict initial object implies strict terminal object
trivial by self-dualityself-dual and strict terminal object implies strict initial object
trivial by self-duality
Examples
- 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
- 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
For these categories the database has no info if they satisfy this property or not.
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