category of metric spaces with ∞ allowed
- notation:
- objects: metric spaces, where the metric is allowed to assume the value
- morphisms: non-expansive maps , meaning for all
- nLab Link
- Related categories: ,
The fact that we allow means that universal constructions work much better.
Properties
Properties from the database
- has a cogenerator
- has disjoint coproducts
- is infinitary distributive
- is locally small
- is locally ℵ₁-presentable
Deduced properties
- is locally essentially small
- has coproducts
- has disjoint finite coproducts
- has finite coproducts
- has finite products
- has binary products
- has a terminal object
- is connected
- is distributive
- has a strict initial object
- has an initial object
- is locally presentable
- is cocomplete
- is complete
- has connected limits
- has filtered limits
- is finitely complete
- has wide pullbacks
- has equalizers
- has products
- has countable products
- has pullbacks
- is Cauchy complete
- has sequential limits
- has a generator
- is well-copowered
- is well-powered
- is inhabited
- has connected colimits
- has filtered colimits
- is finitely cocomplete
- has wide pushouts
- has coequalizers
- has countable coproducts
- has binary coproducts
- has pushouts
- has sequential colimits
Non-Properties
Non-Properties from the database
- is not Malcev
- is not balanced
- is not cartesian closed
- does not have exact filtered colimits
- is not skeletal
- does not have a strict terminal object
Deduced Non-Properties*
- is not discrete
- is not locally finitely presentable
- is not finitary algebraic
- is not trivial
- is not essentially discrete
- is not thin
- is not essentially small
- is not small
- is not essentially finite
- is not finite
- is not left cancellative
- is not pointed
- does not have zero morphisms
- is not self-dual
- is not an elementary topos
- is not a Grothendieck topos
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not a groupoid
- is not mono-regular
- does not have a subobject classifier
- is not right cancellative
- is not epi-regular
*This also uses the deduced properties.
Unknown properties
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Special morphisms
- Isomorphisms: bijective isometries
- Monomorphisms: injective non-expansive maps
- Epimorphisms: