delooping of an infinite group
- notation:
- objects: a single object
- morphisms: the elements of an infinite group
- nLab Link
- Related categories: ,
Every group yields a groupoid with a single object, morphisms given by the elements of , and composition given by the group operation. In this example, we consider the case of an infinite group .
Properties
Properties from the database
Deduced properties
- is essentially small
- is locally small
- is locally essentially small
- is well-copowered
- is well-powered
- has filtered limits
- has sequential limits
- is left cancellative
- is Cauchy complete
- is mono-regular
- has pullbacks
- has wide pullbacks
- is self-dual
- is inhabited
- is balanced
- is epi-regular
- has filtered colimits
- has sequential colimits
- has pushouts
- has wide pushouts
- is right cancellative
- has a cogenerator
Non-Properties
Non-Properties from the database
- is not essentially finite
- does not have zero morphisms
Deduced Non-Properties*
- is not finite
- is not trivial
- is not essentially discrete
- is not discrete
- is not thin
- does not have products
- does not have coequalizers
- is not complete
- does not have finite products
- is not finitely complete
- is not pointed
- does not have a terminal object
- does not have countable products
- does not have exact filtered colimits
- is not infinitary distributive
- is not distributive
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not finitary algebraic
- is not an elementary topos
- is not cartesian closed
- is not a Grothendieck topos
- does not have a subobject classifier
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- does not have equalizers
- does not have binary products
- does not have connected limits
- does not have an initial object
- does not have a strict initial object
- is not Malcev
- does not have coproducts
- does not have disjoint coproducts
- is not cocomplete
- is not finitely cocomplete
- does not have finite coproducts
- does not have disjoint finite coproducts
- does not have binary coproducts
- does not have connected colimits
- does not have a strict terminal object
- does not have countable coproducts
*This also uses the deduced properties.
Unknown properties
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Special morphisms
- Isomorphisms: every morphism
- Monomorphisms: every morphism
- Epimorphisms: every morphism