delooping of an infinite group
- notation:
- objects: a single object
- morphisms: the elements of an infinite group
- Related categories: ,
- nLab Link
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 .
Satisfied Properties
Properties from the database
- has a generator
- is a groupoid
- is skeletal
- is small
- is strongly connected
Deduced properties
- is essentially small
- is locally small
- is locally essentially small
- is well-copowered
- is well-powered
- has a generating set
- is inhabited
- has directed limits
- has sequential limits
- is left cancellative
- is Cauchy complete
- has coreflexive equalizers
- has reflexive coequalizers
- is mono-regular
- has pullbacks
- is self-dual
- is locally cartesian closed
- is balanced
- is connected
- has cofiltered limits
- has wide pullbacks
- has a cogenerating set
- has directed colimits
- has filtered colimits
- has sifted colimits
- has sequential colimits
- is epi-regular
- has pushouts
- has cosifted limits
- has wide pushouts
- is right cancellative
- has a cogenerator
Unsatisfied Properties
Properties from the database
- is not essentially finite
- is not trivial
Deduced properties*
- is not finite
- does not have binary products
- does not have finite products
- does not have products
- does not have a terminal object
- does not have countable products
- is not complete
- is not finitely complete
- does not have finite powers
- does not have countable powers
- does not have powers
- does not have biproducts
- does not have exact filtered colimits
- is not infinitary distributive
- is not distributive
- is not regular
- is not lextensive
- does not have an initial object
- is not pointed
- does not have a strict initial object
- is not extensive
- is not infinitary extensive
- does not have zero morphisms
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not discrete
- is not thin
- does not have coequalizers
- does not have binary copowers
- does not have equalizers
- does not have connected limits
- is not one-way
- is not direct
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not cartesian closed
- is not an elementary topos
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not a Grothendieck topos
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not Malcev
- is not unital
- does not have binary coproducts
- does not have finite coproducts
- does not have coproducts
- does not have countable coproducts
- is not cocomplete
- is not finitely cocomplete
- does not have connected colimits
- does not have finite copowers
- does not have countable copowers
- does not have copowers
- does not have disjoint products
- does not have disjoint finite products
- is not infinitary codistributive
- is not codistributive
- does not have a strict terminal object
- is not coextensive
- is not infinitary coextensive
- is not coregular
- does not have binary powers
- is not inverse
- is not co-Malcev
- is not counital
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
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Special morphisms
- isomorphisms: every morphism
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as isomorphisms