category of groups
- Notation:
- Objects: groups
- Morphisms: group homomorphisms
- Related categories: , , , ,
- nLab Link
This is the prototype of a finitary algebraic category.
Satisfied Properties
Assigned properties
- is locally small
- is pointed
- is finitary algebraic
- is mono-regular
- is conormal
- is Malcev
- has effective cocongruences
- has an extremal generator
Deduced properties
- is finitely complete
- is unital
- is locally strongly finitely presentable
- is well-copowered
- has an extremal generating set
- has a generator
- has an initial object
- has zero morphisms
- is balanced
- is locally essentially small
- has a terminal object
- is epi-regular
- is locally finitely presentable
- is cocomplete
- is a generalized variety
- is regular
- is multi-algebraic
- has equalizers
- has finite products
- is connected
- has a multi-terminal object
- is ℵ₁-filtered
- has a generating set
- is inhabited
- has a multi-initial object
- is cofiltered
- is ℵ₁-cofiltered
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has sifted colimits
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- is multi-cocomplete
- has effective congruences
- is strongly connected
- has coreflexive equalizers
- is Cauchy complete
- has kernels
- is filtered
- has binary products
- has finite powers
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is cosifted
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has connected limits
- has quotients of congruences
- is Barr-exact
- is semi-strongly connected
- has disjoint finite products
- has cartesian filtered colimits
- is sifted
- has reflexive coequalizers
- has binary powers
- has pullbacks
- has finite coproducts
- has coquotients of cocongruences
- has cokernels
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- is well-powered
- is complete
- is locally multi-presentable
- has directed colimits
- has wide pullbacks
- has disjoint finite coproducts
- has cosifted limits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has pushouts
- is locally poly-presentable
- has products
- is multi-complete
- has disjoint coproducts
- has cofiltered limits
- has sequential colimits
- has countable copowers
- has binary copowers
- has powers
- has ℵ₂-small products
- has disjoint products
- has directed limits
- has ℵ₁-cofiltered limits
- has sequential limits
- has countable products
- has ℵ₂-small powers
- has countable powers
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not normal
- does not have a cogenerator
- is not coregular
- is not counital
- does not satisfy CIP
- does not satisfy CSP
- does not have a regular quotient object classifier
- does not have cofiltered-limit-stable epimorphisms
- does not have cocartesian cofiltered limits
Deduced properties*
- is not abelian
- is not Grothendieck abelian
- is not additive
- is not discrete
- does not have a cogenerating set
- is not extensive
- does not have biproducts
- is not preadditive
- does not have a subobject classifier
- is not gaunt
- is not direct
- is not a quasitopos
- is not a Grothendieck topos
- is not co-Malcev
- is not cocartesian coclosed
- is not locally cocartesian coclosed
- is not Barr-coexact
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- does not have an extremal cogenerator
- is not left cancellative
- does not have a quotient object classifier
- is not thin
- is not inverse
- is not self-dual
- is not split abelian
- is not trivial
- is not essentially discrete
- does not have a strict terminal object
- is not countably extensive
- is not locally cartesian closed
- is not a groupoid
- is not finite
- does not have a regular subobject classifier
- is not regular-subobject-trivial
- is not core-thin
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not an elementary topos
- is not a pretopos
- is not core-connected
- does not have a strict initial object
- does not have an extremal cogenerating set
- is not regular-quotient-trivial
- does not have a natural numbers object
- is not locally copresentable
- is not cartesian closed
- is not distributive
- is not infinitary extensive
- is not small
- is not countable
- is not subobject-trivial
- is not coaccessible
- is not codistributive
- is not coextensive
- is not countably distributive
- is not countably codistributive
- is not countably coextensive
- is not infinitary distributive
- is not infinitary codistributive
- is not infinitary coextensive
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: trivial group
- initial object: trivial group
- products: direct products with pointwise operations
- coproducts: free products
Special morphisms
- isomorphisms: bijective morphisms
- monomorphisms: injective morphisms
- epimorphisms: surjective morphisms
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: surjective morphisms
Functors
The database has stored 12 functors whose (co-)domain is the category of groups.
- abelianization functor for groups
- enveloping group functor
- forgetful functor for groups
- forgetful functor from abelian groups to groups
- forgetful functor from finite groups to groups
- forgetful functor from groups to monoids
- forgetful functor from groups to pointed sets
- free group functor
- fundamental group functor
- group of units functor
- simple-group probing functor
- trivial functor from the category of groups
Morphisms
The database has stored 2 morphisms in the category of groups.