walking isomorphism
- notation:
- objects: two objects and
- morphisms: identities, and two morphisms and that are mutually inverse
- Related categories: , ,
- nLab Link
The name of this category comes from the fact that it consists of two objects and an isomorphism between them, and a functor out of this category is the same as an isomorphism in the target category. The walking isomorphism is actually equivalent to the trivial category.
Satisfied Properties
Properties from the database
Deduced properties
- is essentially finite
- is small
- is essentially small
- is locally small
- is locally essentially small
- is well-copowered
- is well-powered
- has a generating set
- is essentially discrete
- is a groupoid
- has directed limits
- has sequential limits
- is left cancellative
- is Cauchy complete
- has filtered colimits
- has directed colimits
- has coreflexive equalizers
- has reflexive coequalizers
- is mono-regular
- has pullbacks
- has sifted colimits
- has wide pullbacks
- has cofiltered limits
- is self-dual
- is locally cartesian closed
- is balanced
- is thin
- is one-way
- has connected limits
- has equalizers
- is finitary algebraic
- has a generator
- is inhabited
- is locally strongly finitely presentable
- is regular
- is finitely complete
- has finite products
- has binary products
- has a terminal object
- has products
- has countable products
- is connected
- is complete
- has powers
- has countable powers
- has finite powers
- has binary powers
- is strongly connected
- is a Grothendieck topos
- is split abelian
- is abelian
- is additive
- is preadditive
- has zero morphisms
- has biproducts
- has finite coproducts
- is unital
- has disjoint finite coproducts
- has coequalizers
- is epi-regular
- is cocomplete
- is locally finitely presentable
- is locally presentable
- is locally ℵ₁-presentable
- has exact filtered colimits
- is cartesian closed
- is distributive
- has a strict initial object
- has an initial object
- is pointed
- has disjoint products
- has coproducts
- has disjoint coproducts
- is infinitary distributive
- is Grothendieck abelian
- has a cogenerator
- is an elementary topos
- has a subobject classifier
- has a regular subobject classifier
- is finitely cocomplete
- is coregular
- is extensive
- is lextensive
- is infinitary extensive
- is co-Malcev
- is Malcev
- has connected colimits
- has cosifted limits
- has countable coproducts
- has binary coproducts
- has sequential colimits
- has pushouts
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- is counital
- has disjoint finite products
- has a strict terminal object
- has a cogenerating set
- is right cancellative
- is codistributive
- is infinitary codistributive
- is coextensive
- is infinitary coextensive
Unsatisfied Properties
Properties from the database
- is not skeletal
Deduced properties*
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: both objects
- initial object: both objects
- products:
- coproducts:
Special morphisms
- isomorphisms: every morphism
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms