trivial category
- notation:
- objects: a single object
- morphisms: only the identity morphism
- Related categories:
- nLab Link
This is the simplest category, consisting of a single object and its identity morphism . It is the terminal object in the category of small categories.
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 direct
- is one-way
- is skeletal
- has sequential limits
- has reflexive coequalizers
- is essentially discrete
- is a groupoid
- has directed limits
- is left cancellative
- is Cauchy complete
- has filtered colimits
- has directed colimits
- has coreflexive equalizers
- 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
- 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 inverse
- is codistributive
- is infinitary codistributive
- is coextensive
- is infinitary coextensive
Unsatisfied Properties
Properties from the database
—
Deduced properties*
—
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: the unique object
- initial object: the unique object
- products:
- coproducts:
Special morphisms
- isomorphisms: every morphism
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms