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 . A concrete representation is the full subcategory of consisting of the empty set. It is the terminal object in the category of small categories.
Satisfied Properties
Properties from the database
Deduced properties
- is essentially small
- is locally small
- has a generating set
- is locally essentially small
- is well-copowered
- is well-powered
- is countable
- is essentially finite
- is essentially countable
- 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
- is finitely accessible
- has coreflexive equalizers
- is mono-regular
- has pullbacks
- 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
- has powers
- has countable powers
- is complete
- has finite powers
- has binary powers
- is multi-complete
- has a multi-terminal object
- is connected
- is strongly connected
- is a Grothendieck topos
- is split abelian
- is abelian
- is additive
- is preadditive
- is normal
- has zero morphisms
- has kernels
- has biproducts
- has finite coproducts
- is unital
- has cokernels
- is conormal
- is cocomplete
- is locally finitely presentable
- is locally ℵ₁-presentable
- is locally presentable
- has exact filtered colimits
- has filtered colimits
- has sifted colimits
- has directed colimits
- has cartesian filtered colimits
- is ℵ₁-accessible
- is accessible
- is a generalized variety
- is locally multi-presentable
- is multi-cocomplete
- is locally finitely multi-presentable
- is locally poly-presentable
- is multi-algebraic
- is cartesian closed
- is distributive
- is codistributive
- has a strict initial object
- has an initial object
- is pointed
- has disjoint products
- has coproducts
- is infinitary distributive
- is countably distributive
- has countable coproducts
- is Grothendieck abelian
- has a cogenerator
- is an elementary topos
- has a subobject classifier
- has a regular subobject classifier
- has disjoint finite coproducts
- has disjoint coproducts
- is epi-regular
- is finitely cocomplete
- is infinitary extensive
- is extensive
- has cocartesian cofiltered limits
- is coregular
- is co-Malcev
- has a natural numbers object
- is semi-strongly connected
- has disjoint finite products
- is Malcev
- is filtered
- is sifted
- has connected colimits
- has cosifted limits
- has copowers
- has binary coproducts
- has coequalizers
- has countable copowers
- has sequential colimits
- has pushouts
- has wide pushouts
- has finite copowers
- has binary copowers
- has a multi-initial object
- is counital
- has a strict terminal object
- is infinitary codistributive
- is countably codistributive
- has a cogenerating set
- is right cancellative
- is locally cocartesian coclosed
- is inverse
- is cocartesian coclosed
- is coaccessible
- is locally copresentable
- has a regular quotient object classifier
- has a quotient object classifier
- is coextensive
- is infinitary coextensive
- is cosifted
- is cofiltered
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