poset [0,1]
- notation:
- objects: real numbers between and
- morphisms: a unique morphism when
- Related categories:
- nLab Link
Every poset can be regarded as a small thin category. This is a specific example. This category is locally -presentable (in fact, every object is -presentable), but not locally finitely presentable (in fact, only is finitely presentable).
Satisfied Properties
Properties from the database
- is locally ℵ₁-presentable
- is self-dual
- 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 locally presentable
- is cocomplete
- is complete
- has connected limits
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has products
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- has finite powers
- has binary powers
- is thin
- is one-way
- has reflexive coequalizers
- is left cancellative
- is locally cartesian closed
- is regular
- is cartesian closed
- is inhabited
- has a generator
- is Malcev
- has connected colimits
- has sifted colimits
- has filtered colimits
- has directed colimits
- is finitely cocomplete
- has cosifted limits
- has sequential colimits
- has coequalizers
- has coproducts
- is infinitary distributive
- is distributive
- has finite coproducts
- has a strict initial object
- has an initial object
- has countable coproducts
- has binary coproducts
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- has a strict terminal object
- has a cogenerating set
- is right cancellative
- has a regular subobject classifier
- has a cogenerator
- is coregular
- is co-Malcev
- is codistributive
- is infinitary codistributive
Unsatisfied Properties
Properties from the database
- is not direct
- is not essentially finite
- is not inverse
- is not locally finitely presentable
Deduced properties*
- is not finite
- is not discrete
- is not trivial
- does not have disjoint finite coproducts
- does not have biproducts
- does not have disjoint coproducts
- is not pointed
- does not have zero morphisms
- is not extensive
- is not infinitary extensive
- is not lextensive
- is not a groupoid
- is not balanced
- is not mono-regular
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not essentially discrete
- is not locally strongly finitely presentable
- is not finitary algebraic
- does not have a subobject classifier
- is not an elementary topos
- is not a Grothendieck topos
- is not unital
- does not have disjoint finite products
- does not have disjoint products
- is not coextensive
- is not infinitary coextensive
- is not epi-regular
- is not counital
*This also uses the deduced satisfied properties.
Unknown properties
There is 1 property for which the database doesn't have an answer if it is satisfied or not. Please help to contribute the data!
Special objects
- terminal object:
- initial object:
- products: infimum
- coproducts: supremum
Special morphisms
- isomorphisms: only the identity morphisms
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms