category of posets
- notation:
- objects: partially ordered sets (aka posets), i.e. sets equipped with a reflexive, transitive, antisymmetric relation
- morphisms: order-preserving functions
- Related categories: ,
- nLab Link
Even though there are many similarities with , the main difference is that the forgetful functor has no right adjoint.
Satisfied Properties
Properties from the database
- is cartesian closed
- has a cogenerator
- is coregular
- has a generator
- is infinitary extensive
- is locally finitely presentable
- is locally small
- is strongly connected
Deduced properties
- has coproducts
- is extensive
- has finite coproducts
- is distributive
- has finite products
- has binary products
- has a terminal object
- is connected
- has finite powers
- has binary powers
- has a strict initial object
- has an initial object
- is infinitary distributive
- has disjoint finite coproducts
- has disjoint coproducts
- is locally essentially small
- has a generating set
- is inhabited
- is locally presentable
- is locally ℵ₁-presentable
- is cocomplete
- is complete
- has connected limits
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has products
- has countable products
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- is lextensive
- is well-copowered
- is well-powered
- has exact filtered colimits
- has filtered colimits
- has directed colimits
- has connected colimits
- has sifted colimits
- has reflexive coequalizers
- is finitely cocomplete
- has cosifted limits
- has countable coproducts
- has binary coproducts
- has coequalizers
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- is not balanced
- is not co-Malcev
- is not Malcev
- is not regular
- does not have a regular subobject classifier
- is not skeletal
- does not have a strict terminal object
Deduced properties*
- is not mono-regular
- is not a groupoid
- is not direct
- is not discrete
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not trivial
- is not thin
- is not pointed
- does not have zero morphisms
- does not have biproducts
- is not left cancellative
- is not one-way
- is not preadditive
- is not additive
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not self-dual
- does not have a subobject classifier
- is not an elementary topos
- is not right cancellative
- is not a Grothendieck topos
- is not locally cartesian closed
- is not unital
- does not have disjoint finite products
- does not have disjoint products
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
- is not epi-regular
- is not inverse
- is not counital
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: singleton poset
- initial object: empty poset
- products: direct products with the evident partial order
- coproducts: disjoint union with the obvious partial order that leaves the distinct summands incomparable
Special morphisms
- isomorphisms: bijective functions that are order-preserving and order-reflecting
- monomorphisms: injective order-preserving functions
- epimorphisms: surjective order-preserving functions
- regular monomorphisms: embeddings
- regular epimorphisms: