category of preordered sets
Despite the many similarities with , the main difference is that the forgetful functor has a right adjoint, which maps to (the chaotic preorder). It is a full subcategory of . Preordered sets correspond to small thin categories.
Satisfied properties
Assigned properties
- is locally small
- is locally finitely presentable
- is cartesian closed
- is semi-strongly connected
- is infinitary extensive
- has pushout-stable regular monomorphisms
- has a regular subobject classifier
- has an extremal generator
- has an extremal cogenerator
Deduced properties
- is cocomplete
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has finite products
- is connected
- has coproducts
- is countably extensive
- has an extremal generating collection
- has a generator
- is locally essentially small
- is finitely complete
- has a cogenerator
- has an extremal cogenerating collection
- has pushouts
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally presentable
- has filtered colimits
- has equalizers
- is inhabited
- is infinitary distributive
- has cartesian filtered colimits
- has countable coproducts
- is extensive
- is countably distributive
- has a generating collection
- has binary products
- has a terminal object
- has finite powers
- has connected colimits
- is finitely cocomplete
- has coequalizers
- is multi-cocomplete
- is cofiltered
- has a cogenerating collection
- has copowers
- has ℵ₂-small coproducts
- is Cauchy complete
- has reflexive coequalizers
- has cokernel pairs
- has a parametrized natural numbers object
- is accessible
- has ℵ₁-filtered colimits
- is complete
- is locally finitely multi-presentable
- has powers
- has a multi-terminal object
- has quotients of congruences
- is distributive
- has coreflexive equalizers
- has finite coproducts
- has disjoint finite coproducts
- has a strict initial object
- is filtered
- is ℵ₁-filtered
- has directed colimits
- has sifted colimits
- has binary powers
- has pullbacks
- is concretizable
- has a multi-initial object
- is cosifted
- has sequential colimits
- has ℵ₂-small copowers
- has countable copowers
- has wide pushouts
- has a natural numbers object
- is well-powered
- is well-copowered
- is locally multi-presentable
- has connected limits
- has countable powers
- has products
- is multi-complete
- has disjoint coproducts
- is sifted
- has an initial object
- has ℵ₂-small powers
- has kernel pairs
- has coquotients of cocongruences
- has binary coproducts
- has finite copowers
- has coequalizers of kernel pairs
- has ℵ₂-small products
- has wide pullbacks
- is total
- has equalizers of cokernel pairs
- is ℵ₁-cofiltered
- has cosifted limits
- has binary copowers
- is cototal
- is locally poly-presentable
- has countable products
- has cofiltered limits
- is coregular
- has cocartesian cofiltered limits
- has sequential limits
- has directed limits
- has ℵ₁-cofiltered limits
Unsatisfied properties
Assigned properties
- is not skeletal
- is not co-Malcev
- does not have cofiltered-limit-stable epimorphisms
- does not have effective cocongruences
Deduced properties*
- is not left cancellative
- is not epi-regular
- is not regular
- is not discrete
- is not regular-subobject-trivial
- is not gaunt
- is not direct
- is not thin
- is not additive
- is not finitely coaccessible
- is not right cancellative
- is not core-thin
- does not have exact cofiltered limits
- is not quotient-trivial
- is not Barr-coexact
- is not regular-quotient-trivial
- is not inverse
- is not self-dual
- is not preadditive
- is not abelian
- does not have a strict terminal object
- is not locally cartesian closed
- is not trivial
- is not essentially discrete
- is not a groupoid
- does not have pullback-stable regular epimorphisms
- is not Barr-exact
- is not subobject-trivial
- is not Malcev
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not essentially finite
- is not an elementary topos
- is not locally finitely copresentable
- is not cocartesian coclosed
- does not have disjoint finite products
- is not conormal
- does not have a quotient object classifier
- does not have a regular quotient object classifier
- is not pointed
- is not locally copresentable
- is not Grothendieck abelian
- is not split abelian
- is not finitary algebraic
- is not multi-algebraic
- is not core-connected
- is not strongly connected
- is not coaccessible
- is not small
- is not finite
- is not countable
- does not have a subobject classifier
- is not a Grothendieck topos
- is not a pretopos
- is not a quasitopos
- is not locally cocartesian coclosed
- does not have disjoint products
- is not codistributive
- is not coextensive
- is not infinitary coextensive
- is not unital
- is not a generalized variety
- is not one-sorted finitary algebraic
- does not have zero morphisms
- is not mono-regular
- is not counital
- is not locally ℵ₁-copresentable
- is not ℵ₁-coaccessible
- is not countably codistributive
- is not countably coextensive
- does not have biproducts
- does not have effective congruences
- does not have kernels
- does not satisfy CIP
- is not normal
- is not infinitary codistributive
- does not have cokernels
- does not satisfy CSP
- is not balanced
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: singleton preordered set
- initial object: empty preordered set
- products: direct products with the evident preorder
- coproducts: disjoint union with the obvious preorder 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: