category of metric spaces with ∞ allowed

Satisfied properties

Assigned properties

• is locally small
• is semi-strongly connected
• is locally ℵ₁-presentable
• has cartesian filtered colimits
• is infinitary extensive
• has an extremal generator
• has an extremal cogenerator

Deduced properties

• is cocomplete
• is ℵ₁-accessible
• is locally presentable
• is connected
• has filtered colimits
• has finite products
• has coproducts
• is countably extensive
• has an extremal generating collection
• has a generator
• is locally essentially small
• has a cogenerator
• has an extremal cogenerating collection
• is accessible
• has ℵ₁-filtered colimits
• is complete
• is inhabited
• has countable coproducts
• is extensive
• is infinitary distributive
• is countably distributive
• has directed colimits
• 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
• has a cogenerating collection
• has copowers
• has ℵ₂-small coproducts
• has a parametrized natural numbers object
• is well-powered
• is Cauchy complete
• is locally multi-presentable
• has connected limits
• is finitely complete
• has equalizers
• has products
• is multi-complete
• has a multi-terminal object
• is distributive
• has finite coproducts
• has disjoint finite coproducts
• has a strict initial object
• is filtered
• is ℵ₁-filtered
• has sifted colimits
• has binary powers
• is concretizable
• has a multi-initial object
• has reflexive coequalizers
• is cosifted
• has sequential colimits
• has ℵ₂-small copowers
• has countable copowers
• has wide pushouts
• has a natural numbers object
• has quotients of congruences
• has disjoint coproducts
• has coreflexive equalizers
• is sifted
• has an initial object
• has powers
• has ℵ₂-small products
• has pullbacks
• has wide pullbacks
• is cofiltered
• has cosifted limits
• has binary coproducts
• has finite copowers
• has pushouts
• is cototal
• is well-copowered
• is locally poly-presentable
• has countable products
• has ℵ₂-small powers
• has cofiltered limits
• has kernel pairs
• has coquotients of cocongruences
• is ℵ₁-cofiltered
• has binary copowers
• has cokernel pairs
• has coequalizers of kernel pairs
• has cocartesian cofiltered limits
• has sequential limits
• has countable powers
• is total
• has equalizers of cokernel pairs
• has directed limits
• has ℵ₁-cofiltered limits

Unsatisfied properties

Assigned properties

• is not skeletal
• is not balanced
• is not Malcev
• is not co-Malcev
• is not cartesian closed
• does not have filtered-colimit-stable monomorphisms
• does not have cofiltered-limit-stable epimorphisms
• does not have effective cocongruences
• is not regular
• does not have a regular subobject classifier

Deduced properties*

• is not thin
• is not additive
• is not finitely accessible
• is not locally finitely multi-presentable
• is not abelian
• is not finitary algebraic
• is not a generalized variety
• is not left cancellative
• is not locally cartesian closed
• is not Barr-exact
• is not discrete
• does not have exact filtered colimits
• is not subobject-trivial
• is not essentially finite
• is not mono-regular
• is not right cancellative
• does not have a subobject classifier
• is not regular-subobject-trivial
• is not gaunt
• is not direct
• is not an elementary topos
• is not a quasitopos
• is not core-thin
• is not Barr-coexact
• does not have exact cofiltered limits
• is not quotient-trivial
• is not epi-regular
• is not regular-quotient-trivial
• is not inverse
• is not locally finitely presentable
• is not preadditive
• is not Grothendieck abelian
• is not split abelian
• is not one-sorted finitary algebraic
• is not multi-algebraic
• does not have effective congruences
• is not trivial
• is not essentially discrete
• does not have a strict terminal object
• is not a groupoid
• is not normal
• is not finite
• is not locally finite
• is not one-way
• is not essentially small
• is not essentially countable
• is not a Grothendieck topos
• is not a pretopos
• 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 core-connected
• is not strongly connected
• is not small
• is not countable
• is not locally cocartesian coclosed
• does not have disjoint products
• is not codistributive
• is not coextensive
• is not self-dual
• is not unital
• does not have zero morphisms
• is not counital
• is not coaccessible
• is not countably codistributive
• is not countably coextensive
• does not have biproducts
• does not have kernels
• does not satisfy CIP
• is not infinitary codistributive
• does not have cokernels
• does not satisfy CSP
• is not infinitary coextensive

*This also uses the deduced satisfied properties.