category of commutative monoids

Satisfied Properties

Assigned properties

• is locally small
• is pointed
• is one-sorted finitary algebraic
• has biproducts

Deduced properties

• has a natural numbers object
• has finite coproducts
• has finite products
• has zero morphisms
• is finitary algebraic
• is well-copowered
• has an extremal generator
• has an initial object
• is locally essentially small
• has a terminal object
• is locally finitely presentable
• is cocomplete
• is a generalized variety
• is regular
• is multi-algebraic
• is connected
• has a multi-terminal object
• is ℵ₁-filtered
• has an extremal generating collection
• has a generator
• has binary products
• has finite powers
• has a multi-initial object
• is ℵ₁-cofiltered
• has binary coproducts
• has finite copowers
• is finitely accessible
• has exact filtered colimits
• is locally ℵ₁-presentable
• has sifted colimits
• is ℵ₁-accessible
• has filtered-colimit-stable monomorphisms
• is locally finitely multi-presentable
• is multi-cocomplete
• has effective congruences
• has coequalizers of kernel pairs
• is finitely complete
• is inhabited
• is filtered
• has a generating collection
• has binary powers
• has connected colimits
• is finitely cocomplete
• has coequalizers
• has coproducts
• is cofiltered
• has binary copowers
• is unital
• is locally presentable
• is accessible
• has ℵ₁-filtered colimits
• has filtered colimits
• has connected limits
• has equalizers
• has quotients of congruences
• is Barr-exact
• is strongly connected
• has cartesian filtered colimits
• is sifted
• has reflexive coequalizers
• has kernel pairs
• is Cauchy complete
• is concretizable
• is total
• is counital
• has cokernels
• is cosifted
• has copowers
• has ℵ₂-small coproducts
• has pushouts
• has wide pushouts
• is well-powered
• is complete
• is locally multi-presentable
• is semi-strongly connected
• has disjoint finite products
• has coreflexive equalizers
• has kernels
• has directed colimits
• has pullbacks
• has wide pullbacks
• has disjoint finite coproducts
• has cosifted limits
• has countable coproducts
• has ℵ₂-small copowers
• has cokernel pairs
• is locally poly-presentable
• has products
• is multi-complete
• has disjoint coproducts
• has cofiltered limits
• has coquotients of cocongruences
• has sequential colimits
• has countable copowers
• satisfies CIP
• has powers
• has ℵ₂-small products
• has equalizers of cokernel pairs
• has disjoint products
• has cocartesian cofiltered limits
• has directed limits
• has ℵ₁-cofiltered limits
• has sequential limits
• has countable products
• has ℵ₂-small powers
• has countable powers

Unsatisfied Properties

Assigned properties

• is not skeletal
• is not balanced
• is not Malcev
• is not co-Malcev
• does not have a cogenerator
• is not coregular
• does not have a regular subobject classifier
• does not have a regular quotient object classifier
• does not satisfy CSP

Deduced properties*

• is not thin
• is not additive
• is not Grothendieck abelian
• is not discrete
• does not have a cogenerating collection
• is not mono-regular
• is not right cancellative
• does not have a subobject classifier
• is not gaunt
• is not direct
• is not a quasitopos
• is not a Grothendieck topos
• is not abelian
• is not locally cocartesian coclosed
• is not Barr-coexact
• does not have cofiltered-limit-stable epimorphisms
• does not have an extremal cogenerator
• is not epi-regular
• is not left cancellative
• does not have a quotient object classifier
• is not inverse
• is not self-dual
• is not preadditive
• is not split abelian
• is not extensive
• is not trivial
• is not essentially discrete
• does not have a strict terminal object
• is not a groupoid
• is not normal
• is not regular-subobject-trivial
• is not subobject-trivial
• is not core-thin
• 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 core-connected
• does not have a strict initial object
• does not have exact cofiltered limits
• is not quotient-trivial
• does not have an extremal cogenerating collection
• is not conormal
• is not regular-quotient-trivial
• does not have a parametrized natural numbers object
• is not locally copresentable
• is not locally cartesian closed
• is not distributive
• is not countably extensive
• is not small
• is not finite
• is not countable
• is not a pretopos
• is not coaccessible
• is not codistributive
• is not coextensive
• is not cartesian closed
• is not countably distributive
• is not infinitary extensive
• is not countably codistributive
• is not cocartesian coclosed
• is not countably coextensive
• is not infinitary distributive
• is not infinitary codistributive
• is not infinitary coextensive

*This also uses the deduced satisfied properties.