CatDat

Properties

Properties from the database

• is locally small

  There is a forgetful functor $\mathbf{Rng} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.
• is finitary algebraic

  Take the algebraic theory of a rng.
• is pointed

  The zero ring is a zero object.
• has disjoint coproducts
• is Malcev

  follows in the same way as for (additive) groups

Deduced properties

• is locally essentially small

  Since it is locally small, we deduce that it is locally essentially small.
• has zero morphisms

  Since it is pointed, we deduce that it has zero morphisms.
• has an initial object

  Since it is pointed, we deduce that it has an initial object.
• has coproducts

  Since it has disjoint coproducts, we deduce that it has coproducts.
• has disjoint finite coproducts

  Since it has disjoint coproducts, we deduce that it has disjoint finite coproducts.
• has finite coproducts

  Since it has disjoint finite coproducts, we deduce that it has finite coproducts.
• is locally finitely presentable

  Since it is finitary algebraic, we deduce that it is locally finitely presentable.
• is finitely complete

  Since it is Malcev, we deduce that it is finitely complete.
• has countable coproducts

  Since it has coproducts, we deduce that it has countable coproducts.
• has binary coproducts

  Since it has finite coproducts, we deduce that it has binary coproducts.
• is connected

  Since it has an initial object, we deduce that it is connected.
• has a terminal object

  Since it is pointed, we deduce that it has a terminal object.
• has finite products

  Since it is finitely complete, we deduce that it has finite products.
• has equalizers

  Since it is finitely complete, we deduce that it has equalizers.
• has binary products

  Since it has finite products, we deduce that it has binary products.
• has pullbacks

  Since it has binary products and has equalizers, we deduce that it has pullbacks.
• is Cauchy complete

  Since it has equalizers, we deduce that it is Cauchy complete.
• is locally presentable

  Since it is locally finitely presentable, we deduce that it is locally presentable.
• has exact filtered colimits

  Since it is locally finitely presentable, we deduce that it has exact filtered colimits.
• is locally ℵ₁-presentable

  Since it is locally finitely presentable, we deduce that it is locally ℵ₁-presentable.
• is inhabited

  Since it is connected, we deduce that it is inhabited.
• has filtered colimits

  Since it has exact filtered colimits, we deduce that it has filtered colimits.
• is well-powered

  Since it is locally presentable, we deduce that it is well-powered.
• is well-copowered

  Since it is locally presentable, we deduce that it is well-copowered.
• is complete

  Since it is locally presentable, we deduce that it is complete.
• is cocomplete

  Since it is locally presentable, we deduce that it is cocomplete.
• has a generator

  Since it is locally presentable, we deduce that it has a generator.
• is finitely cocomplete

  Since it is cocomplete, we deduce that it is finitely cocomplete.
• has wide pushouts

  Since it is cocomplete, we deduce that it has wide pushouts.
• has connected colimits

  Since it is cocomplete, we deduce that it has connected colimits.
• has coequalizers

  Since it is cocomplete, we deduce that it has coequalizers.
• has pushouts

  Since it has binary coproducts and has coequalizers, we deduce that it has pushouts.
• has sequential colimits

  Since it has coequalizers and has countable coproducts, we deduce that it has sequential colimits.
• has filtered limits

  Since it is complete, we deduce that it has filtered limits.
• has wide pullbacks

  Since it is complete, we deduce that it has wide pullbacks.
• has connected limits

  Since it is complete, we deduce that it has connected limits.
• has products

  Since it is complete, we deduce that it has products.
• has countable products

  Since it has products, we deduce that it has countable products.
• has sequential limits

  Since it has equalizers and has countable products, we deduce that it has sequential limits.

Non-Properties

Non-Properties from the database

• is not preadditive
• is not balanced
• does not have a cogenerator
• is not skeletal

  trivial

Deduced Non-Properties*

• is not small

  Assume for a contradiction that it is small. Since it is small, we deduce that it is essentially small. Since it is essentially small and has products, we deduce that it is thin. Since it is thin and is inhabited, we deduce that it has a cogenerator. This is a contradiction since we already know that it does not have a cogenerator.
• is not cartesian closed

  Assume for a contradiction that it is cartesian closed. Since it is pointed and is cartesian closed, we deduce that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. Since it is a Grothendieck topos, we deduce that it has a cogenerator. This is a contradiction since we already know that it does not have a cogenerator.
• is not additive

  Assume for a contradiction that it is additive. Since it is additive, we deduce that it is preadditive. This is a contradiction since we already know that it is not preadditive.
• is not abelian

  Assume for a contradiction that it is abelian. Since it is abelian, we deduce that it is additive. This is a contradiction since we already know that it is not additive.
• is not an elementary topos

  Assume for a contradiction that it is an elementary topos. Since it is an elementary topos, we deduce that it is cartesian closed. This is a contradiction since we already know that it is not cartesian closed.
• is not a Grothendieck topos

  Assume for a contradiction that it is a Grothendieck topos. Since it is a Grothendieck topos, we deduce that it is an elementary topos. This is a contradiction since we already know that it is not an elementary topos.
• does not have a strict initial object

  Assume for a contradiction that it has a strict initial object. Since it has a strict initial object and is pointed, we deduce that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
• does not have a strict terminal object

  Assume for a contradiction that it has a strict terminal object. Since it has a strict terminal object and is pointed, we deduce that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
• is not self-dual

  Assume for a contradiction that it is self-dual. Since it is locally presentable and is self-dual, we deduce that it is thin. Since it is thin and is inhabited, we deduce that it has a cogenerator. This is a contradiction since we already know that it does not have a cogenerator.
• is not a groupoid

  Assume for a contradiction that it is a groupoid. Since it is a groupoid, we deduce that it is self-dual. This is a contradiction since we already know that it is not self-dual.
• is not essentially small

  Assume for a contradiction that it is essentially small. Since it is essentially small and has products, we deduce that it is thin. Since it is thin, we deduce that it is left cancellative. Since it is left cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
• is not thin

  Assume for a contradiction that it is thin. Since it is thin, we deduce that it is left cancellative. Since it is left cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
• is not discrete

  Assume for a contradiction that it is discrete. Since it is discrete, we deduce that it is skeletal. This is a contradiction since we already know that it is not skeletal.
• is not essentially discrete

  Assume for a contradiction that it is essentially discrete. Since it is essentially discrete, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
• is not finite

  Assume for a contradiction that it is finite. Since it is finite, we deduce that it is small. This is a contradiction since we already know that it is not small.
• is not essentially finite

  Assume for a contradiction that it is essentially finite. Since it is essentially finite and has finite products, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
• is not trivial

  Assume for a contradiction that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
• does not have a subobject classifier

  Assume for a contradiction that it has a subobject classifier. Since it has a subobject classifier, we deduce that it is mono-regular. Since it is mono-regular, we deduce that it is balanced. This is a contradiction since we already know that it is not balanced.
• is not infinitary distributive

  Assume for a contradiction that it is infinitary distributive. Since it is infinitary distributive, we deduce that it is distributive. Since it is distributive, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
• is not distributive

  Assume for a contradiction that it is distributive. Since it is distributive, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
• is not Grothendieck abelian

  Assume for a contradiction that it is Grothendieck abelian. Since it is Grothendieck abelian, we deduce that it is abelian. This is a contradiction since we already know that it is not abelian.
• is not left cancellative

  Assume for a contradiction that it is left cancellative. Since it is left cancellative and has coequalizers, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
• is not right cancellative

  Assume for a contradiction that it is right cancellative. Since it is right cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
• is not split abelian

  Assume for a contradiction that it is split abelian. Since it is split abelian, we deduce that it is abelian. This is a contradiction since we already know that it is not abelian.
• is not mono-regular

  Assume for a contradiction that it is mono-regular. Since it is mono-regular, we deduce that it is balanced. This is a contradiction since we already know that it is not balanced.
• is not epi-regular

  Assume for a contradiction that it is epi-regular. Since it is epi-regular, we deduce that it is balanced. This is a contradiction since we already know that it is not balanced.

*This also uses the deduced properties.