CatDat

Properties

Properties from the database

• is an elementary topos
• is essentially small

Deduced properties

• is well-powered

  Since it is essentially small, we deduce that it is well-powered.
• is well-copowered

  Since it is essentially small, we deduce that it is well-copowered.
• is locally essentially small

  Since it is essentially small, we deduce that it is locally essentially small.
• is cartesian closed

  Since it is an elementary topos, we deduce that it is cartesian closed.
• is finitely complete

  Since it is an elementary topos, we deduce that it is finitely complete.
• has a subobject classifier

  Since it is an elementary topos, we deduce that it has a subobject classifier.
• is finitely cocomplete

  Since it is an elementary topos, we deduce that it is finitely cocomplete.
• has disjoint finite coproducts

  Since it is an elementary topos, we deduce that it has disjoint finite coproducts.
• is epi-regular

  Since it is an elementary topos, we deduce that it is epi-regular.
• has finite products

  Since it is cartesian closed, we deduce that it has finite products.
• is mono-regular

  Since it has a subobject classifier, we deduce that it is mono-regular.
• is balanced

  Since it is mono-regular, we deduce that it is balanced.
• has finite coproducts

  Since it is finitely cocomplete, we deduce that it has finite coproducts.
• has coequalizers

  Since it is finitely cocomplete, we deduce that it has coequalizers.
• has an initial object

  Since it has finite coproducts, we deduce that it has an initial object.
• has binary coproducts

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

  Since it has binary coproducts and has coequalizers, we deduce that it has pushouts.
• is connected

  Since it has an initial object, we deduce that it is connected.
• is Cauchy complete

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

  Since it is finitely complete, we deduce that it has equalizers.
• has a terminal object

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

  Since it is cartesian closed and has finite coproducts, we deduce that it is distributive.
• has a strict initial object

  Since it is cartesian closed and has an initial object, we deduce that it has a strict initial object.
• is inhabited

  Since it is connected, we deduce that it is inhabited.

Non-Properties

Non-Properties from the database

• is not locally small
• does not have a strict terminal object
• does not have countable products
• does not have countable coproducts
• is not skeletal

  trivial
• is not Malcev

Deduced Non-Properties*

• is not small

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

  Assume for a contradiction that it is complete. Since it is complete, we deduce that it has products. Since it has products, we deduce that it has countable products. This is a contradiction since we already know that it does not have countable products.
• is not cocomplete

  Assume for a contradiction that it is cocomplete. Since it is cocomplete, we deduce that it has coproducts. Since it has coproducts, we deduce that it has countable coproducts. This is a contradiction since we already know that it does not have countable coproducts.
• does not have zero morphisms

  Assume for a contradiction that it has zero morphisms. Since it has zero morphisms and has an initial object, we deduce that it is pointed. Since it is pointed and is cartesian closed, we deduce that it is trivial. Since it is trivial, we deduce that it is split abelian. Since it is split abelian, we deduce that it is abelian. Since it is abelian, we deduce that it is Malcev. This is a contradiction since we already know that it is not Malcev.
• is not preadditive

  Assume for a contradiction that it is preadditive. Since it is preadditive, we deduce that it has zero morphisms. This is a contradiction since we already know that it does not have zero morphisms.
• 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 pointed

  Assume for a contradiction that it is pointed. Since it is pointed, we deduce that it has zero morphisms. This is a contradiction since we already know that it does not have zero morphisms.
• is not locally finitely presentable

  Assume for a contradiction that it is locally finitely presentable. Since it is locally finitely presentable, we deduce that it is locally presentable. Since it is locally presentable, we deduce that it is complete. This is a contradiction since we already know that it is not complete.
• is not locally presentable

  Assume for a contradiction that it is locally presentable. Since it is locally presentable, we deduce that it is complete. This is a contradiction since we already know that it is not complete.
• is not locally ℵ₁-presentable

  Assume for a contradiction that it is locally ℵ₁-presentable. Since it is locally ℵ₁-presentable, we deduce that it is locally presentable. This is a contradiction since we already know that it is not locally presentable.
• 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 locally presentable. This is a contradiction since we already know that it is not locally presentable.
• does not have products

  Assume for a contradiction that it has products. Since it has products and has equalizers, we deduce that it is complete. This is a contradiction since we already know that it is not complete.
• does not have coproducts

  Assume for a contradiction that it has coproducts. Since it has coproducts and has coequalizers, we deduce that it is cocomplete. This is a contradiction since we already know that it is not cocomplete.
• does not have filtered limits

  Assume for a contradiction that it has filtered limits. Since it has finite products and has filtered limits, we deduce that it has products. This is a contradiction since we already know that it does not have products.
• does not have filtered colimits

  Assume for a contradiction that it has filtered colimits. Since it has finite coproducts and has filtered colimits, we deduce that it has coproducts. This is a contradiction since we already know that it does not have coproducts.
• does not have sequential limits

  Assume for a contradiction that it has sequential limits. Since it has finite products and has sequential limits, we deduce that it has countable products. This is a contradiction since we already know that it does not have countable products.
• does not have sequential colimits

  Assume for a contradiction that it has sequential colimits. Since it has finite coproducts and has sequential colimits, we deduce that it has countable coproducts. This is a contradiction since we already know that it does not have countable coproducts.
• does not have connected limits

  Assume for a contradiction that it has connected limits. Since it has connected limits, we deduce that it has wide pullbacks. Since it has wide pullbacks and has a terminal object, we deduce that it is complete. This is a contradiction since we already know that it is not complete.
• does not have connected colimits

  Assume for a contradiction that it has connected colimits. Since it has connected colimits, we deduce that it has wide pushouts. Since it has wide pushouts and has an initial object, we deduce that it is cocomplete. This is a contradiction since we already know that it is not cocomplete.
• is not self-dual

  Assume for a contradiction that it is self-dual. Since it is self-dual and has a strict initial object, we deduce that it has a strict terminal object. This is a contradiction since we already know that it does not have a strict terminal object.
• 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 thin

  Assume for a contradiction that it is thin. Since it is thin and is finitely complete, we deduce that it is Malcev. This is a contradiction since we already know that it is not Malcev.
• is not discrete

  Assume for a contradiction that it is discrete. Since it is discrete, we deduce that it is locally small. This is a contradiction since we already know that it is not locally small.
• 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 finitary algebraic

  Assume for a contradiction that it is finitary algebraic. Since it is finitary algebraic, we deduce that it is locally finitely presentable. This is a contradiction since we already know that it is not locally finitely presentable.
• 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 finitary algebraic. This is a contradiction since we already know that it is not finitary algebraic.
• is not infinitary distributive

  Assume for a contradiction that it is infinitary distributive. Since it is infinitary distributive, we deduce that it has coproducts. This is a contradiction since we already know that it does not have coproducts.
• does not have exact filtered colimits

  Assume for a contradiction that it has exact filtered colimits. Since it has exact filtered colimits, we deduce that it has filtered colimits. This is a contradiction since we already know that it does not have filtered colimits.
• 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.
• does not have disjoint coproducts

  Assume for a contradiction that it has disjoint coproducts. Since it has disjoint coproducts, we deduce that it has coproducts. This is a contradiction since we already know that it does not have coproducts.
• 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 equalizers, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
• does not have wide pullbacks

  Assume for a contradiction that it has wide pullbacks. Since it has wide pullbacks and has equalizers, we deduce that it has connected limits. This is a contradiction since we already know that it does not have connected limits.
• does not have wide pushouts

  Assume for a contradiction that it has wide pushouts. Since it has wide pushouts and has coequalizers, we deduce that it has connected colimits. This is a contradiction since we already know that it does not have connected colimits.
• 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.

*This also uses the deduced properties.