CatDat

Properties

Properties from the database

• is locally small

  There is a forgetful functor $\mathbf{Top} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.
• is complete

  Take the limit of the underlying sets and endow it with the coarsest topology making all projections continuous.
• is cocomplete

  Take the colimit of the underlying sets and endow it with the finest topology making all inclusions continuous.
• is well-powered

  This is clear from the classification of monomorphisms as injective continuous maps.
• is well-copowered

  This is clear from the classification of epimorphisms as surjective continuous maps.
• is infinitary distributive

  The canonical continuous map $\coprod_i (X \times Y_i) \to X \times \coprod_{i \in I} Y_i$ is bijective since $\mathbf{Set}$ is infinitary distributive, and one easily checks with the definitions of product and coproduct topologies that it maps open sets to open sets.
• has a generator

  The one-point space is a generator since it represents the forgetful functor $\mathbf{Top} \to \mathbf{Set}$.
• has disjoint coproducts

  This follows easily from the corresponding fact for sets.

Deduced properties

• is locally essentially small

  Since it is locally small, we deduce that it is locally essentially small.
• is finitely complete

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

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

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

  Since it has products, we deduce that it has countable products.
• 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 connected

  Since it has a terminal object, we deduce that it is connected.
• 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 Cauchy complete

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

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

  Since it is infinitary distributive, we deduce that it is distributive.
• has a strict initial object

  Since it is distributive, we deduce that it has a strict initial object.
• is inhabited

  Since it is connected, we deduce that it is inhabited.
• is finitely cocomplete

  Since it is cocomplete, we deduce that it is finitely cocomplete.
• has filtered colimits

  Since it is cocomplete, we deduce that it has filtered colimits.
• 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 countable coproducts

  Since it has coproducts, we deduce that it has countable coproducts.
• 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.
• has sequential colimits

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

Non-Properties

Non-Properties from the database

• is not cartesian closed
• is not locally presentable
• does not have a strict terminal object
• is not balanced
• does not have a cogenerator
• does not have exact filtered colimits
• 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 essentially small. Since it is essentially small and has products, we deduce that it is thin. Since it is essentially small and is thin and is complete and is infinitary distributive, we deduce that it is cartesian closed. This is a contradiction since we already know that it is not cartesian closed.
• 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 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. 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.
• 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. This is a contradiction since we already know that it is not locally presentable.
• 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 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.
• 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 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 essentially small and is thin and is complete and is infinitary distributive, we deduce that it is cartesian closed. This is a contradiction since we already know that it is not cartesian closed.
• 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 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 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.
• 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 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 equalizers, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
• 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.