The category of topological spaces has all limits and colimits

Further, these limits and colimits are preserved by the forgetful functor --- that is, the underlying types are just the limits in the category of types.

def TopCat.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.Cone F

A choice of limit cone for a functor F : J ⥤ TopCat. Generally you should just use limit.cone F, unless you need the actual definition (which is in terms of Types.limitCone).

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def TopCat.limitConeInfi {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.Cone F

A choice of limit cone for a functor F : J ⥤ TopCat whose topology is defined as an infimum of topologies infimum. Generally you should just use limit.cone F, unless you need the actual definition (which is in terms of Types.limitCone).

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def TopCat.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.IsLimit (TopCat.limitCone F)

The chosen cone TopCat.limitCone F for a functor F : J ⥤ TopCat is a limit cone. Generally you should just use limit.isLimit F, unless you need the actual definition (which is in terms of Types.limitConeIsLimit).

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def TopCat.limitConeInfiIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.IsLimit (TopCat.limitConeInfi F)

The chosen cone TopCat.limitConeInfi F for a functor F : J ⥤ TopCat is a limit cone. Generally you should just use limit.isLimit F, unless you need the actual definition (which is in terms of Types.limitConeIsLimit).

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instance TopCat.topCat_hasLimitsOfSize : CategoryTheory.Limits.HasLimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} TopCat

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• TopCat.topCat_hasLimitsOfSize = ⋯

instance TopCat.topCat_hasLimits : CategoryTheory.Limits.HasLimits TopCat

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• TopCat.topCat_hasLimits = TopCat.topCat_hasLimitsOfSize

instance TopCat.forgetPreservesLimitsOfSize : CategoryTheory.Limits.PreservesLimitsOfSize.{v, v, max u v, max u v, (max u v) + 1, (max u v) + 1} (CategoryTheory.forget TopCat)

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instance TopCat.forgetPreservesLimits : CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget TopCat)

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• TopCat.forgetPreservesLimits = TopCat.forgetPreservesLimitsOfSize

def TopCat.colimitCocone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.Cocone F

A choice of colimit cocone for a functor F : J ⥤ TopCat. Generally you should just use colimit.cocone F, unless you need the actual definition (which is in terms of Types.colimitCocone).

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def TopCat.colimitCoconeIsColimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J TopCat) : CategoryTheory.Limits.IsColimit (TopCat.colimitCocone F)

The chosen cocone TopCat.colimitCocone F for a functor F : J ⥤ TopCat is a colimit cocone. Generally you should just use colimit.isColimit F, unless you need the actual definition (which is in terms of Types.colimitCoconeIsColimit).

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instance TopCat.topCat_hasColimitsOfSize : CategoryTheory.Limits.HasColimitsOfSize.{v, v, max u v, max (u + 1) (v + 1)} TopCat

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• TopCat.topCat_hasColimitsOfSize = ⋯

instance TopCat.topCat_hasColimits : CategoryTheory.Limits.HasColimits TopCat

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• TopCat.topCat_hasColimits = TopCat.topCat_hasColimitsOfSize

instance TopCat.forgetPreservesColimitsOfSize : CategoryTheory.Limits.PreservesColimitsOfSize.{v, v, max v u_1, max v u_1, (max v u_1) + 1, (max v u_1) + 1} (CategoryTheory.forget TopCat)

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• One or more equations did not get rendered due to their size.

instance TopCat.forgetPreservesColimits : CategoryTheory.Limits.PreservesColimits (CategoryTheory.forget TopCat)

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• TopCat.forgetPreservesColimits = TopCat.forgetPreservesColimitsOfSize

def TopCat.isTerminalPUnit : CategoryTheory.Limits.IsTerminal (TopCat.of PUnit.{u + 1} )

The terminal object of Top is PUnit.

Equations

• TopCat.isTerminalPUnit = CategoryTheory.Limits.IsTerminal.ofUnique (TopCat.of PUnit.{?u.10 + 1} )

def TopCat.terminalIsoPUnit : ⊤_ TopCat ≅ TopCat.of PUnit.{u + 1}

The terminal object of Top is PUnit.

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• TopCat.terminalIsoPUnit = CategoryTheory.Limits.terminalIsTerminal.uniqueUpToIso TopCat.isTerminalPUnit

def TopCat.isInitialPEmpty : CategoryTheory.Limits.IsInitial (TopCat.of PEmpty.{u + 1} )

The initial object of Top is PEmpty.

Equations

• TopCat.isInitialPEmpty = CategoryTheory.Limits.IsInitial.ofUnique (TopCat.of PEmpty.{?u.10 + 1} )

def TopCat.initialIsoPEmpty : ⊥_ TopCat ≅ TopCat.of PEmpty.{u + 1}

The initial object of Top is PEmpty.

Equations

• TopCat.initialIsoPEmpty = CategoryTheory.Limits.initialIsInitial.uniqueUpToIso TopCat.isInitialPEmpty