category of pointed topological spaces
- notation:
- objects: pointed topological spaces
- morphisms: pointed continuous functions
- Related categories: ,
- nLab Link
This category plays an important role in algebraic topology and homotopy theory. Although it may appear similar to , adding a base point drastically changes its categorical properties. In particular, it introduces a zero object.
Satisfied Properties
Properties from the database
- has coequalizers
- has a cogenerator
- is complete
- has coproducts
- is coregular
- is counital
- has disjoint finite coproducts
- has disjoint finite products
- has a generator
- is locally small
- is pointed
- has a regular subobject classifier
- is well-copowered
- is well-powered
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has products
- has countable products
- has finite products
- has binary products
- has a terminal object
- is connected
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has powers
- has countable powers
- has finite powers
- has binary powers
- has an initial object
- has zero morphisms
- has disjoint coproducts
- has finite coproducts
- is locally essentially small
- has a generating set
- is inhabited
- is strongly connected
- has cosifted limits
- has reflexive coequalizers
- has countable coproducts
- has binary coproducts
- is cocomplete
- has connected colimits
- has sifted colimits
- has filtered colimits
- has directed colimits
- is finitely cocomplete
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
- has disjoint products
- has a cogenerating set
Unsatisfied Properties
Properties from the database
- is not balanced
- is not co-Malcev
- does not have exact filtered colimits
- is not locally presentable
- is not regular
- is not skeletal
- is not unital
Deduced properties*
- does not have biproducts
- is not mono-regular
- is not a groupoid
- is not direct
- is not discrete
- is not additive
- is not preadditive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not trivial
- is not thin
- does not have a strict terminal object
- does not have a strict initial object
- is not left cancellative
- is not right cancellative
- is not distributive
- is not infinitary distributive
- is not cartesian closed
- is not extensive
- is not infinitary extensive
- is not lextensive
- is not one-way
- is not essentially small
- is not small
- is not finite
- is not essentially finite
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not an elementary topos
- does not have a subobject classifier
- is not locally cartesian closed
- is not a Grothendieck topos
- is not Malcev
- is not codistributive
- is not infinitary codistributive
- is not coextensive
- is not infinitary coextensive
- is not epi-regular
- is not inverse
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
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Special objects
- terminal object: singleton space with the unique base point
- initial object: singleton space with the unique base point
- products: direct product with the product topology and the obvious base point
- coproducts: wedge sum, aka one-point union
Special morphisms
- isomorphisms: pointed homeomorphisms
- monomorphisms: injective pointed continuous maps
- epimorphisms: surjective pointed continuous maps
- regular monomorphisms: embeddings
- regular epimorphisms: surjective pointed quotient maps