Pointed space

The class of all pointed spaces forms a category Top_{${\displaystyle \bullet }$} with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, (${\displaystyle \{\bullet \}\downarrow }$ Top) where ${\displaystyle \{\bullet \}}$ is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted ${\displaystyle \{\bullet \}/}$Top.) Objects in this category are continuous maps ${\displaystyle \{\bullet \}\to X.}$ Such maps can be thought of as picking out a basepoint in ${\displaystyle X.}$ Morphisms in (${\displaystyle \{\bullet \}\downarrow }$ Top) are morphisms in Top for which the following diagram commutes:

It is easy to see that commutativity of the diagram is equivalent to the condition that ${\displaystyle f}$ preserves basepoints.

As a pointed space, ${\displaystyle \{\bullet \}}$ is a zero object in Top_{${\displaystyle \{\bullet \}}$}, while it is only a terminal object in Top.

There is a forgetful functor Top_{${\displaystyle \{\bullet \}}$} ${\displaystyle \to }$ Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space ${\displaystyle X}$ the disjoint union of ${\displaystyle X}$ and a one-point space ${\displaystyle \{\bullet \}}$ whose single element is taken to be the basepoint.

• A subspace of a pointed space ${\displaystyle X}$ is a topological subspace ${\displaystyle A\subseteq X}$ which shares its basepoint with ${\displaystyle X}$ so that the inclusion map is basepoint preserving.
• One can form the quotient of a pointed space ${\displaystyle X}$ under any equivalence relation. The basepoint of the quotient is the image of the basepoint in ${\displaystyle X}$ under the quotient map.
• One can form the product of two pointed spaces ${\displaystyle \left(X,x_{0}\right),}$ ${\displaystyle \left(Y,y_{0}\right)}$ as the topological product ${\displaystyle X\times Y}$ with ${\displaystyle \left(x_{0},y_{0}\right)}$serving as the basepoint.
• The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the 'one-point union' of spaces.
• The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such as compactly generated weak Hausdorff ones.
• The reduced suspension ${\displaystyle \Sigma X}$ of a pointed space ${\displaystyle X}$ is (up to a homeomorphism) the smash product of ${\displaystyle X}$ and the pointed circle ${\displaystyle S^{1}.}$
• The reduced suspension is a functor from the category of pointed spaces to itself. This functor is left adjoint to the functor ${\displaystyle \Omega }$ taking a pointed space ${\displaystyle X}$ to its loop space ${\displaystyle \Omega X}$.

• Category of groups – category in mathematicsPages displaying wikidata descriptions as a fallback
• Category of metric spaces – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphismsPages displaying wikidata descriptions as a fallback
• Category of sets – Category in mathematics where the objects are sets
• Category of topological spaces – category whose objects are topological spaces and whose morphisms are continuous mapsPages displaying wikidata descriptions as a fallback
• Category of topological vector spaces – Topological category

• Gamelin, Theodore W.; Greene, Robert Everist (1999) [1983]. Introduction to Topology (second ed.). Dover Publications. ISBN 0-486-40680-6.
• Mac Lane, Saunders (September 1998). Categories for the Working Mathematician (second ed.). Springer. ISBN 0-387-98403-8.

• mathoverflow discussion on several base points and groupoids