Category of Pointed Spaces
The class of all pointed spaces forms a category Top• with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ({•} ↓ Top) where {•} is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted {•}/Top.) Objects in this category are continuous maps {•} → X. Such morphisms can be thought of as picking out a basepoint in X. Morphisms in ({•} ↓ 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 f preserves basepoints.
As a pointed space {•} is a zero object in Top• while it is only a terminal object in Top.
There is a forgetful functor Top• → Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space X the disjoint union of X and a one point space {•} whose single element is taken to be the basepoint.
Read more about this topic: Pointed Space
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