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Pointed space - Category of pointed spaces

Pointed space - Category of pointed spaces: Encyclopedia II - Pointed space - 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 ...

Pointed space: Encyclopedia II - Pointed space - Category of pointed spaces

Pointed space - 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.

Note that 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.

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Category theory, Homotopy theory, Topology, algebraic topology, category, category of topological spaces, class, comma category, commutes, continuous maps, coproduct, coslice category, discrete space, disjoint union, equivalence relation, forgetful functor, fundamental group, homotopy theory, inclusion map, left adjoint, mathematics, morphisms, product, quotient, reduced suspension, smash product, terminal object, topological product, topological space, topological subspace, wedge sum, zero object

Adapted from the Wikipedia article "Category of pointed spaces", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki

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