CW complex

In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties); but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: CW stands for closure-finite weak topology.

'The' homotopy category

The idea of a homotopy category is to start with a topological space category, that is, one in which objects are topological spaces and morphisms are continuous mappings, and abstractly to replace the sets Mor(X, Y) of morphisms by sets of equivalence classes Hot(X, Y) that are defined by the homotopy relation. So, the objects remain the same; but the morphisms have been gathered into collections. Under favourable conditions Mor(X, Y) is itself a function space and the procedure is to take its set of components under path-connection as a simpler version: this provides the intuitive picture.

Related Topics:
Morphism - Homotopy - Favourable conditions - Function space

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The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category. In fact, for technical 'administrative' reasons a homotopy category must keep track of basepoints in each space: for example the fundamental group of a connected space is, properly speaking, dependent on the basepoint chosen. A topological space with a distinguished basepoint is called a pointed space. The need to use basepoints has a significant effect on the products (and other limits) appropriate to use. For example, in homotopy theory, the smash product X ∧ Y of spaces X and Y is used.

Related Topics:
Basepoint - Fundamental group - Connected space - Pointed space - Smash product

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To a large extent the business of homotopy theory is to describe the homotopy category; in fact it turns out that calculating Hot(X, Y) is hard, as a general problem, and much effort has been put into the most interesting cases, for example where X and Y are spheres (the homotopy groups of spheres).

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Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion, but half a century since Whitehead has left this definition of homotopy category in good shape. One basic result is that the representable functors on the homotopy category have a simple characterisation (the Brown representability theorem).

Related Topics:
Representable functor - Brown representability theorem

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One important later development was that of spectra in homotopy theory, essentially the derived category idea in a form useful for topologists.

Related Topics:
Spectra - Derived category

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