For these purposes a closed *cell* is a topological space homeomorphic to a simplex, or equally a ball (sphere plus interior) or cube in *n* dimensions. Only the topological nature matters: but one does want to keep track of the subspace on the 'surface' (the sphere that bounds the ball), and its complement, the interior points. A general *cell complex* would be a topological space *X* that is covered by cells; or to put it another way, we start with a space that is the disjoint union of some collection of cells, and take *X* as a quotient space, for some equivalence relation. This is too general a concept.

Table of contents |

2 CW complexes are defined inductively 3 'The' homotopy category |

A cell is attached by gluing a closed *n*-dimensional ball *D ^{n}* to the

Assume that *X* is to be a Hausdorff space: for the purposes of homotopy theory this loses nothing important. Then since closed cells are compact spaces, we can be sure that their images in *X* are also compact, closed subspaces. From now on, we refer to 'closed cells', and 'open cells', as subspaces of X, the open cell being the image of the distinguished interior.

A 0-cell is just a point; if we only have 0-cells building up a Hausdorff space, it must be a discrete space. The general CW-complex definition can proceed by induction, using this as the base case.

The first restriction is the *closure-finite* one: each closed cell should be covered by a finite union of open cells.

The other restriction is to do with the possibility of having infinitely many cells, of unbounded dimension. The space X will be presented as a limit of subspaces *X*_{i} for i = 0, 1, 2, 3, … . How do we infer a topological structure for X? This is a colimit, in category theory terms. From the continuity of each mapping *X*_{i} to *X*, a closed set in *X* must have a closed inverse image in each *X*_{i}; and so must intersect each closed cell in a closed subset. We can turn this round, and say that a subset C of X is by definition closed precisely when the intersection of C with the closed cells in X is always closed.

With all those preliminaries, the definition of CW-complex runs like this: given *X*_{0} a discrete space, and inductively constructed subspaces *X*_{i} obtained from *X*_{i-1} by attaching some collection of i-cells, the resulting colimit space *X* is called a **CW-complex** provided it is given the weak topology, and the closure-finite condition is satisfied for its closed cells.

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 Map(*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 Map(*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.

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 base-points in each space: for example the fundamental group of a connected space is, properly speaking, dependent on the base-point chosen. A base point is in effect a mapping {pt} -> *X* for each space *X*; morphisms should respect base points, and all homotopies too. The need to use base points has a significant effect on the products (and other limits) appropriate to use. For example in homotopy theory the *smash product* of spaces *X* and *Y* is used, which is *XxY* with *Xx*{y} and {x}*xY* collapsed to one base point.

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.

Auxiliary constructions may mean that 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 (Brown’s representability theorem).

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