Shape of compacta as extension of weak homotopy of finite spaces

Abstract

We construct a category that classifies compact Hausdorff spaces by their shape and finite topological spaces by their weak homotopy type.

A combinatorial description of shape theory

“…From an algebraic point of view, this result is not surprising (see Proposition 2.11). Nevertheless, every finite connected space has trivial shape (see [10]). Given a compact metric space X and a finite approximation (U 4 n (A n ), q n,n+1 ) of it, we can apply other functors.…”

“…In that way, we may have a sort of computational or combinatorial shape theory. There are other previous results that point out some of the relations between finiteness and shape theory, see [22,23] or [10]. In [23] an intrinsic description of shape theory is given using sequences of continuous maps defined on open dense subsets of the compact metric spaces and having finite images.…”

“…In [22] it is given an intrinsic representation of Čech homology of compacta in terms of inverse limits of discrete approximations. In [10] it is constructed a category that classifies compacta by their shape and finite topological spaces by their weak homotopy type.…”