Shape theory (mathematics)
Shape theory is a branch of topology, generalizing the idea of homotopy theory to cases with unfavorable local properties. The overall goal of shape theory is adapt the methods and results from homotopy theory to more general spaces, such as compact metric spaces or compact Hausdorff spaces.
Shape theory was founded by the Polish mathematician Karol Borsuk in 1968.
Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. This is a compact subset of the plane produced by "closing up" a topologist's sine curve with an arc.
Borsuk's original shape theory has been replaced by a more systematic approach by inverse systems, pioneered by Sibe Mardešić, and independently, by Timothy Porter. In abstract terms, one starts with a dense subcategory of good objects, and approximates general objects by inverse systems of good objects in the best way in the sense of certain universality property. Thus the object is replaced by pro-object in dense category in appropriate way.
For some purposes, like dynamical systems, more sophisticated invariants were developed under the name strong shape. Generalizations to noncommutative geometry, e.g. the shape theory for operator algebras have been found.
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