# Shape theory (mathematics)

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Branch of topology

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**Shape theory** is a branch of [topology](/source/Topology) that provides a more global view of the topological spaces than [homotopy theory](/source/Homotopy_theory). The two coincide on [compacta](/source/Compact_set) dominated homotopically by finite [polyhedra](/source/Simplicial_complex). Shape theory associates with the [Čech homology](/source/%C4%8Cech_cohomology) theory while homotopy theory associates with the [singular homology](/source/Singular_homology) theory.

## Background

Shape theory was invented and published by [D. E. Christie](https://en.wikipedia.org/w/index.php?title=D._E._Christie&action=edit&redlink=1) in 1944; it was reinvented, further developed and promoted by the Polish mathematician [Karol Borsuk](/source/Karol_Borsuk) in 1968. Actually, the name *shape theory* was coined by Borsuk.

### Warsaw circle

The Warsaw circle

Borsuk lived and worked in [Warsaw](/source/Warsaw), hence the name of one of the fundamental examples of the area, the **Warsaw circle**.[1] It is a compact subset of the plane produced by "closing up" a [topologist's sine curve](/source/Topologist's_sine_curve) (also called a *Warsaw sine curve*) with an arc. The [homotopy groups](/source/Homotopy_group) of the Warsaw circle are all [trivial](/source/Trivial_group), just like those of a point, and so any map between the Warsaw circle and a point induces a [weak homotopy equivalence](/source/Weak_homotopy_equivalence). However these two spaces are not [homotopy equivalent](/source/Homotopy_equivalence). So by the [Whitehead theorem](/source/Whitehead_theorem), the Warsaw circle does not have the homotopy type of a [CW complex](/source/CW_complex).

## Historical development

Borsuk's shape theory was generalized onto arbitrary (non-[metric](/source/Metric_space)) compact spaces, and even onto general categories, by Włodzimierz Holsztyński in year 1968/1969, and published in [Fund. Math.](/source/Fundamenta_Mathematicae) **70**, 157–168, y. 1971 (see Jean-Marc Cordier, Tim Porter, (1989) below). This was done in a *continuous style*, characteristic for the Čech homology rendered by [Samuel Eilenberg](/source/Samuel_Eilenberg) and [Norman Steenrod](/source/Norman_Steenrod) in their monograph *Foundations of Algebraic Topology*. Due to the circumstance[*[clarification needed](https://en.wikipedia.org/wiki/Wikipedia:Please_clarify)*], Holsztyński's paper was hardly noticed, and instead a great popularity in the field was gained by a later paper by [Sibe Mardešić](/source/Sibe_Marde%C5%A1i%C4%87) and Jack Segal, Fund. Math. **72**, 61–68, y.1971. Further developments are reflected by the references below, and by their contents.

For some purposes, like [dynamical systems](/source/Dynamical_system), more sophisticated invariants were developed under the name **strong shape**. Generalizations to [noncommutative geometry](/source/Noncommutative_geometry), e.g. the shape theory for [operator algebras](/source/Operator_algebra) have been found.

## See also

- [List of topologies](/source/List_of_topologies)

## References

1. **[^](#cite_ref-1)** "[The Polish Circle and some of its unusual properties](http://math.ucr.edu/~res/math205B-2012/polishcircle.pdf)". Math 205B-2012 Lecture Notes, University of California Riverside. Retrieved November 16, 2023. See also the accompanying diagram "[Constructions on the Polish Circle](http://math.ucr.edu/~res/math205B-2012/polishcircleA.pdf)"

- [Mardešić, Sibe](/source/Sibe_Marde%C5%A1i%C4%87) (1997). ["Thirty years of shape theory"](http://hrcak.srce.hr/file/2848) ([PDF](/source/PDF)). *Mathematical Communications*. **2**: 1–12.

- [shape theory](https://ncatlab.org/nlab/show/shape%20theory) at the [*n*Lab](/source/NLab)

- Jean-Marc Cordier and Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, [Ellis Horwood](/source/Simon_%26_Schuster). Reprinted Dover (2008)

- Aristide Deleanu and [Peter John Hilton](/source/Peter_Hilton), On the categorical shape of a functor, [Fundamenta Mathematicae](/source/Fundamenta_Mathematicae) 97 (1977) 157–176.

- Aristide Deleanu and [Peter John Hilton](/source/Peter_Hilton), Borsuk's shape and Grothendieck categories of pro-objects, [Mathematical Proceedings of the Cambridge Philosophical Society](/source/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society) 79 (1976) 473–482.

- [Sibe Mardešić](/source/Sibe_Marde%C5%A1i%C4%87) and Jack Segal, Shapes of compacta and ANR-systems, [Fundamenta Mathematicae](/source/Fundamenta_Mathematicae) 72 (1971) 41–59

- [Karol Borsuk](/source/Karol_Borsuk), Concerning homotopy properties of compacta, [Fundamenta Mathematicae](/source/Fundamenta_Mathematicae) 62 (1968) 223–254

- [Karol Borsuk](/source/Karol_Borsuk), Theory of Shape, Monografie Matematyczne Tom 59, Warszawa 1975.

- D. A. Edwards and H. M. Hastings, [Čech Theory: its Past, Present, and Future](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rmjm/1250128825&page=record), [Rocky Mountain Journal of Mathematics](/source/Rocky_Mountain_Journal_of_Mathematics), Volume 10, Number 3, Summer 1980

- D. A. Edwards and H. M. Hastings, (1976), [Čech and Steenrod homotopy theories with applications to geometric topology](http://alpha.math.uga.edu/~davide/Cech_and_Steenrod_Homotopy_Theories_with_Applications_to_Geometric_Topology.pdf), [Lecture Notes in Mathematics](/source/Lecture_Notes_in_Mathematics) 542, [Springer-Verlag](/source/Springer_Science%2BBusiness_Media).

- Tim Porter, Čech homotopy I, II, [Journal of the London Mathematical Society](/source/London_Mathematical_Society), 1, 6, 1973, pp. 429–436; 2, 6, 1973, pp. 667–675.

- J.T. Lisica and [Sibe Mardešić](/source/Sibe_Marde%C5%A1i%C4%87), Coherent prohomotopy and strong shape theory, Glasnik Matematički 19(39) (1984) 335–399.

- Michael Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catég. 38 (1997), no. 1, 3–66, [numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1997__38_1_3_0)

- Marius Dădărlat, Shape theory and asymptotic morphisms for C*-algebras, [Duke Mathematical Journal](/source/Duke_Mathematical_Journal), 73(3):687–711, 1994.

- Marius Dădărlat and Terry A. Loring, Deformations of topological spaces predicted by E-theory, In Algebraic methods in operator theory, p. 316–327. [Birkhäuser](/source/Birkh%C3%A4user) 1994.

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