**Shape theory** is a branch of topology, which provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.

Shape theory was reinvented, further developed and promoted by the Polish mathematician Karol Borsuk in 1968. Actually, the name *shape theory* was coined by Borsuk.

Borsuk lived and worked in Warsaw, hence the name of one of the fundamental examples of the area, the Warsaw circle. It is a compact subset of the plane produced by "closing up" a topologist's sine curve with an arc. The homotopy groups of the Warsaw circle are all trivial, just like those of a point, and so any map between them induces a weak homotopy equivalence. However the two spaces are not homotopy equivalent. So by the Whitehead theorem, the Warsaw circle does not have the homotopy type of a CW complex.

Borsuk's shape theory was generalized onto arbitrary (non-metric) compact spaces, and even onto general categories, by Włodzimierz Holsztyński in year 1968/1969, and published in Fund. Math. **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 and Norman Steenrod in their monograph *Foundations of Algebraic Topology*. Due to the circumstance^{[ clarification needed ]}, Holsztyński's paper was hardly noticed, and instead a great popularity in the field was gained by a later paper by Sibe Mardešić 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, 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.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Kazimierz Kuratowski** was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

**Karol Borsuk** was a Polish mathematician. His main interest was topology.

**Samuel Eilenberg** was a Polish-American mathematician who co-founded category theory and Homological Algebra.

**Daniel Gray** "**Dan**" **Quillen** was an American mathematician.

**Eduard Čech** was a Czech mathematician born in Stračov. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification and the notion of Čech cohomology. He was the first to publish a proof of Tychonoff's theorem in 1937.

* Fundamenta Mathematicae* is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics. It is published by the Mathematics Institute of the Polish Academy of Sciences.

In mathematics, the **homotopy category** is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In mathematics, specifically in algebraic topology, the **Eilenberg–Steenrod axioms** are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

In mathematics, **higher category theory** is the part of category theory at a *higher order*, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.

In category theory, a branch of mathematics, a **section** is a right inverse of some morphism. Dually, a **retraction** is a left inverse of some morphism. In other words, if *f* : *X* → *Y* and *g* : *Y* → *X* are morphisms whose composition *f*o*g* : *Y* → *Y* is the identity morphism on *Y*, then *g* is a section of *f*, and *f* is a retraction of *g*.

**Ronald Brown** is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.

**James Dugundji** was an American mathematician, a professor of mathematics at the University of Southern California.

**Goro Nishida** was a Japanese mathematician. He was a leading member of the Japanese school of homotopy theory, following in the tradition of Hiroshi Toda.

**Sibe Mardešić** was a Croatian mathematician.

In mathematics, **nonabelian algebraic topology** studies an aspect of algebraic topology that involves higher-dimensional algebras.

- Mardešić, Sibe (1997). "Thirty years of shape theory" (PDF).
*Mathematical Communications*.**2**: 1–12. - shape theory in
*nLab* - Jean-Marc Cordier and Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008)
- Aristide Deleanu and Peter John Hilton, On the categorical shape of a functor, Fundamenta Mathematicae 97 (1977) 157 - 176.
- Aristide Deleanu and Peter John Hilton, Borsuk's shape and Grothendieck categories of pro-objects, Mathematical Proceedings of the Cambridge Philosophical Society 79 (1976) 473–482.
- Sibe Mardešić and Jack Segal, Shapes of compacta and ANR-systems, Fundamenta Mathematicae 72 (1971) 41–59
- Karol Borsuk, Concerning homotopy properties of compacta, Fundamenta Mathematicae 62 (1968) 223-254
- 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, 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, Lecture Notes in Maths. 542, Springer-Verlag.
- Tim Porter, Čech homotopy I, II, Journal of the London Mathematical Society, 1, 6, 1973, pp. 429–436; 2, 6, 1973, pp. 667–675.
- J.T. Lisica and Sibe Mardešić, 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
- Marius Dădărlat, Shape theory and asymptotic morphisms for C*-algebras, Duke Math. J., 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 1994.

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