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Variations on a theme of homotopy. (English. French summary) Zbl 1295.55001

This is a more or less loose chat about, what the author calls, the philosophical, historical and (added by the reviewer) political aspects of the good old notion of homotopy. He starts with considerations about different properties of a cone \(-\times I: \mathcal{C} \longrightarrow \mathcal{C}\) (for an appropriate category \(\mathcal{C}\)), a discussion of the properties of such a cone ensuring that this homotopy is, e.g., symmetric and terminates with simplicial model structures in the sense of Quillen. Furthermore the author touches 2-categories, homotopical coherence and cubical enrichment, as well as pro-homotopy (i.e., homotopy in a pro-category with inverse systems as objects), étale homotopy (in algebraic geometry) and proper homotopy theory. At some occasion the author makes an attempt to develop some ancient ideas of strong shape, totally omitting any results of the last 30 years, referring instead to Mardešić’s book for more bibliographic information. There appears no S-duality, no mentioning of the important contributions of Elon Lima, etc.

MSC:

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55-03 History of algebraic topology
01A60 History of mathematics in the 20th century
55P99 Homotopy theory
55U99 Applied homological algebra and category theory in algebraic topology
55P55 Shape theory
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References:

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