∞-groupoids modeled on simplicial sets, whose fibrant objects are the Kan complexes.
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This is the classical model of homotopy theory familiar from traditional point-set topology, such as covering space-theory. ∞-groupoids modeled by topological spaces. Hence, in a self-reflective manner, there are many different but equivalent incarnations of homotopy theory. Conversely, homotopy theory may be understood as the non-abelian generalization of homological algebra. Indeed, chain homotopy is a special case of the general concept of homotopy, and hence homological algebra forms but a special abelian corner within homotopy theory. below) these are the non-abelian generalization of the chain complexes used in homological algebra. This perspective makes explicit that homotopy types are the unification of plain sets with the concept of gauge- symmetry groups.Īn efficient way of handling ∞-groupoids is in their explicit guise as Kan complexes (Def. Hence homotopy types are equivalently ∞-groupoids. The plain sets are recovered as the special case of 0-groupoids.ĭue to the higher orders n n appearing here, mathematical structures based not on sets but on homotopy types are also called higher structures. If there are higher order gauge-of-gauge transformations/ homotopies of homotopies between the transformations in such a groupoid, one speaks of 2-groupoids, 3-groupoids, … n-groupoids, and eventually of ∞-groupoids.
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Since, generally, there is more than one element in a homotopy type, these are like “groups with several elements”, and as such they are called groupoids (Def. This way homotopy theory subsumes parts of topological group theory. When homotopy types are represented by topological spaces, then ∞-groups are represented by topological groups.
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If there are higher order gauge-of-gauge transformations/ homotopies of homotopies between these symmetry group-elements, then one speaks of 2-groups, 3-groups, … n-groups, and eventually of ∞-groups. This way homotopy theory subsumes group theory. In the special case of a homotopy type with a single element x x, the gauge transformations necessarily go from x x to itself and hence form a group of symmetries of x x. A central result of homotopy theory is the proof of the homotopy hypothesis, which says that under this identification homotopy types are equivalent to topological spaces viewed, in turn, up to “ weak homotopy equivalence”. Homotopy theory is gauged mathematics.Ī classical model for homotopy types are simply topological spaces: Their points represent the elements, the continuous paths between points represent the gauge transformations, and continuous deformations of paths represent higher gauge transformations. Hence the theory of homotopy types – homotopy theory – is much like set theory, but with the concept of gauge transformation/ homotopy built right into its foundations.
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Instead, such a collection of elements with higher gauge transformations/ higher homotopies between them is called a homotopy type. This shows that what x x an y y here are elements of is not really a set in the sense of set theory. This principle applies also to gauge transformations/ homotopies themselves, and thus leads to gauge-of-gauge transformations or homotopies of homotopiesĪnd so on to ever higher gauge transformations or higher homotopies: In mathematics this is called a homotopy.