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Many objects of central interest in algebraic topology can be seen as functors, which are difficult to analyze directly, so the idea is to replace them with simpler functors which are sufficiently good approximations for certain purposes. The calculus of functors was developed by
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The notion of a sheaf and sheafification of a presheaf date to early category theory, and can be seen as the linear form of the calculus of functors. The quadratic form can be seen in the work of
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of spheres in 1965, where he defined a "metastable range" in which the problem is simpler. This was identified as the quadratic approximation to the embeddings functor in
Goodwillie and Weiss.
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by using a sequence of increasingly accurate polynomial functions. In a similar way, with the calculus of functors method, you can approximate the behavior of certain kind of
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and can be thought of as "successive approximations", just as in a Taylor series one can progressively discard higher order terms.
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on the space (formally, as a functor on the category of open subsets of the space), and sheaves are the linear functors.
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in a series of three papers in the 1990s and 2000s, and has since been expanded and applied in a number of areas.
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Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a
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As this example illustrates, the linear approximation of a functor (on a topological space) is its
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T. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.
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meaning that the approximating functor approximates the original functor "in dimension up to
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T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, Part II,
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In the calculus of functors method, the sequence of approximations consists of (1) functors
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617:{\displaystyle F\to \cdots \to T_{k+1}F\to T_{k}F\to \cdots \to T_{1}F\to T_{0}F,}
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of topological spaces with continuous morphisms. This kind of functor, called a
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There are three branches of the calculus of functors, developed in the order:
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is, so you can study the topology of the increasingly accurate approximations
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whose first derivative in the sense of calculus of functors is the functor of
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by approximating them by a sequence of simpler functors; it generalizes the
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T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.
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T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.
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Homotopy calculus has seen far wider application than the other branches.
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M. Weiss, Embeddings from the point of view of immersion theory, Part I,
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by using a sequence of increasingly accurate polynomial
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points in the given space. The difference between the
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Enlacements de sphères en codimension supérieure à 2
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936:Syllabus for Math 283: Calculus of Functors
408:{\displaystyle \eta _{k}\colon F\to T_{k}F}
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22:mathematics
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966:John Klein
852:References
784:-connected
189:Definition
93:immersions
78:embeddings
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384::
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991:Functors
957:Archived
814:Branches
779:must be
636:excisive
268:presheaf
266:-valued
230:and let
217:functors
176:presheaf
82:manifold
68:Examples
54:calculus
42:presheaf
34:functors
836:History
666:th and
654:on the
652:sheaves
206:functor
80:of one
292:(X), F
288:(X), F
276:X∈O(M)
246:. Let
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940:(PDF)
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296:(X),
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260:Top
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