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In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes the sheafification of a presheaf. This sequence of approximations is formally similar to the Taylor series of a smooth function, hence the term "''calculus'' of functors". 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 Thomas Goodwillie in a series of three papers in the 1990s and 2000s,〔T. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.〕〔T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.〕〔T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.〕 and has since been expanded and applied in a number of areas. == Examples == A motivational example, of central interest in geometric topology, is the functor of embeddings of one manifold ''M'' into another manifold ''N,'' whose first derivative in the sense of calculus of functors is the functor of immersions. As every embedding is an immersion, one obtains an inclusion of functors – in this case the map from a functor to an approximation is an inclusion, but in general it is simply a map. As this example illustrates, the linear approximation of a functor (on a topological space) is its sheafification, thinking of the functor as a presheaf on the space (formally, as a functor on the category of open subsets of the space), and sheaves are the linear functors. This example was studied by Goodwillie and Michael Weiss.〔M. Weiss, Embeddings from the point of view of immersion theory, Part I, Geometry and Topology 3 (1999), 67-101.〕〔T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, Part II, Geometry and Topology 3 (1999), 103-118.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Calculus of functors」の詳細全文を読む スポンサード リンク
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