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Deriving calculus with cotriples. (English) Zbl 1028.18004
For a functor of spaces and a space satisfying certain connectivity conditions, T. Goodwillie [“Calculus III: The Taylor series of a homotopy functor”, in preparation] has obtained a tower of functors and natural transformations that acts like a Taylor series for the functor expanded about the space. This construction is an important tool in homotopy theory and algebraic homotopy, and is related to A. Dold and D. Puppe’s stable derived functions [Ann. Inst. Fourier 11, 201-312 (1961; Zbl 0098.36005)] and S. Eilenberg and S. MacLane’s Q-constructions [Trans. Am. Math. Soc. 71, 294-330 (1951; Zbl 0043.25403), Ann. Math., II. Ser. 60, 49-139 (1954; Zbl 0055.41704)].
In this paper the authors construct a Taylor tower for functors from basepointed to abelian categories, by using cotriples associated to cross effect functors. They also study the layers $$D_nF=\text{fiber}(P_nF\to P_{n-1}F)$$, obtaining a result analogous to a result of Goodwillie for the layers in the Taylor tower of a functor of spaces. Then they study the limit of the tower, determining a connectivity condition on the cross effects of $$F$$ that guarantees convergence. They define the notion of differential for a functor and prove chain and product rules for them. Finally the authors determine the functors that play the role of exponential functors, and compute a version of the Taylor tower for them.
Although this paper was inspired by the works of Goodwillie, Dold-Puppe and MacLane, it is self contained, and requires only some basic homological algebra and category theory.

##### MSC:
 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 55P65 Homotopy functors in algebraic topology 55U15 Chain complexes in algebraic topology 18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads 19D10 Algebraic $$K$$-theory of spaces
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