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,[1][2][3] and has since been expanded and applied in a number of areas.
[4][5] Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth function f around a point x by using a sequence of increasingly accurate polynomial functions.
To be specific, let M be a smooth manifold and let O(M) be the category of open subspaces of M, i.e., the category where the objects are the open subspaces of M, and the morphisms are inclusion maps.
This kind of functor, called a Top-valued presheaf on M, is the kind of functor you can approximate using the calculus of functors method: for a particular open set X∈O(M), you may want to know what sort of a topological space F(X) is, so you can study the topology of the increasingly accurate approximations F0(X), F1(X), F2(X), and so on.
for each integer k. These natural transforms are required to be compatible, meaning that the composition
and thus form a tower and can be thought of as "successive approximations", just as in a Taylor series one can progressively discard higher order terms.
The approximating functors are required to be "k-excisive" – such functors are called polynomial functors by analogy with Taylor polynomials – which is a simplifying condition, and roughly means that they are determined by their behavior around k points at a time, or more formally are sheaves on the configuration space of k points in the given space.
Further, if one wishes to reconstruct the original functor, the resulting approximations must be n-connected for n increasing to infinity.
[citation needed] 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 André Haefliger on links of spheres in 1965, where he defined a "metastable range" in which the problem is simpler.
[6] This was identified as the quadratic approximation to the embeddings functor in Goodwillie and Weiss.