I will discuss Goodwillie’s calculus of functors on topological spaces. To mimic
\nthe set-up in real analysis, topological spaces are considered small if their nontrivial homotopy
\ngroups start only in higher dimensions. They can be considered close only in relation to a map
\nbetween them, but a map allows us to construct the difference between two spaces, and two
\nspaces are close if the difference between them is small. Spaces can be summed (in different
\nways) by taking twisted products of them. It is straightforward to construct the analogs of
\nconstant, linear, and higher degree homogenous functors, and they can be assembled into
\n\\polynomials” and \\in\fnite sums”. There are notions of differentiability and higher derivatives,
\nof Taylor towers, and of analytic functions.
\nWhat might look like a game of analogies is an extremely useful tool because when one looks
\nat functors that map topological spaces not into the category of topological spaces, but into the
\ncategory of spectra (the stabilized version of the category of spaces, which will be explained),
\nmany of them are, in fact, analytic, so they can be constructed from the homogenous functors
\nof different degrees. And we can use appropriate analogs of calculus theorems to understand
\nthem better. I will conclude with some recent work of Randy McCarthy and myself, applying
\nGoodwillie’s calculus to algebraic K-theory calculations.<\/p>\n","protected":false},"excerpt":{"rendered":"
I will discuss Goodwillie’s calculus of functors on topological spaces. To mimic the set-up in real analysis, topological spaces are considered small if their nontrivial homotopy groups start only in higher dimensions. They can be considered close only in relation…
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