Abstract: (Joint work with Brenda Johnson.) The many theories of “calculus” introduced in algebraic topology over the past couple of decades–e.g., Goodwillie’s calculus of homotopy functors, the Goodwillie-Weiss manifold calculus, the orthogonal calculus, and the Johnson-McCarthy cotriple calculus–all have a similar flavor, though the objects studied and exact methods applied are not the same. We have constructed a common, relatively simple category-theoretic framework, which we call precalculus, into which some of the above-mentioned examples fit and which naturally leads us to define new flavors of calculus as well.

In this talk I will define the theory of precalculi, then explain how to derive a full-blown calculus from a precalculus. I will also describe the precalculi underlying certain standard calculi, then indicate promising directions in which to look for new and useful calculi arising from precalculi.