We propose to continue our investigations in several versions of "functorial calculus". Each of these is a technique that exploits multirelative connectivity estimates to describe continuous functors in some context in terms of special values; for example a functor of spaces might be recovered from its values at highly connected spaces, or a functor of subspaces of a manifold from its values at low-dimensional subspaces, or a functor of real inner product spaces from its values at high-dimensional spaces. Part of the proposal is to refine the purely homotopy-theoretic version of "calculus". Other parts are concerned with combining several versions and applying them to various questions in both high- and low-dimensional differential topology.
Each "functorial calculus" mentioned above is so called because of a not-entirely-fanciful resemblance to the ordinary diferential calculus of Newton and Leibniz. Sometimes a fact about numbers is best proved by placing it in a context where the number is part of a huge family of numbers -- a numerical function. Properties of the function then lead, by general theorems of calculus that may seem a bit magical when encountered for the first time, to a computation of the number. So it is here: sometimes a fact about some geometrically defined object is best proved by placing it in a context where the object is part of a huge family of such objects -- a functor -- and using some magic of a more modern kind. This analogy may show something of the flavor of the work; the content is harder to get at, because most of the "geometric" objects in question are connected to everyday reality by rather long chains of abstract ideas.
