Comparing the homotopy calculi

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Topology is the study of an abstract notion of shape known as a topological space. Simple examples include the circle, the sphere and the torus (an American doughnut), indeed any object in real life represents a space. Topology also encompasses the study of relations between these spaces, known as maps. For example, the circle can be thought of as the equator of the sphere. But we can send the circle to the sphere in number other ways, for example by drawing a loop on the sphere (that may cross itself).Algebraic topology focuses on the properties of spaces and maps which are left unchanged by continuous deformations, known as homotopies. The intuition is that while you can crush a cylinder to a disc, you may not rip a hole into a shape. As an example, any map from a circle into a sphere is homotopic to a constant map, but the same is not true for maps from a circle to a torus. The combination of geometry and algebra and the ubiquity of spaces has helped algebraic topology to become a fascinating area of mathematics that can apply its powerful techniques to many kinds of problems in a wide variety of other mathematical disciplines.One of the fundamental constructions in algebraic topology are functors. These are machines which take one kind of mathematical object as an input and give another kind as an output. As a simple example, there is a functor which accepts a topological space as an input, and as output gives back two disjoint copies of the input space. Being so central to algebraic topology, a good method for studying how certain functors work is important. Goodwillie and Weiss developed two such methods in the 1990s. These methods, known as functor calculus and orthogonal calculus, have produced a number of exciting results in the short time since their invention. Both of these two forms of homotopy calculus work by taking a functor and splitting it into a series of approximations. The difference between one stage and the next is in each case a particularly well-structured space known as a spectrum. This structure makes spectra much easier to study than spaces. The advantage is therefore that we can prove statements about the original functor and study its properties by looking at its approximations and using our knowledge of spectra to move from one approximation to the next.The functor calculus and the orthogonal calculus are known to be related, but no formal study of this relation has ever been undertaken. Consequently the relation is only vaguely alluded to and no good description exists. The purpose of this project is to make this relation clear and use it to construct a general form of homotopy calculus. This general form will allow the results of Goodwillie and Weiss to be applied to other contexts and other areas of mathematics, such as algebra. The better foundations will encourage the users to work on deeper, more interesting results. A clear description of the relation also has a number of uses. For example, it will assist with calculations as now both forms can be used and compared. Overall this project will improve some already useful tools in algebraic topology, extend them to new areas of mathematics and help with important calculations.