The Theory of Polynomial Functors

Abstract: Polynomial functors were introduced by Professors Eilenberg and Mac Lane in 1954, who used them to study certain homology rings. Strict polynomial functors were invented by Professors Friedlander and Suslin in 1997, in order to develop the theory of group schemes.The first real investigation of their intrinsic properties was performed in 1988, when Professor Pirashvili showed that polynomial functors are equivalent to modules over a certain ring. A similar study was conducted on strict polynomial functors in 2003 by Dr. Salomonsson in his doctoral thesis. A radically different method of attack was initiated by Dr. Dreckman and Professors Pirashvili, Franjou, and Baues in the year 2000. Their approach was to combinatorially encode polynomial functors, and utilised for this purpose the category of sets and surjections. Dr. Salomonsson would later repeat the feat for strict polynomial functors, employing instead the category of multi-sets.This thesis proposes the following:1:o. To generalise the notion of polynomial functor to more general base rings than Z, so that it smoothly agree with the existing definition of strict polynomial functor, allowing for easy comparison. This results in the definition of numerical functors.2:o. To make an extensive study of numerical maps of modules, to see how they fit into Professor Roby's framework of strict polynomial maps.3:o. To conduct a survey of numerical rings.4:o. To develop the theories of numerical and strict polynomial functors so that they run in parallel.5:o. To show how also numerical functors may be interpreted as modules over a certain ring.6:o. To expound the theory of mazes, which will be seen to vastly generalise the category of surjections employed by Professor Pirashvili et al., since they turn out to encode, not only polynomial or numerical functors, but all module functors over any base ring.7:o. To simplify Dr. Salomonsson's construction involving multi-sets, making it more amenable to a comparison with mazes.8:o. To prove comparison theorems interrelating numerical and strict polynomial functors.9:o. And, finally, to indicate how polynomial functors may be used to extend the operad concept.