Qimh Xantcha: Polynomial and strict polynomial functors

Polynomial functors were introduced in 1954 by Eilenberg and Mac Lane, who used them to study certain homology rings. Strict polynomial functors were invented in 1997 by Friedlander and Suslin, who employed them to develop the theory of group schemes. Since then, the two spurious concepts have evolved side by side.

A successful attack on these functors was instigated by the quartet Baues, Dreckman, Franjou and Pirashvili in 2000. Their method was to combinatorially encode integral (non-strict) polynomial functors by means of the category of finite sets and surjections. Evidently inspired by this approach, Salomonsson would, a few years later, in his doctoral thesis of 2003, repeat the feat for strict polynomial functors, this time utilizing the category of finite multi-sets.

In our thesis, we investigate the so-called labyrinth category, which is seen to vastly generalise the category of surjections employed by Pirashvili et al., encoding, as it does, all module functors over any base ring. The polynomiality of a functor can then easily be read off.

Armed with these two combinatorial descriptions, of polynomial and strict polynomial functors, respectively, we are finally in a position to answer the fundamental question: When is a polynomial functor strict polynomial?