Partial groups and maps between p-completed classifying spaces

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One of the most fundamental questions one can ask in any mathematical discipline is, given two objects of a similar nature, what is the set of all maps between them, at least up to some equivalence relation. In homotopy theory, for instance, once can ask specifically, given two topological spaces, what is the set of all homotopy classes of maps between them. Similarly, in group theory the question becomes, given two groups G and H, what is the set of all homomorphisms, or representations between them. The theory of p-local finite groups sits squarely in the intersection between homotopy theory and group theory. It was introduced by Broto, Oliver and the PI in the early 2000's with the aim to create a unified framework in which the p-local homotopy theory of classifying spaces of finite group can be studied and generalised. They obtained a number of fundamental results, and created a field of study which attracted significant attention among mathematicians working both in group and representation theory, and in algebraic topology. In particular the setup of p-local groups enabled them to obtain an algebraic classification of the group of all homotopy classes of self homotopy equivalences of a classifying space of a p-local finite group. This, along with numerous other successes in the subject, raised the expectation that p-local group theory will become instrumental in understanding homotopy classes of more general maps between classifying spaces. To date however, all attempts to reach this goal have failed. A p-local finite group consists of three bits of data, a finite p-group S, and two categories, F and L. The category F called a saturated fusion system over S, has subgroups of S as its objects and homomorphisms between them as morphisms, such that certain axioms are satisfied. Such categories arise naturally in group theory and modular representation theory. The category L, called a centric linking systems associated to F, is in a precise sense an enrichment of F which allows one to associate a topologically meaningful classifying space with F. Broto, Oliver and the PI developed an obstruction theory to the existence and uniqueness of a centric linking system associated to a given fusion system F, but were not able to show that such linking systems do exist, and indeed are unique if they do. Chermak (the VR) recently introduced a further abstraction of the concept of a linking system which he named partial groups, or more specifically localities. A careful study of these objects allowed him to deliver a spectacular positive solution to the existence/uniqueness problem.Partial groups, unlike linking systems, come equipped with a very natural concept of morphisms between them. The further refinement - localities - on the other hand still do not possess a good concept of morphisms between them, but there are several natural ideas which deserve serious consideration. It is the close relationship between localities and p-local finite groups which suggest that revisiting the theory from this perspective may give information about mapping spaces between classifying spaces of p-local groups. This project is aimed at a careful study of localities, morphisms between them, and the interaction with linking systems/p-local groups, in order to gain insight into the question of how to classify maps or homotopy classes of maps between their classifying spaces.