The homotopy theory of fusion systems

We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such $X$ has a Sylow subgroup $BS\longrightarrow X$, maps $BQ\longrightarrow X$ for a $p$-group $Q$ are described via homomorphisms $Q\longrightarrow S$, and $H^*(X;\mathbb{F} _p)$ is isomorphic to a certain ring of ``stable elements'' in $H^*(BS;\mathbb{F} _p)$. These spaces arise as the ``classifying spaces'' of certain algebraic objects which we call ``$p$-local finite groups''. Such an object consists of a system of fusion data in $S$, as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.