Rational homotopy type, rational proper homotopy type and rational proper homotopy type at infinity

The Sullivan approach to rational homotopy theory can be thought of as being applied to connected nilpotent simplicial sets whose Bbb{Q}-rank is finite, that is, each of the cohomology groups H^n(X,Bbb{Q}) is a finite-dimensional Bbb{Q}-vector space. In this paper the author shows how, by the use of ind- and pro-categorical methods, derived in part from shape theory and proper homotopy theory, one can extend Sullivan's theory to connected locally nilpotent simplicial sets of arbitrary Bbb{Q}-rank. As an application of the results and of the theory developed for their proofs, the author extends rational homotopy theoretic methods to give rational proper homotopy theory and rational proper homotopy theory at infinity, plus several other variants. Minimal model techniques are extended to all these settings.