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

Sullivan approach to Rational homotopy theory of connected nilpotent simplicial sets of finite ℚ-rank is extended to connected locally nilpotent simplicial sets of arbitrary ℚ-rank. Rational proper homotopy type and rational proper homotopy type at infinity of connected, one-ended, proper nilpotent and nilpotent at infinity, locally finite simplicial sets are also defined. In particular, the notion of minimal algebras and minimal models in these setting are introduced in such a way that the indecomposable elements for such a minimal model are identified in each case, with the dual, as ℚ-vector space, of the corresponding version of the homotopy groups of the given simplicial set. © 2010 Topology Proceedings.