Inequilogical spaces, directed homology and noncommutative geometry

Abstract

We introduce a preordered version of D. Scott's equilogical spaces [29], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper.

This setting is compared with other structures which have been recently used for Directed Algebraic Topology: spaces equipped with an order, or a local order, or distinguished paths, or distinguished cubes.