# Inequilogical spaces, directed homology and noncommutative geometry

## Marco Grandis

We introduce a preordered version of D. Scott's equilogical spaces \cite{Sc}, called {\it 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 {\it 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.

Homology, Homotopy and Applications, Vol. 6(2004), No. 1, pp.413-437

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