Abstract. This is a brief study of the homology of cubical sets, with two main purposes.First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes.But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologically-trivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C * -algebras.MSC: 55U10, 55Nxx, 81R60.

This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim can be stated as 'modelling non-reversible phenomena' and its domain should be distinguished from that of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. Its homotopical tools (corresponding in the classical case to ordinary homotopies, fundamental group and fundamental groupoid) should be similarly 'non-reversible': directed homotopies, fundamental monoid and fundamental category. Homotopy constructions occur here in a directed version, which gives rise to new 'shapes', like directed cones and directed spheres. Applications will deal with domains where privileged directions appear, including rewrite systems, traffic networks and biological systems. The most developed examples can be found in the area of concurrency.

The notion of weak subobject, or variation, was introduced in [Gr4] as an extension of the notion of subobject, adapted to homotopy categories or triangulated categories, and well linked with their weak limits. We study here some formal properties of this notion. The variations in X can be identified with the (distinguished) subobjects in the Freyd completion FrX, the free category with epimonic factorisation system over X, which extends the Freyd embedding of the stable homotopy category of spaces in an abelian category [Fr2]. If X has products and weak equalisers, as HoTop and various other homotopy categories, FrX is complete; similarly, if X has zero-object, weak kernels and weak cokernels, as the homotopy category of pointed spaces, then FrX is a homological category [Gr1]; finally, if X is triangulated, FrX is abelian and the embedding X FrX is the universal homological functor on X, as in the original case [Fr2]. These facts have consequences on the ordered sets of variations.Mathematics Subject Classification: 18A20, 18A35, 18E, 55P, 18E30.Normal or regular variations have appeared in Eckmann -Hilton [EH] and Freyd [Fr4-5], under the equivalent form of "principal right ideals" of maps, to deal with weak kernels or weak equalisers. Recently, in connection with proof theory, Lawvere [La] has considered a "proof-theoretic power set X (A)", defined as the "poset-reflection of the slice category X/A", which amounts to Var(A). A different approach to "subobjects" in homotopy categories can be found in Kieboom [Ki]. FrX is related with the regular and Barr-exact completions of a category with limits or weak limits, studied in Carboni [Ca] and Carboni -Vitale [CV]. Finally, let us recall that the pseudo algebras for the 2-monad X X 2 are known to correspond to the factorisation systems over X (Coppey [Co]; Korostenski -Tholen [KT]); similar relations link the induced 2-monad X FrX with the epi-monic factorisation systems over X (2.3).The author gratefully acknowledges helpful remarks from F.W. Lawvere and P. Freyd.

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Le strutture ottenibili per incollamento di ~ spazi elementari ~, come le varieth, i ]ibrati, le variet~ fogliettate, possono essere definite da ~ tttlanti di incollamento ~ e, formalmente, come eategorie arriechite su o2po~'tune categorie ordinate. O. -Introduction.0.1. Glueing structures, for instance maniiolds, fibre bundles, vector bundles or loliations, can be obtained by patching togother a family (Ui) of suitable ~ elementary spaces >> by means of partial bijections u~: U~ -> Uj expressing the glueing conditions and forming a sort of <~ glueing atlas >>, instead of the more usual atlas of charts.The goal oi this paper is to treat these structures as enriched categories over ~ totally cohesive >> categories, that is ordered categories having binary meets and arbitrary joins of pairwise <~ compatible ~> morphisms. The morphisms of these <~ generalized manifolds ~> are obtained as <~ compatible >~ modules between enriched categories, which Can be composed precisely because of the existence of compatible joins. The condition of Cauchy-completeness corresponds to the maximality of the glueing atlas; however, since our morphisms are modules, the procedure of Cauchycompletion just produces an isomorphic object.This approach to glueing structures is clearly related to Ehresmann's one, based on pseudogroups of transformations (e.g. see [El, E2]).On the other hand, our setting inscribes in Lawvere's remark that interesting mathematical structures not only organize in categories, but are themselves categories, enriched over some suitable base: a monoidal category as in Lawvere's original formulation [La], or more generally a bieategory as in BETTI [Be]. The bases we actually use are suitable ordered categories (very particular bieategories).Last, this work is closely related with the notion of (~ glueing data ~>, considered