To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we call a 2-dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2-dimensional Postnikov complexes which satisfy certain restricted "exact horn-lifting" conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1-simplices. Those complexes which have, at minimum, their degenerate 2-simplices always invertible and have an invertible 2-simplex # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1-simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids -- all 2-cells are isomorphisms and all 1-cells are equivalences. Contents

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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.

We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1

Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence of a universal symmetric cubical monoid ; in fact, K is the classifying category of a monoidal algebraic theory of such monoids. Analogous results are given for the restricted cubical site I of ordinary cubical sets (just faces and degeneracies) and for the intermediate site J (including connections). We also consider briefly the reversible analogue, !K.

This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of symmetric versions of an adjunction—and use them and their combinations to get ‘directed models ’ of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory.

Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes. An image, represented by a metric space X, can be explored at a variable resolution # > 0, by equipping it with a structure t # X of simplicial complex depending on #; this complex can be further analysed by homotopy groups # n (X) = #n (t # X) and homology groups H n (X) = Hn (t # X). Loosely

Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes. An image, represented by a metric space X, can be explored at a variable resolution ɛ> 0, by equipping it with a structure tɛX of simplicial complex depending on ɛ; this complex can be further analysed by homotopy groups π ɛ n(X) = πn(tɛX) and homology groups H ɛ n(X) = Hn(tɛX). Loosely speaking, these objects detect singularities which can be captured by an n-dimensional grid, with edges bound by ɛ; this works equally well for continuous or discrete regions of euclidean spaces. Similarly, a directed image, represented by an “asymmetric metric space”, produces a family of directed simplicial complexes sɛX and can be explored by the fundamental n-category ↑Π ɛ n(X) of the latter. The same directed tools can be applied to combinatorial models of concurrent automata, like Chu-spaces.