Symmetric cubical sets

We introduce a new cubical model for homotopy types. More precisely, we will define a category Q_Σ with the following features: Q_Σ is a prop containing the classical box category as a subcategory; the category of presheaves of sets on Q_Σ models the homotopy category; and combinatorial symmetric monoidal model categories with cofibrant unit have homotopically well-behaved enrichments.     

Cocomplete toposes whose exact completions are toposes

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere-Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Grothendieck toposes whose exact completion is a Grothendieck topos.We also show that for any topological space X, the exact completion of is a topos if and only if X is discrete. The corollary in this case characterizes the Grothendieck toposes with enough points whose exact completions are toposes.     

Locally anisotropic toposes

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.     

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.     

Elementary axioms for local maps of toposes

We present a complete elementary axiomatization of local maps of toposes.     

Equivariant simplicial cohomology with local coefficients and its classification

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.     

A characterization of the left exact categories whose exact completions are toposes

We characterize the categories with finite limits whose exact completions are toposes and discuss some examples and counter-examples.     

Varieties of simplicial groupoids I: Crossed complexes

It is usual to use algebraic models for homotopy types. Simplicial groupoids provide such a model. Other partial models include the crossed complexes of Brown and Higgins. In this paper, the simplicial groupoids that correspond to crossed complexes are shown to form a variety within the category of all simplicial groupoids and the corresponding verbal subgroupoid is identified.     

Homotopy theory of simplicial sheaves in completely decomposable topologies

There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works.     

Torsion in the matching complex and chessboard complex

Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vre?ica and ?ivaljevi? established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn,n is a 3-group of exponent at most 9. When , the bottom nonvanishing homology of Mn,n is shown to be Z₃. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

On the subdivision of small categories

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent.     

The behaviour of the $$\overline{W}$$ -construction on the homotopy theory of bisimplicial sets

The codiagonal functor \(\overline{W}\) transfers a Quillen closed model structure on the bisimplicial set category from the ordinary model category of simplicial sets. This bisimplicial model structure is different from the so called Moerdijk model structure, which is similarly transferred from simplicial sets but through the diagonal functor. We show the mutual relationship of these two closed model structures on the category of bisimplicial sets.     

The complement of polyhedral product spaces and the dual simplicial complexes

In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral join complexes. Then we compute the universal algebra of polyhedral join complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.     

Topological robustness of non-saddle sets

We study in this paper preservation of dynamical and shape theoretical properties under continuation for parametrized families of flows. We show that, although attractors continue, the same does not hold for non-saddle sets. However, when they continue, their shape is preserved in quite general settings, which include differentiable families of flows and regular non-saddle sets for general flows, not necessarily differentiable.We also study how the continuation of a non-saddle set influences that of its dual non-saddle set.     

Relativized Grothendieck topoi

In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.