Categorically Algebraic Foundations for Homotopical Algebra

We investigate a structure for an abstract cylinder endofunctor I whichproduces a good basis for homotopical algebra. It essentially consists ofthe usual operations (faces, degeneracies, connections, symmetries, verticalcomposition) together with a transformation w: I² I², whichwe call lens collapse after its realisation in the standard topologicalcase. This structure, if somewhat heavy, has the interest of beingcategorically algebraic, i.e., based on operations onfunctors. Consequently, it can be naturally lifted from a category A to itscategories of diagrams A^S and its slice categories A\X,A/X.Further, the dual structure, based on a cocylinder (or path) endofunctor Pcan be lifted to the category of A-valued sheaves on a site, wheneverthe path functor P preserves limits, and to the category Mon A of internalmonoids, with respect to any monoidal structure of A consistent with P.     

Local Projective Model Structures on Simplicial Presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis,A1) and SH((Sch/k)cdh,A1) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of -closed classes.     

Flasque Model Structures for Simplicial Presheaves

It is well known that there are two useful families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties. (Received: February 2006)     

An Intrinsic Homotopy Theory for Simplicial Complexes, with Applications to Image Analysis

A simplicial complex is a set equipped with a down-closed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe spaces whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced.The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure t X of simplicial complex at each resolution >0; the resulting homotopy group _n (X) detects those singularities which can be captured by an n-dimensional grid, with edges bound by this works equally well for continuous or discrete regions of Euclidean spaces. Its computation is based on direct, intrinsic methods.     

Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a system of open neighbourhoods at infinity while an exterior map is a continuous map which is continuous at infinity. The category of spaces and proper maps is a subcategory of the category of exterior spaces.In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T,T} of suitable exterior spaces. For this goal, for a given exterior space T, we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown–Grossman, erin–Steenrod, p-cylindrical, Baues–Quintero and Farrell–Taylor–Wagoner groups, as well as the standard (Hurewicz) homotopy groups.The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T,T}, it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.     

The Steenrod Algebra and Theories Associated to Hopf Algebras

We show that the homotopy category of products of Z/p-Eilenberg–Mac Lane spaces is an -algebra which algebraically is determined by the Steenrod algebra considered as a Hopf algebra with unstable structure.     

Finite Block Theory and Hopf Algebra Actions

A ring is said to have finite block theory if it can be written as the finite direct sum of indecomposable subrings. In the paper, algebras R are acted on by Hopf algebras H. We prove a series of going up and going down results analyzing when R and its subalgebra of invariants R ^H have finite block theory. We also provide counterexamples when the hypotheses of our main results are weakened. Presented by D. Passman     

A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

The concept of acombinatorial(W; P; U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP  Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W, P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W; P; U)-geometries and suggest two equivalent definitions of a (W; P; U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W; P; U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients.