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Catégories dérivables

Denis-Charles Cisinski (2010)

Bulletin de la Société Mathématique de France

Ces notes sont consacrées à la construction de dérivateurs à partir d’une nouvelle notion de catégorie de modèles assez générale pour recouvrir les théories de Quillen, Thomason et Brown. On développe en particulier la théorie des catégories exactes dérivables (par exemple les catégories de Frobenius et les catégories biWaldhausen compliciales vérifiant de bonnes propriétés de stabilité homotopique), lesquelles donnent lieu à des dérivateurs triangulés. On donne une caractérisation combinatoire...

Fundamental pro-groupoids and covering projections

Luis Hernández-Paricio (1998)

Fundamenta Mathematicae

We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs (X) and an induced category pro (π crs (X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro (π crs (X), Sets). We also prove that the latter category is equivalent to pro (π CX,...

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska (2001)

Fundamenta Mathematicae

Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space ( B p ( G ) ) / G of the classifying space of the category associated to the G-poset p ( G ) of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor X E / ( N E . . . N E ) defined over the poset ( s d p ( G ) ) / G , where sd is the barycentric subdivision. We also...

On the homotopy type of (n-1)-connected (3n+1)-dimensional free chain Lie algebra

Mahmoud Benkhalifa, Nabilah Abughzalah (2005)

Open Mathematics

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ⊒Q, then p=∞. Denote by DGL nnp, n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL nnp. In this work we intend to answer the following two questions: Given an object (L(V), ϖ) in DGL n3n+2 and denote by S(L(V), ϖ) the class of objects homotopy equivalent...

Orbit spaces, Quillen's Theorem A and Minami's formula for compact Lie groups

Assaf Libman (2011)

Fundamenta Mathematicae

Let G be a compact Lie group. We present a criterion for the orbit spaces of two G-spaces to be homotopy equivalent and use it to obtain a quick proof of Webb’s conjecture for compact Lie groups. We establish two Minami type formulae which present the p-localised spectrum Σ B G as an alternating sum of p-localised spectra Σ B H for subgroups H of G. The subgroups H are calculated from the collections of the non-trivial elementary abelian p-subgroups of G and the non-trivial p-radical subgroups of G. We...

Topos based homology theory

M. V. Mielke (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper we extend the Eilenberg-Steenrod axiomatic description of a homology theory from the category of topological spaces to an arbitrary category and, in particular, to a topos. Implicit in this extension is an extension of the notions of homotopy and excision. A general discussion of such homotopy and excision structures on a category is given along with several examples including the interval based homotopies and, for toposes, the excisions represented by “cutting out” subobjects. The...

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