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Ilias Amrani

Publications and source records attributed to Ilias Amrani.

12 recordsLinked to original sources
arXiv2017

A remark on the Farrell-Jones conjecture

Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring $R$ and a thick subcategory $\mathsf{C}$ of perfect $R$-complexes such that the Waldhausen $K$-theory space $\mathrm{K}(\mathsf{C})$ is equivalent to a rational Eilenberg-Maclane space.

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arXiv2015

Analogy between the cyclotomic trace map $K \rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry

In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of noncommuative schemes over $\mathbf{F}_{p}$. Our analogy suggests the existence of a categorification of the $l$-adic cohomology trace formula in the noncommutative setting for $l\neq p$. Finally, we write down the corresponding dictionary.

math.AT↗
arXiv2014

Rational homotopy theory of function spaces and Hochschild cohomology

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $Ω_{f} Map(X,Y)$. For $n> 0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\mathbb{Q}$-vector spaces $π_{n} Ω_{f}Map(X,Y_{\mathbb{Q}}) \rightarrow HH^{-n}(C^{\ast}(Y),C^{\ast}(X)_{f})$, where $HH^{\ast}$ is the (negative) Hochschild cohomology and $C^{\ast}(X)_{f}$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\ast}(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\ast}: C^{\ast}(Y)\rightarrow C^{\ast}(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., $π_{1} Aut(X_{\mathbb{Q}})_{id}$.

math.AT↗
arXiv2014

The mapping space of unbounded differential graded algebras

In this paper, we give a concrete description of the higher homotopy groups (n>0) of the mapping space Map_{Alg}(R,S) for R and S unbounded differential graded algebras (DGA) over a commutative ring k. In the connective case, we describe the relation between the higher (negative) Hochschild cohomology $HH^{-n+1}(R,S)$ and higher homotopy groups $π_{n} Map_{Alg}(R,S)$, when $n>1$.

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arXiv2012

Moduli space of fibrations in the category of simplicial presheaves

We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. As a new result, we describe the moduli space of M-bundles over a fixed space X, when M is a simplicial monoid. Moreover, the moduli space of M-bundles is classified by the classifying space of the simplicial submonoid generated by homotopy invertible elements of M. We give a general interpretation of generalized cohomology theories (connective) in terms of classification of principle bundles. We also construct categorical model for the classifying space BG and EG when G is a simplicial (topological) monoid group like.

math.AT↗
arXiv2012

Stabilization of the category of simplicial objects in CAT

In this article, we define two equivalent new model structures on $\mathbf{sCat}$ the category of simplicial objects in $\mathbf{Cat}$. Then we construct the corresponding stable model category of spectra $Sp(\mathbf{sCat})$ and make some links with the algebraic $K$-theory via the mapping space.

math.AT↗
arXiv2012

Homotopy Theory of T-algebras over Top-Cat ?

In this article, we interconnect two different aspects of higher category theory, in one hand the theory of infinity categories and on an other hand the theory of 2-categories.We construct an explicit functorial path objet in the model category of topological categories. We discuss some properties and consequences of such path object. We also explain the construction of a 2-monad which algebras are (symmetric) monoidal topological categories. Finally, we explain the relationship with the eventual model structure on the category of T-algebras.

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arXiv2011

Grothendieck's Homotopy Hypothesis

We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [Δ^{op}, Cat_{Top}]. Moreover, we prove that the diagonal model structures is left proper and cellular. We also prove that the category of \infty-groupoids (the full subcategory of topological categories) has a cofibrantly generated model structure and is Quillen equivalent to the model category of simplicial sets, which proves the Grothendieck's homotopy hypothesis.

math.AT↗
arXiv2011

A Model Structure on the Category of Topological Categories

In this article, we construct a cofibrantly generated Quillen model structure on the category of small topological categories $\mathbf{Cat}_{\mathbf{Top}}$. It is Quillen equivalent to the Joyal model structure of $(\infty,1)$-categories and the Bergner model structure on $\mathbf{Cat}_{\mathbf{sSet}}$.

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