author={John F. Jardine},
  journal={Homology, Homotopy and Applications},
  • J. Jardine
  • Published 2006
  • Mathematics
  • Homology, Homotopy and Applications
This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves with an equivalent homotopy category. The theory specializes, for example, to the homotopy theories of… 
Cubical setting for discrete homotopy theory, revisited
We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy
Cocycles in Local Higher Category Theory
We develop a model structure on presheaves of small simplicially enriched categories on a site $\mathscr{C}$, for which the weak equivalences are 'stalkwise' weak equivalences for the Bergner model
Descent theory and mapping spaces
The purpose of this paper is to develop a theory of $$(\infty , 1)$$ ( ∞ , 1 ) -stacks, in the sense of Hirschowitz–Simpson’s ‘Descent Pour Les n–Champs’, using the language of quasi-category theory
Homology and cohomology of cubical sets with coefficients in systems of objects
This paper continues the research of the author on the homology of cubical and semi-cubical sets with coefficients in systems. The main result is the theorem that the homology of cubical sets with
Homotopy groups of cubical sets
We define and study homotopy groups of cubical sets. To this end, we give four definitions of homotopy groups of a cubical set, prove that they are equivalent, and further that they agree with their
Homotopy theory of C*-algebras
In this work we construct from ground up a homotopy theory of C-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model
Cubical models of $(\infty, 1)$-categories
We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with
Local Homotopy Theory
Preface.- 1 Introduction.- Part I Preliminaries.- 2 Homotopy theory of simplicial sets.- 3 Some topos theory.- Part II Simplicial presheaves and simplicial sheaves.- 4 Local weak equivalences.- 5
An interval of a presheaf category  consists of a presheaf I endowed with two global sections ∂ 0 , ∂
  • Mathematics
  • 2018
We prove that the category of trees Ω is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category
Fibred sites and stack cohomology
AbstractThe usual notion of the site associated to a stack is expanded to a definition to a site $$\mathcal{C}/A$$ fibred over a presheaf of categories A on a site $$\mathcal{C}$$. If the presheaf of


Localization Theories for Simplicial Presheaves
Abstract Most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are
Les Pr'efaisceaux comme mod`eles des types d''homotopie
Grothendieck introduced in Pursuing Stacks the notion of test category . These are by definition small categories on which presheaves of sets are models for homotopy types of CW-complexes. A well
Local Projective Model Structures on Simplicial Presheaves
We give a model structure on the category of simplicial presheaves on some essentially small Grothendieck site T . When T is the Nisnevich site it specializes to a proper simplicial model category
Stacks and the homotopy theory of simplicial sheaves
Stacks are described as sheaves of groupoids G satisfying an eective descent condition, or equivalently such that the clas- sifying object BG satisÞes descent. The set of simplicial sheaf homotopy
On the homotopy theory of sheaves of simplicial groupoids
  • A. JoyalM. Tierney
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1996
The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as
Motivic symmetric spectra.
This paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results together provide a categorical model for the motivic stable category which has
A1-homotopy theory of schemes
© Publications mathématiques de l’I.H.É.S., 1999, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://
Algebraic $K$-theory and etale cohomology
© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www.
Letter to the...
The discussion by Dr. Hardy was intended to infer indirectly that by improving cerebral circulation, infarction can be prevented, and it is pleased that his work may help to elucidate this belief.
Higher algebraic K-theory I, Springer Lecture
  • Notes in Math
  • 1973