Sheaves on Involutive Quantaloids

We extend the well-known notions of a singleton, complete -set, presheaf and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary involutive quantaloids. We show that sheaves on and complete -sets come to the same thing. This paper can be considered as a symmetric version of an earlier work of the author.     

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.     

TheK-theory of finite fields, revisited

A new, short proof is given for the Quillen theorem that calculates theK-theory of finite fields. This proof uses the Gabber rigidity theorem and the homotopy theory of simplicial presheaves.Research supported by NSERC.     

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.     

Parametrized spaces model locally constant homotopy sheaves

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π₁-sets. We then use these two equivalences to study base change functors for parametrized spaces.     

Sheaves on quantaloids

We generalize the well-known notions of a singleton, completeQ-set, presheaf, and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary quantaloids. We show that every completeQ-set can be viewed as a sheaf and vice versa. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 133–171, April–June, 2000.     

单纯广群的准层和同伦纤维

显示了在设置C上的单纯广群的准层的范畴是个封闭模型范畴.证明了在一个单纯广群的准层G上的单纯函子X是局部弱等价于同伦纤维.