1. CMB 2011 (vol 55 pp. 319)
|The Verdier Hypercovering Theorem|
This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant.
Keywords:simplicial presheaf, hypercover, cocycle
Categories:14F35, 18G30, 55U35
2. CMB 2006 (vol 49 pp. 407)
|Intermediate Model Structures for Simplicial Presheaves |
This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.
Categories:18G30, 18F20, 55U35
3. CMB 2004 (vol 47 pp. 321)
|Classifying Spaces for Monoidal Categories Through Geometric Nerves |
The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories.
Keywords:monoidal category, pseudo-simplicial category,, simplicial set, classifying space, homotopy type
Categories:18D10, 18G30, 55P15, 55P35, 55U40