Fibred sites and stack cohomology

J. F. Jardine

Home/Archive/arXiv

arXiv · math/0406483

Fibred sites and stack cohomology

J. F. Jardine

View original source ↗Read source document ↗Export metadata ↓

Abstract

The usual notion of a site fibred over a stack is expanded to a definition of a site C/A fibred over a presheaf of categories A. Presheaves of simplicial sets on the site fibred over a presheaf of categories A are contravariant enriched diagrams defined on A, taking values in simplicial sets. The standard model structure for presheaves of simplicial sets induces a coarse equivariant structure for enriched contravariant A-diagrams. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site C/G is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on C/G. In particular, stack cohomology can be calculated on the fibred site for a representing presheaf of groupoids.

Explore related subjects

math.AT math.AG math.KT

Keep this discovery

BibTeX RIS

J. F. Jardine. 2004-06-23. Fibred sites and stack cohomology. https://arxiv.org/abs/math/0406483

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

math.AT↗

arXiv2026

The homotopy types of directed path and trace spaces

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

math.AT↗

arXiv2026

Moduli spaces of geometric functorial field theories

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

math.AT↗