We give a survey of the ideas of descent and nilpotence, beginning with the theory of thick
subcategories. We focus on examples arising from chromatic homotopy theory (such as …

We define notions of direct and inverse limits in an $ n $-category. We prove that the $ n+ 1$-
category $ nCAT'$ of fibrant $ n $-categories admits direct and inverse limits. At the end we …

arXiv:math/0412512v4 [math.AG] 24 May 2007 Page 1 arXiv:math/0412512v4 [math.AG] 24
May 2007 Notes on Grothendieck topologies, fibered categories and descent theory …

For the average Joe, and the median Jane, the nilpotence theorem refers to a result in stable
homotopy theory, conjectured by Ravenel and proved by Devinatz, Hopkins and Smith in …

We introduce the notions of normal tensor functor and exact sequence of tensor categories.
We show that exact sequences of tensor categories generalize strictly exact sequences of …

Higher Homotopy van Kampen Theorems allow the computation as colimits of certain
homotopical invariants of glued spaces. One corollary is to describe homotopical excision in …

We prove that the homotopy theory of Joyal's tribes is equivalent to that of fibration
categories. As a consequence, we deduce a variant of the conjecture asserting that Martin …

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for
homotopy pullbacks and note that these results imply similar results for zigzags in the …

The goal of this paper is to address the problem of building a path object for the category of
Grothendieck (weak)∞-groupoids. This is the missing piece for a proof of Grothendieck's …

We study topological full groups attached to groupoid models for left regular representations
of Garside categories. Groups arising in this way include Thompson's group $ V $ and many …