Daniel Dugger

K-theory and derived equivalences

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

Replacing model categories with simplicial ones

In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories.

Motivic cell structures

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Kunneth theorems for cellular objects. AMS Classification 55U35; 14F42

Mapping spaces in quasi-categories

We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.

Postnikov extensions of ring spectra

!P2R! P1R! P0R! in the homotopy category of ring spectra. The levels come equipped with compatible maps R! PnR, and the n‐th level is characterized by having i.PnR/D 0 for i > n, together with the fact that i.R/! i.PnR/ is an isomorphism for i n. In this paper we produce k ‐invariants for the levels of this tower and explain their role in the following problem: if one only knows Pn 1R together with n.R/ as a 0.R/‐bimodule, what are the possibilities for PnR? Corollary 1.4 shows in what sense the possibilities are classified by k ‐invariants.

A curious example of triangulated-equivalent model categories which are not Quillen equivalent

The paper gives a new proof that the model categories of stable modules for the rings Z/p and Z/p[ǫ]/(ǫ) are not Quillen equivalent. The proof uses homotopy endomorphism ring spectra. Our considerations lead to an example of two differential graded algebras which are derived equivalent but whose associated model categories of modules are not Quillen equivalent. As a bonus, we also obtain derived equivalent dgas with non-isomorphic K-theories.

Rigidification of quasi-categories

We give a new construction for rigidifying a quasi-category into a simplicial category, and prove that it is weakly equivalent to the rigidification given by Lurie. Our construction comes from the use of necklaces, which are simplicial sets obtained by stringing simplices together. As an application of these methods, we use our model to reprove some basic facts from Lurie [13] about the rigidification process. 55U40; 18G30, 18B99

Low-dimensional Milnor–Witt stems over ℝ

This article computes some motivic stable homotopy groups over R. For 0 <= p - q <= 3, we describe the motivic stable homotopy groups of a completion of the motivic sphere spectrum. These are the first four Milnor-Witt stems. We start with the known Ext groups over C and apply the rho-Bockstein spectral sequence to obtain Ext groups over R. This is the input to an Adams spectral sequence, which collapses in our low dimensional range.

Coherence for invertible objects and multigraded homotopy rings

We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skew-commutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for stable homotopy groups.

Etale homotopy and sums-of-squares formulas

This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p>2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized etale cohomology theory called etale BP2.