Daniel C. Isaksen

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

A model structure on the category of pro-simplicial sets

We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.

Etale realization on the A1-homotopy theory of schemes

Abstract We compare Friedlanders definition of the etale topological type for simplicial schemes to another definition involving realizations of pro-simplicial sets. This can be expressed as a notion of hypercover descent for etale homotopy. We use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over S to the category of pro-spaces. After completing away from the characteristics of the residue fields of S, we get a functor from the Morel–Voevodsky A 1 -homotopy category of schemes to the homotopy category of pro-spaces.

Duality and Pro-Spectra

Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spec- tra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the oppo- site of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier- Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocaliza- tion model structures generalizing known results for cofibrantly generated model categories. AMS Classification 55P42; 55P25, 18G55, 55U35, 55Q55

Strict Model Structures for Pro-categories

We show that if ς is a proper model category, then the pro-category pro-ς has a proper model structure in which the weak equivalences are the levelwise weak equivalences. The structure is simplicial when ς is simplicial. This is related to a major result of [10] and is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and [19].

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.

OBSTRUCTION THEORY IN MODEL CATEGORIES

Abstract Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.

The η–inverted ℝ–motivic sphere

We use an Adams spectral sequence to calculate the R-motivic stable homotopy groups after inverting eta. The first step is to apply a Bockstein spectral sequence in order to obtain h_1-inverted R-motivic Ext groups, which serve as the input to the eta-inverted R-motivic Adams spectral sequence. The second step is to analyze Adams differentials. The final answer is that the Milnor-Witt (4k-1)-stem has order 2^{u+1}, where u is the 2-adic valuation of 4k. This answer is reminiscent of the classical image of J. We also explore some of the Toda bracket structure of the eta-inverted R-motivic stable homotopy groups.

Large Annihilators in Cayley–Dickson Algebras

Cayley–Dickson algebras are nonassociative ℝ-algebras that generalize the well-known algebras ℝ, ℂ, ℍ, and 𝕆. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2 n -dimensional Cayley–Dickson algebra has dimension at most 2 n − 4n + 4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Although a complete description of zero-divisors seems to be out of reach, we can describe precisely the elements whose annihilators have dimension 2 n − 4n + 4.

Eigentheory of Cayley-Dickson algebras

Abstract We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.