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Abstract. The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure [5]. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for classification of polynomial and homogeneous functors. Finally we show that the n-th derivative induces a Quillen map between the n-homogeneous model structure on small functors from pointed simplicial sets to spectra and the category of spectra with Σn-action. We consider also a finitary version of the n-homogeneous model structure and the n-homogeneous model structure on functors from pointed finite simplicial sets to spectra. In these two cases the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie [12]. 1.

Abstract. An n-truncated model structure on simplicial (pre-)sheaves is described having as weak equivalences maps that induce isomorphisms on certain homotopy sheaves only up to degree n. Starting from one of Jardine’s intermediate model structures we construct such an n-type model structure via Bousfield-Friedlander localization and exhibit useful generating sets of trivial cofibrations. Injectively fibrant objects in these categories are called n-hyperstacks. The whole setup can consequently be viewed as a description of the homotopy theory of higher hyperstacks. More importantly, we construct analogous n-truncations on simplicial groupoids and prove a Quillen equivalence between these settings. We achieve a classification of n-types of simplicial presheaves in terms of (n −1)-types of presheaves of simplicial groupoids. Our classification holds for general n. Therefore this can also be viewed as the homotopy theory of (pre-)sheaves of (weak) higher groupoids. Contents

Abstract. This is a foundational account of the étale topology of generalized Witt vectors and of related constructions. The theory of the usual, “p-typical” Witt vectors of p-adic schemes of finite type is already reasonably well developed. The main point here is to generalize this theory in two different ways. We allow not just p-typical Witt vectors but also, for example, those taken with respect to any set of primes in any ring of integers in any global field. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of the Greenberg transform. We investigate whether many standard geometric properties of spaces and maps are preserved by Witt vector functors.

We generalize and greatly simplify the approach of Lydakis and Dundas-Röndigs-Østvær to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V = M = S ∗ pointed simplicial sets and L = S1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We

Abstract. Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi. 1.

Abstract. In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every contravariant functor from spaces to spaces which takes coproducts to products up to homotopy, and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie’s classification of linear functors [15].