We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group G and the category of G-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group G, under certain conditions on the fixed point functors of the subgroups of G. These conditions hold in many examples, though not in the category of chain complexes, where we nevertheless establish and generalize to collections an equivariant Whitehead Theoremà la Kropholler and Wall for the normalized chain complexes of simplicial G-sets.

Considering the results of this pilot study, it can be concluded that the German version of the SIS is an instrument measuring perceived stigmatization in cancer patients with very good psychometric properties. However, further validation studies are considered necessary.

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category D is an infinite loop space, the zeroth space of the K-theory spectrum KD. This spectrum has stable homotopy groups concentrated in levels 0, 1, and 2. In this paper, we describe part of the Postnikov data of KD in terms of categorical structure. We use this to show that there is no strict skeletal Picard 2-category whose K-theory realizes the 2-truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard 2-category ΣC from a Picard 1-category C, and show that it commutes with K-theory in that KΣC is stably equivalent to ΣK C. Z/2 → π 1 (i.e., a stable quadratic map from π 0 to π 1 ) corresponding to the symmetry. Such a characterization is implied by the following result.Theorem 1.1 ([JO12, Theorem 2.2]). Every Picard category is equivalent to one which is both strict and skeletal.We call this phenomenon strict skeletalization. This theorem is quite surprising given that it is false without the symmetry. Indeed, Baez and Lauda [BL04] give a good account of the failure of strict skeletalization for 2-groups (the non-symmetric version of Picard 1-categories), and how it leads to a cohomological classification for 2-groups. Johnson and Osorno [JO12] show, in effect, that the relevant obstructions are unstable phenomena which become trivial upon stabilization.When we turn to the question of building models for specific homotopy types, the strict and skeletal ones are the simplest: given a stable 1-type X , a strict and skeletal model will have objects equal to the elements of π 0 X and automorphisms of every object equal to the elements of π 1 X , with no morphisms between distinct objects. All that then remains is to define the correct symmetry isomorphisms, and these are determined entirely by the map k 0 .As an example, a strict and skeletal model for the 1-truncation of the sphere spectrum has objects the integers, each hom-set of automorphisms the integers mod 2, and k 0 given by the identity map on Z/2 corresponding to the fact that the generating object 1 has a nontrivial symmetry with itself. One might be tempted to build a strict and skeletal model for the 2-type of the sphere spectrum (the authors here certainly were, and such an idea also appears in [Bar14, Example 5.2]). But here we prove that this is not possible for the sphere spectrum, and in fact a large class of stable 2-types. Theorem 1.2 (Theorem 3.14). Let D be a strict skeletal Picard 2-category with k 0 surjective. Then the 0-connected cover of K D splits as a product of Eilenberg-Mac Lane spectra.

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