Algebraic models for higher categories

We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction we propose algebraic Kan complexes as an algebraic model for oo-groupoids and algebraic quasi-categories as an algebraic model for (oo,1)-categories. We furthermore give an explicit proof of the homotopy hypothesis.Comment: 23 pages, minor change

Dendroidal sets as models for connective spectra

Dendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.Comment: 27 pages, final version, accepted for publication in Journal of K-theor

Equivariance In Higher Geometry

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-prestacks. Finally we show that, for a 2-stack, the pullback functor along a Morita-equivalence of Lie groupoids is an equivalence of bicategories. Our results have direct applications to gerbes and 2-vector bundles. For instance, they allow to construct equivariant gerbes from local data and can be used to simplify the description of the local data. We illustrate the usefulness of our results in a systematic discussion of holonomies for unoriented surfaces.Comment: 42 pages, minor correction

Presentably symmetric monoidal infinity-categories are represented by symmetric monoidal model categories

We prove the theorem stated in the title. More precisely, we show the stronger statement that every symmetric monoidal left adjoint functor between presentably symmetric monoidal infinity-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories.Comment: v3: 17 pages, references updated and exposition improved, accepted for publication in Algebraic and Geometric Topolog

T-Duality via Gerby Geometry and Reductions

We consider topological T-duality of torus bundles equipped with S^{1}-gerbes. We show how a geometry on the gerbe determines a reduction of its band to the subsheaf of S^{1}-valued functions which are constant along the torus fibres. We observe that such a reduction is exactly the additional datum needed for the construction of a T-dual pair. We illustrate the theory by working out the example of the canonical lifting gerbe on a compact Lie group which is a torus bundles over the associated flag manifold. It was a recent observation of Daenzer and van Erp (arXiv1211.0763) that for certain compact Lie groups and a particular choice of the gerbe, the T-dual torus bundle is given by the Langlands dual group.Comment: 57 pages (revised version

Four Equivalent Versions of Non-Abelian Gerbes

We recall and partially improve four versions of smooth, non-abelian gerbes: Cech cocycles, classifying maps, bundle gerbes, and principal 2-bundles. We prove that all these four versions are equivalent, and so establish new relations between interesting recent developments. Prominent partial results we prove are a bijection between continuous and smooth non-abelian cohomology, and an explicit equivalence between bundle gerbes and principal 2-bundles as 2-stacks.Comment: 65 pages, v2: minor corrections, new Corollary 7.2 about equivariant gerbes; v3: again minor corrections; v3 is the final and published versio

Algebraic KK-theory of planar cuspidal curves

In this paper, we evaluate the algebraic $K$-groups of a planar cuspidal curve over a perfect $\mathbb{F}_p$-algebra relative to the cusp point. A conditional calculation of these groups was given earlier by Hesselholt, assuming a conjecture on the structure of certain polytopes. Our calculation here, however, is unconditional and illustrates the advantage of the new setup for topological cyclic homology by Nikolaus-Scholze, which is used throughout. The only input necessary for our calculation is the evaluation by the Buenos Aires Cyclic Homology group and by Larsen of the structure of Hochschild complex of the coordinate ring as a mixed complex, that is, as an object of the infinity category of chain complexes with circle action.Comment: 9 page

Beyond Frozen Conflict Scenarios for the Separatist Disputes of Eastern Europe. CEPS Paperback

This book forms part of a wider project on the relations between the European Union and Georgia, Moldova and Ukraine, and in particular the Association Agreements and Deep and Comprehensive Free Trade Areas (DCFTAs) between these three states and the European Union. The wider project was begun in 2015 in the aftermath of the Maidan uprising at the beginning of 2014, which had been provoked when President Yanukovich reneged over the signing of Ukraine’s Association Agreement with the EU. Following Yanukovich’s flight to Russia, the Association Agreement was duly signed later in 2014. The agreements with Georgia, Moldova and Ukraine have a substantial common content, while differing in various details. Overall, they provide an association model of unprecedented extent and depth. Democratic political values are at the heart of the agreements, while the economic content goes far beyond classic free trade agreements to include a wholesale approximation of EU internal market regulatory law. The purpose of our wider project was first of all to explain the complex content of the Association Agreements and DCFTAs, which was achieved in a series of comprehensive handbooks published at www.3dcftas.eu. However, the agreements contain only short and simple articles on conflict prevention and management, without meaningful operational content. This was notwithstanding the fact that the EU considers itself, for its own historical reasons, to have a special vocation in conflict prevention and resolution. In addition, Georgia and Moldova were already the sites of unresolved separatist conflicts originating around the collapse of the Soviet Union three decades ago, namely Abkhazia and South Ossetia in Georgia, and Transdniestria in Moldova, to which we have added the case of the Nagorny Karabakh between Armenia and Azerbaijan. On top of this legacy, the Maidan uprising led to the Russian annexation of Crimea and its hybrid war in the Donetsk and Luhansk oblasts of the Donbas. The Donbas thus joined the old ‘frozen conflicts’. In the light of the above, CEPS took the initiative to examine all five unresolved conflicts, to assess where these disputes seem to be heading, and what different scenarios could be imagined for their future, including how the European Union might become more engaged. Indeed, while none of the conflicts are resolved, none are for that matter ‘frozen’. Our first practical priority was to find an author to undertake a comprehensive study of the Donbas, since conditions there make it practically impossible for any analysts from the government-controlled part of Ukraine or from Europe to safely enter these territories for research purposes. We were therefore very fortunate to find Nikolaus von Twickel who had recently been travelling in the Donbas as part of the OSCE Mission there, and is now an independent analyst. For the other four ‘old’ conflicts we were also most fortunate to bring in Thomas de Waal, who has been a leading scholar of the region for some decades, and was willing to bring the stories of these conflicts up to date. The two authors were able to address the complete set of conflicts with a consistent analytical approach, as will be evident from reading the sets of scenarios. We express our warm appreciation towards Sweden and the Swedish International Development Agency (Sida) for their support to the entire project. This volume looks at future prospects for the string of unresolved conflicts that continue to plague the post-Soviet world. Four of them date back to the period when the USSR began to break up in the late 1980s. A new conflict, with many different elements and some similarities, was added to the list in 2014: the Donbas in eastern Ukraine. The open confrontation between Russia and Ukraine over the Donbas and Crimea not only destroyed relations between Moscow and Kyiv but changed politics across the region, shaking up the dynamics of the four existing protracted territorial conflicts over Abkhazia, Nagorny Karabakh, South Ossetia and Transdniestria. The five post-Soviet conflicts are often called ‘frozen’, but this is a misnomer. Although the peace processes around them often look frozen, the situations themselves are anything but frozen and are constantly changing. Two of them, over the Donbas and Nagorny Karabakh, are either ongoing or close to violence. Each dispute has its own history, character and context, which has grown more distinctive over time and has been further shaped by the confrontation over Ukraine. Each continues to evolve. Here we chart scenarios for how these conflicts may develop further with the aim of focusing policymakers’ thinking on which tendencies are dangerous and which ones can be encouraged. There are many moving parts to these situations and complacency is not an option