Topological Representation of Geometric Theories

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a `syntax-semantics' duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantical topological groupoid of models and isomorphisms of a theory and a direct proof that this groupoid represents its classifying topos. Using this representation, a contravariant adjunction is constructed between theories and topological groupoids. The restriction of this adjunction yields a contravariant equivalence between theories with enough models and semantical groupoids. Technically a variant of the syntax-semantics duality constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids---reflecting the use of `indexed' models in the representation theorem---which in several respects simplifies the construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised version can be found at http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting of which is not allowed by Wiley). Changes in v2: updated comment

A note on the Penon definition of nn-category

We show that doubly degenerate Penon tricategories give symmetric rather than braided monoidal categories. We prove that Penon tricategories cannot give all tricategories, but we show that a slightly modified version of the definition rectifies the situation. We give the modified definition, using non-reflexive rather than reflexive globular sets, and show that the problem with doubly degenerate tricategories does not arise.Comment: 14 pages, to appear in Cahiers de Topologie et Geometrie Differentielle Categorique

Chopping Hungary Up by the 1920 Peace Dictate of Trianon. Causes, Events and Consequences

The regime that emerged with the peace treaties concluded after World War I chopped up the Kingdom of Hungary and rewarded its neighbours, helping them to establish themselves on the basis of the principle of national self-determination, by giving them two-thirds of Hungary’s areas and one-third of its Hungarian-speaking population. During the settlement that followed World War II, the great powers repeatedly forced this shocking decision upon Hungarians drifted to the losing side. This essay sums up the causes and events of this difficult-to-survive historical traumatism and its adverse impacts on the Central European region, in retrospect after more than a century

Period of Mystery: Or Syntax and the Semantic Pause

Paper by Adam Makka