2,083 research outputs found

    Completeness Theorems via the Double Dual Functor

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    The aim of this paper is to apply properties of the double dual endofunctor on the category of bounded distributive lattices and some extensions thereof to obtain completeness of certain non-classical propositional logics in a unified way. In particular, we obtain completeness theorems for Moisil calculus, n-valued Łukasiewicz calculus and Nelson calculus. Furthermore we show some conservativeness results by these methods.Facultad de Ciencias Exacta

    Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion

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    Motivated by the recent interest in models of guarded (co-)recursion we study its equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading to the description of classifying theories for guarded recursion and hence completeness results involving our axioms of guarded fixpoint operators in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589

    A trace formula approach to control theorems for overconvergent automorphic forms

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    We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.Comment: 25 pages. Main results strengthened, higher level version include

    Kan extensions and the calculus of modules for \infty-categories

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    Various models of (,1)(\infty,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an \infty-cosmos. In a generic \infty-cosmos, whose objects we call \infty-categories, we introduce modules (also called profunctors or correspondences) between \infty-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from AA to BB is an \infty-category equipped with a left action of AA and a right action of BB, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed \infty-cosmoi, to limits and colimits of diagrams valued in an \infty-category, as introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom circularity removed; v3. final journal version to appear in Alg. Geom. To

    A general framework for homotopic descent and codescent

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    In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as \infty-category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for (,1)(\infty, 1)-categories. We provide general criteria, reminiscent of Mandell's theorem on EE_{\infty}-algebra models of pp-complete spaces, under which homotopic (co)descent is satisfied. Furthermore, we construct general descent and codescent spectral sequences, which we interpret in terms of derived (co)completion and homotopic (co)descent. We show that a number of very well-known spectral sequences, such as the unstable and stable Adams spectral sequences, the Adams-Novikov spectral sequence and the descent spectral sequence of a map, are examples of general (co)descent spectral sequences. There is also a close relationship between the Lichtenbaum-Quillen conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic K-theory to \'etale K-theory. Moreover, there are intriguing analogies between derived cocompletion (respectively, completion) and homotopy left (respectively, right) Kan extensions and their associated assembly (respectively, coassembly) maps.Comment: Discussion of completeness has been refined; statement of the theorem on assembly has been corrected; numerous small additions and minor correction
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