2,083 research outputs found
Completeness Theorems via the Double Dual Functor
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
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
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 -categories
Various models of -categories, including quasi-categories,
complete Segal spaces, Segal categories, and naturally marked simplicial sets
can be considered as the objects of an -cosmos. In a generic
-cosmos, whose objects we call -categories, we introduce
modules (also called profunctors or correspondences) between
-categories, incarnated as as spans of suitably-defined fibrations with
groupoidal fibers. As the name suggests, a module from to is an
-category equipped with a left action of and a right action of ,
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 -cosmoi,
to limits and colimits of diagrams valued in an -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
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 -category-theoretic, as our framework
is constructed in the universe of simplicially enriched categories, which are a
model for -categories.
We provide general criteria, reminiscent of Mandell's theorem on
-algebra models of -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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