9,530 research outputs found
Elimination of quotients in various localisations of premodels into models
The contribution of this article is quadruple. It (1) unifies various schemes
of premodels/models including situations such as presheaves/sheaves,
sheaves/flabby sheaves, prespectra/-spectra, simplicial topological
spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors
in categories/strong stacks and, to some extent, functors from a limit sketch
to a model category versus the homotopical models for the limit sketch; (2)
provides a general construction from the premodels to the models; (3) proposes
technics that allows one to assess the nature of the universal properties
associated with this construction; (4) shows that the obtained localisation
admits a particular presentation, which organises the structural and relational
information into bundles of data. This presentation is obtained via a process
called an elimination of quotients and its aim is to facilitate the handling of
the relational information appearing in the construction of higher dimensional
objects such as weak -categories, weak -groupoids and
higher moduli stacks.Comment: The text is the same as in v6; this version contains corrections to
the published MDPI paper, the main reason for this change is that the diagram
of Proposition 3.1 was meant to be a 3 dimensional diagram (while only the
front face appeared in the published paper). The wording of some sentences
and the diagram of Example 6.42 are changed accordingly. A typo in the table
of Ex. 6.42 is correcte
Triangulated surfaces in triangulated categories
For a triangulated category A with a 2-periodic dg-enhancement and a
triangulated oriented marked surface S we introduce a dg-category F(S,A)
parametrizing systems of exact triangles in A labelled by triangles of S. Our
main result is that F(S,A) is independent on the choice of a triangulation of S
up to essentially unique Morita equivalence. In particular, it admits a
canonical action of the mapping class group. The proof is based on general
properties of cyclic 2-Segal spaces.
In the simplest case, where A is the category of 2-periodic complexes of
vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya
category of the surface S. Therefore, our construction can be seen as
implementing a 2-dimensional instance of Kontsevich's program on localizing the
Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded
version, comments welcom
Knowledge Spaces and Learning Spaces
How to design automated procedures which (i) accurately assess the knowledge
of a student, and (ii) efficiently provide advices for further study? To
produce well-founded answers, Knowledge Space Theory relies on a combinatorial
viewpoint on the assessment of knowledge, and thus departs from common,
numerical evaluation. Its assessment procedures fundamentally differ from other
current ones (such as those of S.A.T. and A.C.T.). They are adaptative (taking
into account the possible correctness of previous answers from the student) and
they produce an outcome which is far more informative than a crude numerical
mark. This chapter recapitulates the main concepts underlying Knowledge Space
Theory and its special case, Learning Space Theory. We begin by describing the
combinatorial core of the theory, in the form of two basic axioms and the main
ensuing results (most of which we give without proofs). In practical
applications, learning spaces are huge combinatorial structures which may be
difficult to manage. We outline methods providing efficient and comprehensive
summaries of such large structures. We then describe the probabilistic part of
the theory, especially the Markovian type processes which are instrumental in
uncovering the knowledge states of individuals. In the guise of the ALEKS
system, which includes a teaching component, these methods have been used by
millions of students in schools and colleges, and by home schooled students. We
summarize some of the results of these applications
12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser
This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
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