6,612 research outputs found
Van Kampen Colimits and Path Uniqueness
Fibred semantics is the foundation of the model-instance pattern of software
engineering. Software models can often be formalized as objects of presheaf
topoi, i.e, categories of objects that can be represented as algebras as well
as coalgebras, e.g., the category of directed graphs. Multimodeling requires to
construct colimits of models, decomposition is given by pullback.
Compositionality requires an exact interplay of these operations, i.e.,
diagrams must enjoy the Van Kampen property. However, checking the validity of
the Van Kampen property algorithmically based on its definition is often
impossible.
In this paper we state a necessary and sufficient yet efficiently checkable
condition for the Van Kampen property to hold in presheaf topoi. It is based on
a uniqueness property of path-like structures within the defining congruence
classes that make up the colimiting cocone of the models. We thus add to the
statement "Being Van Kampen is a Universal Property" by Heindel and
Soboci\'{n}ski the fact that the Van Kampen property reveals a presheaf-based
structural uniqueness feature
Being Van Kampen in Presheaf Topoi is a Uniqueness Property
Fibred semantics is the foundation of the model-instance pattern of software engineering. Software models can often be formalized as objects of presheaf topoi, e.g. the category of directed graphs. Multimodeling requires to construct colimits of diagrams of single models and their instances, while decomposition of instances of the multimodel is given by pullback. Compositionality requires an exact interplay of these operations, i.e., the diagrams must enjoy the Van Kampen property. However, checking the validity of the Van Kampen property algorithmically based on its definition is often impossible.
In this paper we state a necessary and sufficient yet easily checkable condition for the Van Kampen property to hold for diagrams in presheaf topoi. It is based on a uniqueness property of path-like structures within the defining congruence classes that make up the colimiting cocone of the models. We thus add to the statement "Being Van Kampen is a Universal Property" by Heindel and Sobocinski presented at CALCO 2009 the fact that the Van Kampen property reveals a set-based structural uniqueness feature
Zariski-van Kampen theorem for higher homotopy groups
This paper gives an extension of the classical Zariski-van Kampen theorem
describing the fundamental groups of the complements of plane singular curves
by generators and relations. It provides a procedure for computation of the
first non-trivial higher homotopy groups of the complements of singular
projective hypersurfaces in terms of the homotopy variation operators
introduced here.Comment: 37 pages, LaTeX2e with amsmath, amsthm and amscd packages. To appear
in J. Inst. Math. Jussieu (2003) with the first proof of Theorem 7.1
significantly developped and new references added. Due to copyright
restrictions, this final version will only be available at Cambridge Journals
Online (http://journals.cambridge.org) when published. Thus the content of
the paper here is the same as that of version 1 of 3 March 200
On the axioms for adhesive and quasiadhesive categories
A category is adhesive if it has all pullbacks, all pushouts along
monomorphisms, and all exactness conditions between pullbacks and pushouts
along monomorphisms which hold in a topos. This condition can be modified by
considering only pushouts along regular monomorphisms, or by asking only for
the exactness conditions which hold in a quasitopos. We prove four
characterization theorems dealing with adhesive categories and their variants.Comment: 20 pages; v2 final version, contains more details in some proof
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem
This paper is the second in a series exploring the properties of a functor
which assigns a homotopy double groupoid with connections to a Hausdorff space.
We show that this functor satisfies a version of the van Kampen theorem, and
so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems.
The methods are analogous to those developed by Brown and Higgins for similar
theorems for other higher homotopy groupoids.
An integral part of the proof is a detailed discussion of commutative cubes
in a double category with connections, and a proof of the key result that any
composition of commutative cubes is commutative. These results have recently
been generalised to all dimensions by Philip Higgins.Comment: 19 pages, uses picte
Characterizing Van Kampen Squares via Descent Data
Categories in which cocones satisfy certain exactness conditions w.r.t.
pullbacks are subject to current research activities in theoretical computer
science. Usually, exactness is expressed in terms of properties of the pullback
functor associated with the cocone. Even in the case of non-exactness,
researchers in model semantics and rewriting theory inquire an elementary
characterization of the image of this functor. In this paper we will
investigate this question in the special case where the cocone is a cospan,
i.e. part of a Van Kampen square. The use of Descent Data as the dominant
categorical tool yields two main results: A simple condition which
characterizes the reachable part of the above mentioned functor in terms of
liftings of involved equivalence relations and (as a consequence) a necessary
and sufficient condition for a pushout to be a Van Kampen square formulated in
a purely algebraic manner.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
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