4 research outputs found
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
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