3,494 research outputs found
Ind- and Pro- definable sets
We describe the ind- and pro- categories of the category of definable sets,
in some first order theory, in terms of points in a sufficiently saturated
model.Comment: 8 pages; Part of author's phd thesi
The categorical limit of a sequence of dynamical systems
Modeling a sequence of design steps, or a sequence of parameter settings,
yields a sequence of dynamical systems. In many cases, such a sequence is
intended to approximate a certain limit case. However, formally defining that
limit turns out to be subject to ambiguity. Depending on the interpretation of
the sequence, i.e. depending on how the behaviors of the systems in the
sequence are related, it may vary what the limit should be. Topologies, and in
particular metrics, define limits uniquely, if they exist. Thus they select one
interpretation implicitly and leave no room for other interpretations. In this
paper, we define limits using category theory, and use the mentioned relations
between system behaviors explicitly. This resolves the problem of ambiguity in
a more controlled way. We introduce a category of prefix orders on executions
and partial history preserving maps between them to describe both discrete and
continuous branching time dynamics. We prove that in this category all
projective limits exist, and illustrate how ambiguity in the definition of
limits is resolved using an example. Moreover, we show how various problems
with known topological approaches are now resolved, and how the construction of
projective limits enables us to approximate continuous time dynamics as a
sequence of discrete time systems.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
2-Vector Spaces and Groupoids
This paper describes a relationship between essentially finite groupoids and
2-vector spaces. In particular, we show to construct 2-vector spaces of
Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding
to functors between groupoids in both a covariant and contravariant way, which
are ambidextrous adjoints. This is used to construct a representation--a weak
functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids)
into 2Vect. In this paper we prove this and give the construction in detail.Comment: 44 pages, 5 figures - v2 adds new theorem, significant changes to
proofs, new sectio
Lifting retracted diagrams with respect to projectable functors
We prove a general categorical theorem that enables us to state that under
certain conditions, the range of a functor is large. As an application, we
prove various results of which the following is a prototype: If every diagram,
indexed by a lattice, of finite Boolean (v,0)-semilattices with
(v,0)-embeddings, can be lifted with respect to the \Conc functor on
lattices, then so can every diagram, indexed by a lattice, of finite
distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this
statement held, this would solve in turn the (still open) problem whether every
distributive algebraic lattice is isomorphic to the congruence lattice of a
lattice. We also outline potential applications of the method to other
functors, such as the functor on von Neumann regular rings
- …