1,537 research outputs found
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-L\"of type theory taking
advantage of its homotopical models. In particular, we can use and construct
objects of homotopy theory and reason about them using higher inductive types.
In this article, we construct the real projective spaces, key players in
homotopy theory, as certain higher inductive types in homotopy type theory. The
classical definition of RP(n), as the quotient space identifying antipodal
points of the n-sphere, does not translate directly to homotopy type theory.
Instead, we define RP(n) by induction on n simultaneously with its tautological
bundle of 2-element sets. As the base case, we take RP(-1) to be the empty
type. In the inductive step, we take RP(n+1) to be the mapping cone of the
projection map of the tautological bundle of RP(n), and we use its universal
property and the univalence axiom to define the tautological bundle on RP(n+1).
By showing that the total space of the tautological bundle of RP(n) is the
n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an
(n+1)-cell attached to it. The infinite dimensional real projective space,
defined as the sequential colimit of the RP(n) with the canonical inclusion
maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises
as the subtype of the universe consisting of 2-element types. Indeed, the
infinite dimensional projective space classifies the 0-sphere bundles, which
one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of
homotopy type theory, including the interplay of type theoretic and homotopy
theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201
A Vectorized Algorithm for Molecular Dynamics of Short Range Interacting Particles
We report on a lattice based algorithm, completely vectorized for molecular
dynamics simulations. Its algorithmic complexity is of the order O(N), where
is the number of particles. The algorithm works very effectively when the
particles have short range interaction, but it is applicable to each kind of
interaction. The code was tested on a Cray ymp el in a simulation of flowing
granular material.Comment: 9 pages, 6 figures, Late
Molecular dynamics of arbitrarily shaped granular particles}
We propose a new model for the description of complex granular particles and
their interaction in molecular dynamics simulations of granular material in two
dimensions. The grains are composed of triangles which are connected by
deformable beams. Particles are allowed to be convex or concave. We present
first results of simulations using this particle model.Comment: uuencoded compressed PostScript, 40 pages, 19 figures (included
Cellular Cohomology in Homotopy Type Theory
We present a development of cellular cohomology in homotopy type theory.
Cohomology associates to each space a sequence of abelian groups capturing part
of its structure, and has the advantage over homotopy groups in that these
abelian groups of many common spaces are easier to compute. Cellular cohomology
is a special kind of cohomology designed for cell complexes: these are built in
stages by attaching spheres of progressively higher dimension, and cellular
cohomology defines the groups out of the combinatorial description of how
spheres are attached. Our main result is that for finite cell complexes, a wide
class of cohomology theories (including the ones defined through
Eilenberg-MacLane spaces) can be calculated via cellular cohomology. This
result was formalized in the Agda proof assistant
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