15:45
Directed algebraic topology
Abstract
In directed algebraic topology, a topological space is endowed
with an extra structure, a selected subset of the paths called the
directed paths or the d-structure. The subset has to contain the
constant paths, be closed under concatenation and non-decreasing
reparametrization. A space with a d-structure is a d-space.
If the space has a partial order, the paths increasing wrt. that order
form a d-structure, but the circle with counter clockwise paths as the
d-structure is a prominent example without an underlying partial order.
Dipaths are dihomotopic if there is a one-parameter family of directed
paths connecting them. Since in general dipaths do not have inverses,
instead of fundamental groups (or groupoids), there is a fundamental
category. So already at this stage, the algebra is less desirable than
for topological spaces.
We will give examples of what is currently known in the area, the kind
of methods used and the problems and questions which need answering - in
particular with applications in computer science in mind.