Title: A Cofibration Category on Closure Spaces,
Abstract: We construct a cofibration category structure on the category of closure
spaces $\mathbf{Cl}$, the category whose objects are sets endowed with a
Čech closure operator and whose morphisms are the continuous maps between
them. We then study various closure structures on metric spaces, graphs, and
simplicial complexes, showing how each case gives rise to an interesting
homotopy theory. In particular, we show that there exists a natural family of
closure structures on metric spaces which produces a non-trivial homotopy
theory for finite metric spaces, i.e. point clouds, the spaces of interest in
topological data analysis. We then give a closure structure to graphs and
simplicial complexes which may be used to construct a new combinatorial (as
opposed to topological) homotopy theory for each skeleton of those spaces. We
show that there is a Seifert-van Kampen theorem for closure spaces, a
well-defined notion of persistent homotopy and an associated interleaving
distance, and, as an illustration of the difference with the topological
setting, we calculate the fundamental group for the circle and the wedge of
circles endowed with different closure structures.