We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of t...

The localic definitions of cosheaves and connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π0 together with certain additional structure and properties that constrain π0 to be the connected components cosheaf. In the inductively generated case, complete spreads (...

Informally speaking, a map is stratified if it is proper and there is a partition S of X into manifolds so that over each manifold S ∈ S the map f(S)→ S is a fiber bundle. One perspective on this question uses the open sets of X to define a functor F̂ : Open(X)→ Set that assigns to each open U the set π0(f (U)) of path components. This functor satisfies a gluing axiom reminiscent of the usual va...

In this paper, we study data flows in directed networks with a hierarchical recurrent structure from a cosheaf theoretical perspective. We utilize the visual parametrization of directed recurrent programs provided in persistence diagrams for cosheaf theoretical constructions. In considering cosheaves on persistence diagrams, we link global network structure and local recurrent process data. An ...

The localic definitions of cosheaves, connectedness and local connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π0 together with certain additional structure and properties that constrain π0 to be the connected components cosheaf. In the inductively generated cas...