Basic topology in terms of simplicial sets with a notion of smallness

We consider simplicial sets equipped with a notion of smallness, and observe that this slight “topological” extension of the “algebraic” simplicial language allows a concise reformulation of a number of classical notions in topology, e.g. continuity, limit of a map or a sequence along a filter, various notions of equicontinuity and uniformconvergence of a sequence of functions; completeness and compactness; in algebraic topology, locally trivial bundles as a direct product after base-change and geometric realisation as a space of discontinuous paths. These reformulations are elementary and can perhaps be used in teaching to give motivated examples of elementary concepts in category theory. Surprisingly, this category is not well-studied and thus these observations raise many easy but open problems, which we like to think are in line with goals of tame topology put by Grothendieck. In the talk, we will work through of a couple of example, briefly mention some others, and indicate a number of open problems, who we like to think are in line with the goals of tame topologyput by Grothendieck.
This talk is based on preliminary notes http://mishap.sdf.org/Skorokhod_Geometric_Realisation_HomSets.pdf, http://mishap.sdf.org/6a6ywke/6a6ywke.pdf