Simplicial sets with a notion of smallness

Abstract

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 uniform convergence 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. In model theory, we observe that indiscernible sequences in a model form a simplicial set with a notion of smallness which can be seen as an analogue of the Stone space of types. These reformulations are presented as a series of exercises, to emphasise their elementary nature and that they indeed can be used as exercises to make a student familiar with computations in basic simplicial and topological language. (Formally, we consider the category of simplicial objects in the category of filters in the sense of Bourbaki.) This work is unfinished and is likely to remain such for a while, hence we release it as is, in the small hope that our reformulations may provide interesting examples of computations in basic simplicial and topological language on material familiar to a student in a first course of topology or category theory.