Simplicial sets with a notion of smallness

TL;DR

This paper introduces a notion of smallness in simplicial sets, offering a unified framework that reformulates classical topological and algebraic concepts, aiming to aid students in understanding foundational ideas through elementary exercises.

Contribution

It presents a novel approach to incorporate smallness into simplicial sets, connecting topology, algebraic topology, and model theory in a simplified, exercise-based format.

Findings

Reformulation of continuity, limits, and convergence in simplicial terms

Representation of locally trivial bundles as direct products after base change

Indiscernible sequences modeled as simplicial sets analogous to Stone spaces

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.