2024/09/30 by Misha Gavrilovich, Gavrilovich, Misha
Business, Management and Accounting · Computer Science · #Algebraic Topology (math.AT) #Category Theory (math.CT) #Digital Image Processing Techniques #FOS: Mathematics #Optics and Image Analysis
paper · pdf · doi:10.48550/arxiv.2409.20464
openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explain how to see finite combinatorics of preorders implicit in the text of basic topological definitions or arguments in (Bourbaki, General topology, Ch.I), and define a concise combinatorial notation such that complete definitions of connectedness, compactness, contractibility, having a generic point, subspace, closed subspace, fit into 2 or 4 bytes. This notation is homotopy theoretic in nature, and is based on the following observation: A number of basic properties of continuous maps and topological spaces are defined using a single category-theoretic operation, taking left or right orthogonal complement with respect to the Quillen lifting property, repeatedly applied to a simple example illustrating the definition or its failure. Moreover, for most of these definitions this example can be chosen to be a map of finite topological spaces (=preorders) of size at most 5. This includes the properties of a space being connected, compact, contractible, discrete, having a generic point, and a map having dense image, being the inclusion of an (open or closed) subspace, or of a component into a disjoint union, and others. Our reformulations illustrate the generative power of the lifting property as a means of defining basic mathematical properties starting from their simplest or typical example. The exposition is accessible to a student.