Finite combinatorics implicit in the basic definitions of topology
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910625399046144 |
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| author | Gavrilovich, Misha |
| author_facet | Gavrilovich, Misha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_20464 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite combinatorics implicit in the basic definitions of topology Gavrilovich, Misha Category Theory Algebraic Topology 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. |
| title | Finite combinatorics implicit in the basic definitions of topology |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2409.20464 |