Arxiv:1807.06986V1 [Math.CT] 18 Jul 2018 1.4
A NAIVE DIAGRAM-CHASING APPROACH TO FORMALISATION OF TAME TOPOLOGY by notes by misha gavrilovich and konstantin pimenov in memoriam: evgenii shurygin ——————————————————————————————————— ... due to internal constraints on possible architectures of unknown to us functional "mental structures". Misha Gromov. Structures, Learning and Ergosystems: Chapters 1-4, 6. Abstract.— We rewrite classical topological definitions using the category- theoretic notation of arrows and are thereby led to their concise reformulations in terms of simplicial categories and orthogonality of morphisms, which we hope might be of use in the formalisation of topology and in developing the tame topology of Grothendieck. Namely, we observe that topological and uniform spaces are simplicial objects in the same category, a category of filters, and that a number of elementary properties can be obtained by repeatedly passing to the left or right orthogonal (in the sense of Quillen model categories) starting from a simple class of morphisms, often a single typical (counter)example appearing implicitly in the definition. Examples include the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms, and, outside of topology, finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective and surjective (homo)morphisms. Contents 1. Introduction. 3 1.1. Main ideas. 3 1.2. Contents. 4 1.3. Speculations. 6 arXiv:1807.06986v1 [math.CT] 18 Jul 2018 1.4. Surjection: an example . 6 1.5. Intuition/Yoga of orthogonality. 7 1.6. Intuition/Yoga of transcription. 8 A draft of a research proposal. Comments welcome at miishapp@sddf:org.
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