34 lines
3.1 KiB
Markdown
34 lines
3.1 KiB
Markdown
---
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title: "Generalized space"
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chunk: 1/1
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source: "https://en.wikipedia.org/wiki/Generalized_space"
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category: "reference"
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tags: "science, encyclopedia"
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date_saved: "2026-05-05T07:23:56.343111+00:00"
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instance: "kb-cron"
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---
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In mathematics, a generalized space is a generalization of a topological space. Impetuses for such a generalization comes at least in two forms:
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A desire to apply concepts like cohomology for objects that are not traditionally viewed as spaces. For example, a topos was originally introduced for this reason.
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A practical need to remedy the deficiencies that some naturally occurring categories of spaces (e.g., ones in functional analysis) tend not to be abelian, a standard requirement to do homological algebra.
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Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I:
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On peut done dire que la notion de topos, dérivé naturel du point de vue faisceautique en Topologie, constitue à son tour un élargissement substantiel de la notion d'espace topologique, un grand nombre de situations qui autrefois n'étaient pas considérées comme relevant de intuition topologique
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However, William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself."
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A generalized space should not be confused with a geometric object that can substitute the role of spaces. For example, a stack is typically not viewed as a space but as a geometric object with a richer structure.
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== Examples ==
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A locale is a sort of a space but perhaps not with enough points. The topos theory is sometimes said to be the theory of generalized locales.
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Jean Giraud's gros topos, Peter Johnstone's topological topos, or more recent incarnations such as condensed sets or pyknotic sets. These attempt to embed the category of (certain) topological spaces into a larger category of generalized spaces, in a way philosophically if not technically similar to the way one generalizes a function to a generalized function. (Note these constructions are more precise than various completions of the category of topological spaces.)
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== Citations ==
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== Bibliography ==
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Lawvere, F. William (1975). "Continuously Variable Sets; Algebraic Geometry = Geometric Logic". Logic Colloquium '73, Proceedings of the Logic Colloquium. Studies in Logic and the Foundations of Mathematics. Vol. 80. pp. 135–156. doi:10.1016/S0049-237X(08)71947-5. ISBN 978-0-444-10642-1.
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Lawvere, F. William (2005), "Categories of spaces may not be generalized spaces as exemplified by directed graphs" (PDF), Reprints in Theory and Applications of Categories (9): 1–7
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Johnstone, Peter T. (1985). "How general is a generalized space?". Aspects of Topology. pp. 77–112. doi:10.1017/CBO9781107359925.004. ISBN 978-0-521-27815-7.
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Grothendieck, A.; Verdier, J. L. (1972), "Topos", Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathematics, vol. 269, Springer, pp. 299–518, doi:10.1007/BFb0081555, ISBN 978-3-540-05896-0 |