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Generalized space
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<h2 id="examples">Examples</h2> <ul><li>A <a href="/facts/Locale_(mathematics)/sNE2vnP8">locale</a> is a sort of a space but perhaps not with enough points.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> The topos theory is sometimes said to be the theory of generalized locales.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li><a href="/facts/Jean_Giraud_(mathematician)/Ca1bx8CV">Jean Giraud</a>'s gros topos, <a href="/facts/Peter_Johnstone_(mathematician)/j3k2A3Xl">Peter Johnstone</a>'s topological topos,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> or more recent incarnations such as <a href="/facts/Condensed_set/pmoY5bgE">condensed sets</a> or <a href="/facts/Pyknotic_set/wSjQ7wmX">pyknotic sets</a>. 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 <a href="/facts/Generalized_function/fCM4VsyT">generalized function</a>. (Note these constructions are more precise than various <a href="/facts/Completions_in_category_theory/gMvrkBxl">completions</a> of the category of topological spaces.)</li></ul> <ul><li>Lawvere, F. William (1975). "Continuously Variable Sets; Algebraic Geometry = Geometric Logic". <i>Logic Colloquium '73, Proceedings of the Logic Colloquium</i>. Studies in Logic and the Foundations of Mathematics. Vol. 80. pp. 135–156. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0049-237X%2808%2971947-5">10.1016/S0049-237X(08)71947-5</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> .</li> <li>Lawvere, <a href="http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf">Categories of spaces may not be generalized spaces as exemplified by directed graphs</a></li> <li>Johnstone, Peter T. (1985). "How general is a generalized space?". <i>Aspects of Topology</i>. pp. 77–112. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9781107359925.004">10.1017/CBO9781107359925.004</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> .</li> <li>Grothendieck, A.; Verdier, J. L. (1972), "Topos", <i>Théorie des Topos et Cohomologie Etale des Schémas</i>, Lecture Notes in Mathematics, vol. 269, Springer, pp. 299–518, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0081555">10.1007/BFb0081555</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> </li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Grothendieck & Verdier 1972 - 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 <a href="https://doi.org/10.1007%2FBFb0081555" target="_blank">https://doi.org/10.1007%2FBFb0081555</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Lawvere 1975 - 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. <a href="https://doi.org/10.1016%2FS0049-237X%2808%2971947-5" target="_blank">https://doi.org/10.1016%2FS0049-237X%2808%2971947-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Locales as geometric objects". MathOverflow. Retrieved 2024-07-22. <a href="https://mathoverflow.net/questions/250263/locales-as-geometric-objects/250307#250307" target="_blank">https://mathoverflow.net/questions/250263/locales-as-geometric-objects/250307#250307</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Johnstone 1985 - 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. <a href="https://doi.org/10.1017%2FCBO9781107359925.004" target="_blank">https://doi.org/10.1017%2FCBO9781107359925.004</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"On a Topological Topos at The n-Category Café". golem.ph.utexas.edu. <a href="https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html" target="_blank">https://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>