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The theory was rounded out by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning, since it involved a Grothendieck topology. The idea of a Grothendieck topology (also known as a site ) has been characterised by John Tate as a bold pun on the two senses of Riemann surface .
A locale is a sort of a space but perhaps not with enough points. [3] The topos theory is sometimes said to be the theory of generalized locales. [4]Jean Giraud's gros topos, Peter Johnstone's topological topos, [5] or more recent incarnations such as condensed sets or pyknotic sets.
An introduction to fibrations, topos theory, the effective topos and modest sets (Technical report). Laboratory for Foundations of Computer Science, University of Edinburgh. CiteSeerX 10.1.1.112.4533. ECS-LFCS-92-208. Bernadet, Alexis; Graham-Lengrand, Stéphane (2013). "A simple presentation of the effective topos". arXiv: 1307.3832 .
Topos Theory. Courier. ISBN 978-0-486-49336-7. For a long time the standard compendium on topos theory. However, even Johnstone describes this work as "far too hard to read, and not for the faint-hearted." Johnstone, Peter T. (2002). Sketches of an Elephant: A Topos Theory Compendium. Vol. 2. Clarendon Press. ISBN 978-0-19-851598-2.
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Francis William Lawvere (/ l ɔː ˈ v ɪər /; February 9, 1937 – January 23, 2023) was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Moerdijk is seen, together with André Joyal, as one of the founders of algebraic set theory. [12] [13] In 1992 he wrote, together with Saunders Mac Lane, a book on topos theory that became the standard reference on the subject: Sheaves in geometry and logic. A first introduction to topos theory. [14]
In a topos mathematics can be done. In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the (n+1)-category nCat is a Grothendieck (n+1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting.