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In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic.
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In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to deformation theory in algebraic geometry and the mathematical foundation of quantum mechanics. In the latter subject, a Bohr topos is a ringed topos that plays the role of a quantum phase space.
References
- ↑ Grothendieck & Verdier 1972
- ↑ Lawvere 1975
- ↑ "Locales as geometric objects". MathOverflow. Retrieved 2024-07-22.
- ↑ Johnstone 1985
- ↑ "On a Topological Topos at The n-Category Café". golem.ph.utexas.edu.
- 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.
- Lawvere, Categories of spaces may not be generalized spaces as exemplified by directed graphs
- 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.
- Grothendieck, A.; Verdier, J. L. (1972), "Topos", Théorie des Topos et Cohomologie Etale des Schémas, Lecture Notes in Mathematics, vol. 269, Springer, doi:10.1007/BFb0081555
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