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00:00:44 1 Grothendieck topoi (topoi in geometry)
00:01:29 1.1 Equivalent definitions
00:02:14 1.1.1 Giraud's axioms
00:02:59 1.1.2 Examples
00:03:44 1.1.2.1 Counterexamples
00:04:28 1.2 Geometric morphisms
00:05:13 1.2.1 Points of topoi
00:05:58 1.2.2 Essential geometric morphisms
00:06:43 1.3 Ringed topoi
00:07:28 1.4 Homotopy theory of topoi
00:08:13 2 Elementary topoi (topoi in logic)
00:08:35 2.1 Introduction
00:09:20 2.2 Formal definition
00:10:27 2.3 Logical functors
00:11:12 2.4 Explanation
00:12:19 2.5 Further examples
00:13:04 3 See also
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SUMMARY
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In mathematics, a topos (UK: , US: ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). 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.