7 Sites and Sheaves

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Section 7.1: Introduction
Section 7.2: Presheaves
- Definition 7.2.1
  - Equation 7.2.1.1
- Definition 7.2.2
- Remark 7.2.3
Section 7.3: Injective and surjective maps of presheaves
- Definition 7.3.1
- Lemma 7.3.2
- Definition 7.3.3
- Lemma 7.3.4
- Definition 7.3.5
Section 7.4: Limits and colimits of presheaves
Section 7.5: Functoriality of categories of presheaves
- Equation 7.5.0.1
- Lemma 7.5.1
- Lemma 7.5.2
- Lemma 7.5.3
- Lemma 7.5.4
- Remark 7.5.5
- Lemma 7.5.6
Section 7.6: Sites
- Definition 7.6.1
- Definition 7.6.2
- Remark 7.6.3
- Example 7.6.4
- Example 7.6.5
- Example 7.6.6
Section 7.7: Sheaves
- Definition 7.7.1
  - Equation 7.7.1.1
- Remark 7.7.2
- Example 7.7.3
- Example 7.7.4
- Definition 7.7.5
- Definition 7.7.6
Section 7.8: Families of morphisms with fixed target
- Definition 7.8.1
- Definition 7.8.2
- Lemma 7.8.3
- Lemma 7.8.4
- Lemma 7.8.5
- Lemma 7.8.6
- Lemma 7.8.7
- Lemma 7.8.8
Section 7.9: The example of G-sets
- Proposition 7.9.1
Section 7.10: Sheafification
- Lemma 7.10.1 slogan
- Example 7.10.2
- Lemma 7.10.3
- Lemma 7.10.4
- Lemma 7.10.5
- Lemma 7.10.6
- Remark 7.10.7
- Lemma 7.10.8
- Definition 7.10.9
- Theorem 7.10.10
- Definition 7.10.11
- Proposition 7.10.12
- Lemma 7.10.13 slogan
- Lemma 7.10.14
- Lemma 7.10.15
- Lemma 7.10.16
Section 7.11: Injective and surjective maps of sheaves
- Definition 7.11.1
- Lemma 7.11.2
- Lemma 7.11.3
Section 7.12: Representable sheaves
- Definition 7.12.1
- Definition 7.12.2
- Definition 7.12.3
  - Equation 7.12.3.1
- Lemma 7.12.4 slogan
- Lemma 7.12.5
- Lemma 7.12.6
Section 7.13: Continuous functors
- Definition 7.13.1
- Lemma 7.13.2
- Lemma 7.13.3
- Lemma 7.13.4
- Lemma 7.13.5
- Remark 7.13.6
Section 7.14: Morphisms of sites
- Definition 7.14.1
- Example 7.14.2
- Example 7.14.3
- Lemma 7.14.4 slogan
- Definition 7.14.5
- Lemma 7.14.6
- Proposition 7.14.7
- Remark 7.14.8
- Remark 7.14.9
- Lemma 7.14.10
Section 7.15: Topoi
- Definition 7.15.1: Topoi
- Lemma 7.15.2
- Remark 7.15.3
  - Equation 7.15.3.1
- Remark 7.15.4
- Remark 7.15.5
Section 7.16: G-sets and morphisms
Section 7.17: Quasi-compact objects and colimits
- Definition 7.17.1
- Lemma 7.17.2
- Lemma 7.17.3
- Definition 7.17.4
- Lemma 7.17.5 slogan
- Lemma 7.17.6 slogan
- Lemma 7.17.7
- Lemma 7.17.8
- Remark 7.17.9
- Lemma 7.17.10
Section 7.18: Colimits of sites
- Situation 7.18.1
- Lemma 7.18.2
- Lemma 7.18.3
  - Equation 7.18.3.1
- Lemma 7.18.4
- Lemma 7.18.5
Section 7.19: More functoriality of presheaves
- Lemma 7.19.1
- Lemma 7.19.2
- Lemma 7.19.3
Section 7.20: Cocontinuous functors
- Definition 7.20.1
- Lemma 7.20.2
  - Equation 7.20.2.1
- Lemma 7.20.3
- Lemma 7.20.4
- Remark 7.20.5
Section 7.21: Cocontinuous functors and morphisms of topoi
- Lemma 7.21.1
- Lemma 7.21.2 slogan
- Example 7.21.3
- Example 7.21.4
- Lemma 7.21.5 slogan
- Lemma 7.21.6
- Lemma 7.21.7
- Lemma 7.21.8
- Example 7.21.9
Section 7.22: Cocontinuous functors which have a right adjoint
- Lemma 7.22.1
- Lemma 7.22.2
- Example 7.22.3
- Lemma 7.22.4
- Example 7.22.5
Section 7.23: Cocontinuous functors which have a left adjoint
- Lemma 7.23.1
Section 7.24: Existence of lower shriek
- Lemma 7.24.1
Section 7.25: Localization
- Definition 7.25.1
- Lemma 7.25.2
- Lemma 7.25.3
- Lemma 7.25.4
- Lemma 7.25.5
- Lemma 7.25.6
- Lemma 7.25.7
- Lemma 7.25.8
  - Equation 7.25.8.1
- Lemma 7.25.9
- Remark 7.25.10
- Remark 7.25.11
Section 7.26: Glueing sheaves
- Lemma 7.26.1 slogan
- Lemma 7.26.2 slogan
- Lemma 7.26.3
- Lemma 7.26.4
- Lemma 7.26.5
- Lemma 7.26.6
- Remark 7.26.7
Section 7.27: More localization
- Lemma 7.27.1
- Lemma 7.27.2
- Lemma 7.27.3
- Lemma 7.27.4
- Lemma 7.27.5
Section 7.28: Localization and morphisms
- Lemma 7.28.1
- Lemma 7.28.2
- Lemma 7.28.3
- Lemma 7.28.4
- Lemma 7.28.5
- Lemma 7.28.6
- Lemma 7.28.7
Section 7.29: Morphisms of topoi
- Lemma 7.29.1
- Definition 7.29.2
- Lemma 7.29.3
- Lemma 7.29.4
- Lemma 7.29.5
- Lemma 7.29.6 reference
- Remark 7.29.7
- Remark 7.29.8
Section 7.30: Localization of topoi
- Lemma 7.30.1
- Lemma 7.30.2
- Lemma 7.30.3
- Definition 7.30.4
- Lemma 7.30.5
- Lemma 7.30.6
- Lemma 7.30.7
Section 7.31: Localization and morphisms of topoi
- Lemma 7.31.1
- Lemma 7.31.2
- Lemma 7.31.3
- Lemma 7.31.4
Section 7.32: Points
- Definition 7.32.1
  - Equation 7.32.1.1
- Definition 7.32.2
- Lemma 7.32.3
  - Equation 7.32.3.1
- Lemma 7.32.4
- Lemma 7.32.5
- Definition 7.32.6
- Lemma 7.32.7
- Lemma 7.32.8
- Lemma 7.32.9
- Lemma 7.32.10
Section 7.33: Constructing points
- Lemma 7.33.1
- Lemma 7.33.2
- Proposition 7.33.3
- Remark 7.33.4
- Example 7.33.5
- Example 7.33.6
- Example 7.33.7
- Example 7.33.8
Section 7.34: Points and morphisms of topoi
- Lemma 7.34.1
- Lemma 7.34.2 slogan
- Lemma 7.34.3
Section 7.35: Localization and points
- Lemma 7.35.1
- Lemma 7.35.2
- Lemma 7.35.3
- Remark 7.35.4
Section 7.36: 2-morphisms of topoi
- Definition 7.36.1
Section 7.37: Morphisms between points
- Lemma 7.37.1
- Definition 7.37.2
Section 7.38: Sites with enough points
- Definition 7.38.1
- Lemma 7.38.2
- Lemma 7.38.3
- Lemma 7.38.4
- Lemma 7.38.5
- Lemma 7.38.6
Section 7.39: Criterion for existence of points
- Lemma 7.39.1
- Lemma 7.39.2
- Proposition 7.39.3 reference
- Lemma 7.39.4
Section 7.40: Weakly contractible objects
- Lemma 7.40.1
- Definition 7.40.2
Section 7.41: Exactness properties of pushforward
- Lemma 7.41.1
- Lemma 7.41.2
- Example 7.41.3
- Lemma 7.41.4
- Example 7.41.5
Section 7.42: Almost cocontinuous functors
- Definition 7.42.1
- Lemma 7.42.2
- Definition 7.42.3
- Lemma 7.42.4
- Lemma 7.42.5
- Lemma 7.42.6
Section 7.43: Subtopoi
- Definition 7.43.1
- Definition 7.43.2
- Lemma 7.43.3
- Definition 7.43.4
- Lemma 7.43.5
- Definition 7.43.6
- Definition 7.43.7
- Lemma 7.43.8
Section 7.44: Sheaves of algebraic structures
- Definition 7.44.1
- Lemma 7.44.2
- Proposition 7.44.3 slogan
- Remark 7.44.4
Section 7.45: Pullback maps
- Definition 7.45.1
- Lemma 7.45.2
- Remark 7.45.3
- Remark 7.45.4
Section 7.46: Comparison with SGA4
Section 7.47: Topologies
- Definition 7.47.1
  - Equation 7.47.1.1
- Lemma 7.47.2
- Definition 7.47.3
- Definition 7.47.4
- Lemma 7.47.5
- Definition 7.47.6
- Lemma 7.47.7
- Definition 7.47.8
- Lemma 7.47.9
- Definition 7.47.10
- Lemma 7.47.11
- Definition 7.47.12
Section 7.48: The topology defined by a site
- Lemma 7.48.1
- Definition 7.48.2
- Remark 7.48.3
- Remark 7.48.4
Section 7.49: Sheafification in a topology
- Lemma 7.49.1
- Definition 7.49.2
- Theorem 7.49.3
  - Eqnarray 7.49.3.1
- Definition 7.49.4
- Proposition 7.49.5
Section 7.50: Topologies and sheaves
- Lemma 7.50.1
- Theorem 7.50.2
- Lemma 7.50.3
Section 7.51: Topologies and continuous functors
Section 7.52: Points and topologies
- Definition 7.52.1