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Commit 2021-10-25 22:55 c363ad63

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feat(category_theory/sites/*): Cover preserving functors (#9691) Split from #9650

Defined cover_preserving functors as functors that push covering sieves to covering sieves.
Defined compatible_preserving functors as functors that push compatible families to compatible families.
Proved that functors that are both cover_preserving and compatible_preserving pulls sheaves to sheaves.

Estimated changes

Created src/category_theory/sites/cover_preserving.lean

added theorem category_theory.compatible_preserving.apply_map

added structure category_theory.compatible_preserving

added theorem category_theory.cover_preserving.comp

added structure category_theory.cover_preserving

added theorem category_theory.id_cover_preserving

added theorem category_theory.presieve.family_of_elements.compatible.functor_pushforward

added theorem category_theory.pullback_is_sheaf_of_cover_preserving

added def category_theory.pullback_sheaf

added def category_theory.sites.pullback

Modified src/category_theory/sites/sheaf.lean

added def category_theory.sheaf_over

Modified src/category_theory/sites/sheaf_of_types.lean

added theorem category_theory.presieve.family_of_elements.comp_of_compatible

added def category_theory.presieve.family_of_elements.functor_pushforward

Modified src/category_theory/sites/sieves.lean

added structure category_theory.presieve.functor_pushforward_structure