[Merged by Bors] - refactor(topology+algebraic_geometry): prove and make use of equalities to simplify definitions #9972

alreadydone · 2021-10-26T03:48:29Z

Prove and make use of equalities whenever possible, even between functors (which is discouraged according to certain philosophy), to replace isomorphisms by equalities, to remove the need of transporting across isomorphisms in various definitions (using eq_to_hom directly), most notably: simplified definition of identity morphism for PresheafedSpace, simplified extensionality lemma for morphisms, simplified definition of composition and the global section functor (takes advantage of defeq and doesn't require proving any new equality).

Other small changes to mathlib:

Define pushforward functor of presheaves in topology/sheaves/presheaf.lean
Add new file functor.lean in topology/sheaves, proves the pushforward of a sheaf is a sheaf, and defines the pushforward functor of sheaves, with the expectation that pullbacks will be added later.
Make one of the arguments in various restricts implicit.
Change statement of to_open_comp_comap in structure_sheaf.lean to be more general (the same proof works!)

The new definitions result in simplified proofs, but apart from the main files opens.lean, presheaf.lean and presheafed_space.lean where proofs are optimized, I did only the minimum changes required to fix the broken proofs, and I expect there to be large room of improvement with the new definitions especially in the files changed in this PR. I also didn't remove the old lemmas and mostly just add new ones, so subsequent cleanup may be desired.

Zulip discussion thread: https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/extensionality.20for.20functors

src/algebraic_geometry/presheafed_space.lean

src/algebraic_geometry/stalks.lean

src/topology/category/Top/opens.lean

src/topology/sheaves/functors.lean

semorrison · 2021-10-27T04:27:31Z

bors merge

…es to simplify definitions (#9972) Prove and make use of equalities whenever possible, even between functors (which is discouraged according to certain philosophy), to replace isomorphisms by equalities, to remove the need of transporting across isomorphisms in various definitions (using `eq_to_hom` directly), most notably: [simplified definition of identity morphism for PresheafedSpace](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR89), [simplified extensionality lemma for morphisms](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR68), [simplified definition of composition](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR96) and [the global section functor](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR228) (takes advantage of defeq and doesn't require proving any new equality). Other small changes to mathlib: - Define pushforward functor of presheaves in topology/sheaves/presheaf.lean - Add new file functor.lean in topology/sheaves, proves the pushforward of a sheaf is a sheaf, and defines the pushforward functor of sheaves, with the expectation that pullbacks will be added later. - Make one of the arguments in various `restrict`s implicit. - Change statement of [`to_open_comp_comap`](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-54364470f443f847742b1c105e853afebc25da68faad63cc5a73db167bc85d06R973) in structure_sheaf.lean to be more general (the same proof works!) The new definitions result in simplified proofs, but apart from the main files opens.lean, presheaf.lean and presheafed_space.lean where proofs are optimized, I did only the minimum changes required to fix the broken proofs, and I expect there to be large room of improvement with the new definitions especially in the files changed in this PR. I also didn't remove the old lemmas and mostly just add new ones, so subsequent cleanup may be desired.

bors · 2021-10-27T08:40:33Z

Pull request successfully merged into master.

Build succeeded:

…es to simplify definitions (#9972) Prove and make use of equalities whenever possible, even between functors (which is discouraged according to certain philosophy), to replace isomorphisms by equalities, to remove the need of transporting across isomorphisms in various definitions (using `eq_to_hom` directly), most notably: [simplified definition of identity morphism for PresheafedSpace](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR89), [simplified extensionality lemma for morphisms](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR68), [simplified definition of composition](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR96) and [the global section functor](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-252fb30c3a3221e6472db5ba794344dfb423898696e70299653d95f635de06adR228) (takes advantage of defeq and doesn't require proving any new equality). Other small changes to mathlib: - Define pushforward functor of presheaves in topology/sheaves/presheaf.lean - Add new file functor.lean in topology/sheaves, proves the pushforward of a sheaf is a sheaf, and defines the pushforward functor of sheaves, with the expectation that pullbacks will be added later. - Make one of the arguments in various `restrict`s implicit. - Change statement of [`to_open_comp_comap`](https://github.com/leanprover-community/mathlib/compare/use_equality?expand=1#diff-54364470f443f847742b1c105e853afebc25da68faad63cc5a73db167bc85d06R973) in structure_sheaf.lean to be more general (the same proof works!) The new definitions result in simplified proofs, but apart from the main files opens.lean, presheaf.lean and presheafed_space.lean where proofs are optimized, I did only the minimum changes required to fix the broken proofs, and I expect there to be large room of improvement with the new definitions especially in the files changed in this PR. I also didn't remove the old lemmas and mostly just add new ones, so subsequent cleanup may be desired.

alreadydone added 2 commits October 25, 2021 14:02

prove and use equalities whenever possible

32e5731

fix broken proofs

e0123b1

alreadydone requested review from semorrison and justus-springer October 26, 2021 03:48

fix lint

931b733

alreadydone requested review from kbuzzard and kckennylau October 26, 2021 04:34

style/typo fixes

8ddb634