[Merged by Bors] - feat(category_theory/sites/sheafification): The sheafification of a presheaf.

[adamtopaz]

We construct the sheafification of a presheaf P taking values in a concrete category D with enough (co)limits, where the forgetful functor preserves the appropriate (co)limits and reflects isomorphisms.

We follow the construction on the stacks project https://stacks.math.columbia.edu/tag/00W1

Note: There are two very long proofs here, so I added several comments explaining what's going on.

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• depends on: [Merged by Bors] - feat(category_theory/sites/plus): If P is a sheaf, then the map from P to P^+ is an isomorphism. #10297

[github-actions]

🎉 Great news! Looks like all the dependencies have been resolved:

• [Merged by Bors] - feat(category_theory/sites/plus): If P is a sheaf, then the map from P to P^+ is an isomorphism. #10297

💡 To add or remove a dependency please update this issue/PR description.

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[jcommelin]

This is looking really good! Too bad it times out.

[adamtopaz]

@jcommelin I think I fixed the timeout issue. This PR should be ready for another look.

[alreadydone]

Are you gonna develop some API for sheafification on topological spaces? Possibly in a future PR? Translation of results from sites to spaces happens in topology/sheaves/sheaf_condition/sites, e.g. in my recent PR.

[adamtopaz]

What sort of results do you want? I guess one should prove that the concrete sheafification for type-valued presheaves on a topological space should agree with this general version of sheafification, but that should follow easily from the universal property.

Whatever happens, it will be in a subsequent PR. This one is already more than big enough.

[alreadydone]

What sort of results do you want?

I was thinking of just the sheafification functor and the adjunction. That should be sufficient, e.g. to prove SheafedSpace has limits under assumptions of this PR. My current main project uses the implicit sheafification via local_predicate, so doesn't depend on this PR, and I can't think of other use cases for now.

[jcommelin]

Thanks 🎉

bors merge

[bors]

Pull request successfully merged into master.

Build succeeded:

• Build mathlib
• Lint mathlib
• Lint style
• Run tests