[Merged by Bors] - feat(category_theory): make defining groupoids easier #2360
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semorrison
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The point of this PR is to lower the burden of proof in showing that a category is a groupoid. Rather than constructing well-defined two-sided inverses everywhere, with `groupoid.of_trunc_split_mono` you'll only need to show that every morphism has some retraction. This makes defining the free groupoid painless. There the retractions are defined by recursion on a quotient, so this saves the work of showing that all the retractions agree. I used `trunc` instead of `nonempty` to avoid choice / noncomputability. I don't understand why the @'s are needed: it seems Lean doesn't know what category structure C has without specifying it?
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…munity#2360) The point of this PR is to lower the burden of proof in showing that a category is a groupoid. Rather than constructing well-defined two-sided inverses everywhere, with `groupoid.of_trunc_split_mono` you'll only need to show that every morphism has some retraction. This makes defining the free groupoid painless. There the retractions are defined by recursion on a quotient, so this saves the work of showing that all the retractions agree. I used `trunc` instead of `nonempty` to avoid choice / noncomputability. I don't understand why the @'s are needed: it seems Lean doesn't know what category structure C has without specifying it?
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May 16, 2020
…munity#2360) The point of this PR is to lower the burden of proof in showing that a category is a groupoid. Rather than constructing well-defined two-sided inverses everywhere, with `groupoid.of_trunc_split_mono` you'll only need to show that every morphism has some retraction. This makes defining the free groupoid painless. There the retractions are defined by recursion on a quotient, so this saves the work of showing that all the retractions agree. I used `trunc` instead of `nonempty` to avoid choice / noncomputability. I don't understand why the @'s are needed: it seems Lean doesn't know what category structure C has without specifying it?
cipher1024
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Mar 15, 2022
…munity#2360) The point of this PR is to lower the burden of proof in showing that a category is a groupoid. Rather than constructing well-defined two-sided inverses everywhere, with `groupoid.of_trunc_split_mono` you'll only need to show that every morphism has some retraction. This makes defining the free groupoid painless. There the retractions are defined by recursion on a quotient, so this saves the work of showing that all the retractions agree. I used `trunc` instead of `nonempty` to avoid choice / noncomputability. I don't understand why the @'s are needed: it seems Lean doesn't know what category structure C has without specifying it?
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The point of this PR is to lower the burden of proof in showing that a category is a groupoid. Rather than constructing well-defined two-sided inverses everywhere, with
groupoid.of_trunc_split_monoyou'll only need to show that every morphism has some retraction. This makes defining the free groupoid painless. There the retractions are defined by recursion on a quotient, so this saves the work of showing that all the retractions agree.I used
truncinstead ofnonemptyto avoid choice / noncomputability.I don't understand why the @'s are needed: it seems Lean doesn't know what category structure C has without specifying it?
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