[Merged by Bors] - feat(category_theory/monad/*): Add category of bundled (co)monads; prove equivalence of monads with monoid objects #3762
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1. Add hcomp_id_app / id_hcomp_app 2. Add Monad.hext 3. Add Mon_.hext 4. Add file equiv_mon.lean
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1. Clean up proof of hcomp_id_app 2. Add has_coe_to_sort instance for Cat 3. Add note about bundled (co)monad category
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(By the way, since this was still labelled as |
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Sorry about this. I'll make sure to replace |
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Other than the two new comments above, LGTM. bors d+ |
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Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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…ove equivalence of monads with monoid objects (#3762) This PR constructs bundled monads, and proves the "usual" equivalence between the category of monads and the category of monoid objects in the endomorphism category. It also includes a definition of morphisms of unbundled monads, and proves some necessary small lemmas in the following two files: 1. `category_theory.functor_category` 2. `category_theory.monoidal.internal` Given any isomorphism in `Cat`, we construct a corresponding equivalence of categories in `Cat.equiv_of_iso`. Co-authored-by: Adam Topaz <adamtopaz@users.noreply.github.com>
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This PR constructs bundled monads, and proves the "usual" equivalence between the category of monads and the category of monoid objects in the endomorphism category.
It also includes a definition of morphisms of unbundled monads, and proves some necessary small lemmas in the following two files:
category_theory.functor_categorycategory_theory.monoidal.internalGiven any isomorphism in
Cat, we construct a corresponding equivalence of categories inCat.equiv_of_iso.