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prove that the yoneda embedding preserves limits #1060

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semorrison opened this issue May 20, 2019 · 3 comments
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prove that the yoneda embedding preserves limits #1060

semorrison opened this issue May 20, 2019 · 3 comments
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feature-request This issue is a feature request, either for mathematics, tactics, or CI help-wanted The author needs attention to resolve issues medium

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@semorrison
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semorrison commented May 20, 2019

We have everything we need for the statement:

import category_theory.yoneda
import category_theory.limits.preserves

open category_theory
open category_theory.limits

universes u

variables {C : Type u} [𝒞 : small_category C]

instance yoneda_preserves_limits : preserves_limits (@yoneda C 𝒞) := sorry

and it should be straightforward to give the construction.

This might be a good exercise for someone wanting to learn how to use the category theory library.

@semorrison semorrison added help-wanted The author needs attention to resolve issues medium labels May 20, 2019
@semorrison
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This should probably go in src/category_theory/limits/yoneda.lean.

@semorrison
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semorrison commented May 20, 2019

A harder exercise is to explain the universe annotations .{u u+1 u+2} on preserves_limits, which I really better write a tutorial about...

(Actually the universe annotations I wrote were completely unnecessary.)

@semorrison
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(I noticed that map_cone_X is a missing simp lemma in category_theory/limits/cones.lean, which should probably be added as part of doing this.)

@bryangingechen bryangingechen added the feature-request This issue is a feature request, either for mathematics, tactics, or CI label Mar 30, 2020
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feature-request This issue is a feature request, either for mathematics, tactics, or CI help-wanted The author needs attention to resolve issues medium
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