Left and right lifts along a displayed cat. #343
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TOTBWF
requested changes
Jan 16, 2024
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Thanks so much for the PR! There are a couple of minor nits, but these are all nice additions :) I particularly like |
TOTBWF
approved these changes
Jan 28, 2024
ncfavier
requested changes
Jan 28, 2024
| @@ -142,6 +144,70 @@ higher level of strictness than usual. | |||
| ni .natural _ _ _ = id-comm | |||
| ``` | |||
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| The distinguished projection `πᶠ` has a canonical choice of lifting. | |||
| Later, we will prove that for any functor $F$ valued in | |||
| $\cE$, $\pi^f$ has a canonical choice of lifting; however, this later | |||
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Suggested change
| $\cE$, $\pi^f$ has a canonical choice of lifting; however, this later | |
| $\cE$, $\pi^f \circ F$ has a canonical choice of lifting; however, this later |
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| module _ {o ℓ o' ℓ'} | ||
| {B : Precategory o ℓ} | ||
| (E : Displayed B o' ℓ') | ||
| where | ||
| open Cat.Reasoning B | ||
| open Displayed E | ||
| open Cat.Displayed.Reasoning E |
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| πᶠ-lifting : Lifting E (πᶠ E) | ||
| πᶠ-lifting .Lifting.F₀' (_ , a) = a | ||
| πᶠ-lifting .Lifting.F₁' f = preserves f |
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Suggested change
| πᶠ-lifting .Lifting.F₁' f = preserves f | |
| πᶠ-lifting .Lifting.F₁' f = f .preserves |
| πᶠ-lifting .Lifting.F-∘' f g = refl | ||
| ``` | ||
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| Let $F: \cC\times \cD\to \cB$ be a functor and $\cE\liesover\cB$ a displayed category. Let $F' : \cC\times \cD\to \cE$ be a lift of $F$ along $F'$. We show that $F'(c-,)$ is a lift of $F(c,-)$ for any $c$, similarly $F'(-,d)$ is a lift of $F(-,d)$ for any $d$. |
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| Let $F: \cC\times \cD\to \cB$ be a functor and $\cE\liesover\cB$ a displayed category. Let $F' : \cC\times \cD\to \cE$ be a lift of $F$ along $F'$. We show that $F'(c-,)$ is a lift of $F(c,-)$ for any $c$, similarly $F'(-,d)$ is a lift of $F(-,d)$ for any $d$. | |
| Let $F: \cC\times \cD\to \cB$ be a functor and $\cE\liesover\cB$ a displayed category. Let $F' : \cC\times \cD\to \cE$ be a lift of $F$. We show that $F'(c-,)$ is a lift of $F(c,-)$ for any $c$, similarly $F'(-,d)$ is a lift of $F(-,d)$ for any $d$. |
Also this should be reflowed to fit in the 72-character width limit.
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| module Bifunctor {o₁ ℓ₁ o₂ ℓ₂ o₃ ℓ₃ o₄ ℓ₄} | ||
| {B : Precategory o₁ ℓ₁} | ||
| (E : Displayed B o₂ ℓ₂) | ||
| (C : Precategory o₃ ℓ₃) | ||
| (D : Precategory o₄ ℓ₄) | ||
| (F : Functor (C ×ᶜ D) B) | ||
| (F' : Lifting E F) | ||
| where | ||
| private | ||
| module C = Precategory C | ||
| module D = Precategory D | ||
| module E = Displayed E | ||
| module F' = Lifting F' |
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| Left' d .Lifting.F₀' c = Lifting.F₀' F' (c , d) | ||
| Left' d .Lifting.F₁' f = Lifting.F₁' F' ( f , D.id ) | ||
| Left' d .Lifting.F-id' = Lifting.F-id' F' |
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Suggested change
| Left' d .Lifting.F₀' c = Lifting.F₀' F' (c , d) | |
| Left' d .Lifting.F₁' f = Lifting.F₁' F' ( f , D.id ) | |
| Left' d .Lifting.F-id' = Lifting.F-id' F' | |
| Left' d .Lifting.F₀' c = F' .Lifting.F₀' (c , d) | |
| Left' d .Lifting.F₁' f = F' .Lifting.F₁' (f , D.id) | |
| Left' d .Lifting.F-id' = F' .Lifting.F-id' |
I guess this isn't in the style guide but please use postfix projections everywhere.
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| ((F'.F₁' (f , D.id) E.∘' | ||
| F'.F₁' (g , D.id)) | ||
| E.≡[]˘⟨ F'.F-∘' (f , D.id) (g , D.id) ⟩ | ||
| (ap (F' .Lifting.F₁') | ||
| (λ i → C._∘_ f g , D.idl (D.id) i) | ||
| E.∙[] refl)) |
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The alignment is a bit awkward here, could you make this a proper reasoning chain ending with ∎?
Also it looks like the proofs of first∘first and second∘second could be simplified by omitting the outer sym and reversing the chain, which would also simplify these displayed proofs (but you don't have to do this).
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Description
Let$F: C\times D\to B$ be a functor and $p : E\to B$ a displayed category. Let $F' : C\times D\to E$ be a lift of $F$ along $F'$ . We show that $F'(c-,)$ is a lift of $F(c,-)$ for any $c$ , similarly $F'(-,d)$ is a lift of $F(-,d)$ for any $d$ .
The intended application of this theorem is to the theory of displayed bicategories.