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Subobject classifiers #802

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Subobject classifiers #802

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738a7eb
Add pullbacks in large precategories
maybemabeline Sep 28, 2023
753b8ed
Add missing file
maybemabeline Sep 28, 2023
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Update src/category-theory/pullbacks-in-large-precategories.lagda.md
maybemabeline Sep 28, 2023
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Deindex variables and unnest module
maybemabeline Sep 29, 2023
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Reformat path algebra
maybemabeline Sep 29, 2023
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2 changes: 2 additions & 0 deletions src/category-theory.lagda.md
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Original file line number Diff line number Diff line change
Expand Up @@ -112,13 +112,15 @@ open import category-theory.pregroupoids public
open import category-theory.presheaf-categories public
open import category-theory.products-in-precategories public
open import category-theory.products-of-precategories public
open import category-theory.pullbacks-in-large-precategories public
open import category-theory.pullbacks-in-precategories public
open import category-theory.representable-functors-categories public
open import category-theory.representable-functors-large-precategories public
open import category-theory.representable-functors-precategories public
open import category-theory.representing-arrow-category public
open import category-theory.sieves-in-categories public
open import category-theory.slice-precategories public
open import category-theory.subobject-classifier-in-large-precategories public
open import category-theory.subprecategories public
open import category-theory.terminal-objects-precategories public
open import category-theory.yoneda-lemma-categories public
Expand Down
62 changes: 44 additions & 18 deletions src/category-theory/monomorphisms-in-large-precategories.lagda.md
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Expand Up @@ -11,6 +11,7 @@ open import category-theory.isomorphisms-in-large-precategories
open import category-theory.large-precategories

open import foundation.action-on-identifications-functions
open import foundation.dependent-pair-types
open import foundation.embeddings
open import foundation.equivalences
open import foundation.identity-types
Expand Down Expand Up @@ -51,6 +52,31 @@ module _
is-prop-type-Prop is-mono-prop-Large-Precategory
```

## The type of monomorphisms in a large precategory

```agda
module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level} (l3 : Level)
(X : obj-Large-Precategory C l1) (Y : obj-Large-Precategory C l2)
where

mono-Large-Precategory : UU (α l3 ⊔ β l3 l1 ⊔ β l3 l2 ⊔ β l1 l2)
mono-Large-Precategory =
Σ
( hom-Large-Precategory C X Y)
( is-mono-Large-Precategory C l3 X Y)

hom-mono-Large-Precategory :
(f : mono-Large-Precategory) → hom-Large-Precategory C X Y
hom-mono-Large-Precategory f = pr1 f

is-mono-mono-Large-Precategory :
(f : mono-Large-Precategory) →
is-mono-Large-Precategory C l3 X Y (hom-mono-Large-Precategory f)
is-mono-mono-Large-Precategory f = pr2 f
```

## Properties

### Isomorphisms are monomorphisms
Expand All @@ -70,34 +96,34 @@ module _
( λ P →
( inv
( left-unit-law-comp-hom-Large-Precategory C g)) ∙
( ( ap
( ap
( λ h' → comp-hom-Large-Precategory C h' g)
( inv (is-retraction-hom-inv-iso-Large-Precategory C f))) ∙
( ( associative-comp-hom-Large-Precategory C
( associative-comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory C f)
( hom-iso-Large-Precategory C f)
( g)) ∙
( ap
( comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory C f))
( P)) ∙
( inv
( associative-comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory C f)
( hom-iso-Large-Precategory C f)
( g)) ∙
( ( ap
( comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory C f))
( P)) ∙
( ( inv
( associative-comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory C f)
( hom-iso-Large-Precategory C f)
( h))) ∙
( ( ap
( λ h' → comp-hom-Large-Precategory C h' h)
( is-retraction-hom-inv-iso-Large-Precategory C f)) ∙
( left-unit-law-comp-hom-Large-Precategory C h)))))))
( h))) ∙
( ap
( λ h' → comp-hom-Large-Precategory C h' h)
( is-retraction-hom-inv-iso-Large-Precategory C f)) ∙
( left-unit-law-comp-hom-Large-Precategory C h))
( λ p →
eq-is-prop
( eq-is-prop
( is-set-hom-Large-Precategory C Z Y
( comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C f)
( g))
( comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C f)
( h))))
( h)))))
( λ p → eq-is-prop (is-set-hom-Large-Precategory C Z X g h))
```
186 changes: 186 additions & 0 deletions src/category-theory/pullbacks-in-large-precategories.lagda.md
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@@ -0,0 +1,186 @@
# Pullbacks in large precategories

```agda
module category-theory.pullbacks-in-large-precategories where
```

<details><summary>Imports</summary>

```agda
open import category-theory.large-precategories

open import foundation.action-on-identifications-functions
open import foundation.cartesian-product-types
open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.unique-existence
open import foundation.universe-levels
```

</details>

## Idea

A pullback of two morphisms `f : hom y x` and `g : hom z x` in a category `C`
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@fredrik-bakke fredrik-bakke Sep 28, 2023 •
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Suggested change
A pullback of two morphisms `f : hom y x` and `g : hom z x` in a category `C`
A **pullback** of two morphisms `f : hom y x` and `g : hom z x` in a [category](category-theory.categories.md) `C`

We write terms in boldface when defining them, and add links when mentioning terms for the first time in a file that are defined elsewhere.

consists of:

- an object `w`
- morphisms `p : hom w y` and `q : hom w z` such that
- `f ∘ p = g ∘ q` together with the universal property that for every object
`w'` and pair of morphisms `p' : hom w' y` and `q' : hom w' z` such that
`f ∘ p' = g ∘ q'` there exists a unique morphism `h : hom w' w` such that
- `p ∘ h = p'`
- `q ∘ h = q'`.

We say that `C` has all pullbacks if there is a choice of a pullback for each
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Suggested change
We say that `C` has all pullbacks if there is a choice of a pullback for each
We say that `C` **has all pullbacks** if there is a choice of a pullback for each

object `x` and pair of morphisms into `x` in `C`.

## Definition

```agda
module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 l4 : Level}
where

is-pullback-Large-Precategory :
(x : obj-Large-Precategory C l1)
(y : obj-Large-Precategory C l2)
(z : obj-Large-Precategory C l3)
(f : hom-Large-Precategory C y x)
(g : hom-Large-Precategory C z x) →
(w : obj-Large-Precategory C l4)
(p : hom-Large-Precategory C w y) →
(q : hom-Large-Precategory C w z) →
comp-hom-Large-Precategory C f p ＝ comp-hom-Large-Precategory C g q →
UU (α l1 ⊔ β l1 l1 ⊔ β l1 l2 ⊔ β l1 l3 ⊔ β l1 l4)
is-pullback-Large-Precategory x y z f g w p q _ =
(w' : obj-Large-Precategory C l1) →
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w' must be able to live at any universe level

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@fredrik-bakke fredrik-bakke Sep 28, 2023 •
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this should mirror the definition of universal-property-pullback in foundation-core.universal-property-pullbacks

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@fredrik-bakke fredrik-bakke Sep 28, 2023 •
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Just to confirm your question earlier, using the above reference, x, y, z, and w should indeed all be able to have different universe levels.

(p' : hom-Large-Precategory C w' y) →
(q' : hom-Large-Precategory C w' z) →
comp-hom-Large-Precategory C f p' ＝ comp-hom-Large-Precategory C g q' →
∃!
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Instead of using unique existence, we prefer to define universal properties in terms of certain evaluation maps being equivalences. Here, that unfolds to requiring that the map that sends a morphism w' -> w, where w is the pullback, to the cone constructed by postcomposing with f and g being an iso in the (large) span precategory with fixed codomains y and z. Maybe Egbert has a different opinion on this, but that is what I would say this definition should be at the moment.

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Maybe to expand a little on the above remark. Notice how this definition makes itself more readily available for use with univalence down the line.

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I'm not sure I understand how this would be an isomorphism in the category of spans. Do you mean that postcomposition with f and g forms equivalence between the slice category C/w and the category of spans with codomains y and z?

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Ah, yes, my bad, you're right!

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Also, I shouldn't have said span category. It's the cone category over the cospan of f and g. And the cone is constructed by postcomposing with p and q, not f and g.

( hom-Large-Precategory C w' w)
( λ h →
( comp-hom-Large-Precategory C p h ＝ p') ×
( comp-hom-Large-Precategory C q h ＝ q'))

pullback-Large-Precategory :
(x : obj-Large-Precategory C l1) →
(y : obj-Large-Precategory C l2) →
(z : obj-Large-Precategory C l3) →
hom-Large-Precategory C y x →
hom-Large-Precategory C z x →
UU (α l1 ⊔ α l4 ⊔ β l1 l1 ⊔ β l1 l2 ⊔ β l1 l3 ⊔
β l1 l4 ⊔ β l4 l1 ⊔ β l4 l2 ⊔ β l4 l3)
pullback-Large-Precategory x y z f g =
Σ (obj-Large-Precategory C l4) λ w →
Σ (hom-Large-Precategory C w y) λ p →
Σ (hom-Large-Precategory C w z) λ q →
Σ (comp-hom-Large-Precategory C f p
＝ comp-hom-Large-Precategory C g q) λ α →
is-pullback-Large-Precategory x y z f g w p q α

has-all-pullbacks-Large-Precategory :
UU (α l1 ⊔ α l2 ⊔ α l3 ⊔ α l4 ⊔ β l1 l1 ⊔ β l1 l2 ⊔ β l1 l3 ⊔
β l1 l4 ⊔ β l2 l1 ⊔ β l3 l1 ⊔ β l4 l1 ⊔ β l4 l2 ⊔ β l4 l3)
has-all-pullbacks-Large-Precategory =
(x : obj-Large-Precategory C l1) →
(y : obj-Large-Precategory C l2) →
(z : obj-Large-Precategory C l3) →
(f : hom-Large-Precategory C y x) →
(g : hom-Large-Precategory C z x) →
pullback-Large-Precategory x y z f g

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
(t : has-all-pullbacks-Large-Precategory C)
(x y z : obj-Large-Precategory C l1)
(f : hom-Large-Precategory C y x)
(g : hom-Large-Precategory C z x)
where

object-pullback-Large-Precategory : obj-Large-Precategory C l2
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obj-pullback-...

object-pullback-Large-Precategory = pr1 (t x y z f g)

pr1-pullback-Large-Precategory :
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Usually, I would want to say that you should take a look at foundation.pullbacks for how to name these. Sadly, that file is in need of refactoring as well. But you can have a look at foundation.cones-over-cospans for how to name these.

hom-Large-Precategory C object-pullback-Large-Precategory y
pr1-pullback-Large-Precategory = pr1 (pr2 (t x y z f g))

pr2-pullback-Large-Precategory :
hom-Large-Precategory C object-pullback-Large-Precategory z
pr2-pullback-Large-Precategory = pr1 (pr2 (pr2 (t x y z f g)))

pullback-square-Large-Precategory-comm :
comp-hom-Large-Precategory C f pr1-pullback-Large-Precategory ＝
comp-hom-Large-Precategory C g pr2-pullback-Large-Precategory
pullback-square-Large-Precategory-comm = pr1 (pr2 (pr2 (pr2 (t x y z f g))))

module _
(w' : obj-Large-Precategory C l1)
(p' : hom-Large-Precategory C w' y)
(q' : hom-Large-Precategory C w' z)
(ε : comp-hom-Large-Precategory C f p'
＝ comp-hom-Large-Precategory C g q')
where

morphism-into-pullback-Large-Precategory :
hom-Large-Precategory C w' object-pullback-Large-Precategory
morphism-into-pullback-Large-Precategory =
pr1 (pr1 (pr2 (pr2 (pr2 (pr2 (t x y z f g)))) w' p' q' ε))

morphism-into-pullback-comm-pr1-Large-Precategory :
comp-hom-Large-Precategory C
pr1-pullback-Large-Precategory
morphism-into-pullback-Large-Precategory ＝
p'
morphism-into-pullback-comm-pr1-Large-Precategory =
pr1 (pr2 (pr1 (pr2 (pr2 (pr2 (pr2 (t x y z f g)))) w' p' q' ε)))

morphism-into-pullback-comm-pr2-Large-Precategory :
comp-hom-Large-Precategory C
pr2-pullback-Large-Precategory
morphism-into-pullback-Large-Precategory ＝
q'
morphism-into-pullback-comm-pr2-Large-Precategory =
pr2 (pr2 (pr1 (pr2 (pr2 (pr2 (pr2 (t x y z f g)))) w' p' q' ε)))

is-unique-morphism-into-pullback-Large-Precategory :
(h' : hom-Large-Precategory C w' object-pullback-Large-Precategory) →
comp-hom-Large-Precategory C pr1-pullback-Large-Precategory h' ＝ p' →
comp-hom-Large-Precategory C pr2-pullback-Large-Precategory h' ＝ q' →
morphism-into-pullback-Large-Precategory ＝ h'
is-unique-morphism-into-pullback-Large-Precategory h' η θ =
ap
( pr1)
( pr2 (pr2 (pr2 (pr2 (pr2 (t x y z f g)))) w' p' q' ε) (h' , η , θ))

module _
{α : Level → Level} {β : Level → Level → Level}
{l1 l2 l3 l4 : Level} (C : Large-Precategory α β)
(x : obj-Large-Precategory C l1)
(y : obj-Large-Precategory C l2)
(z : obj-Large-Precategory C l3)
(f : hom-Large-Precategory C y x)
(g : hom-Large-Precategory C z x)
(w : obj-Large-Precategory C l4)
Comment on lines +164 to +169
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You probably want x y z and w to be implicit arguments here, unless Agda is unable to infer them generally

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Good point! I'll change this in the other file too.

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Also, when they are implicit, I would place w right after z instead

(p : hom-Large-Precategory C w y)
(q : hom-Large-Precategory C w z)
(ε : comp-hom-Large-Precategory C f p ＝ comp-hom-Large-Precategory C g q)
where

is-prop-is-pullback-Large-Precategory :
is-prop (is-pullback-Large-Precategory C x y z f g w p q ε)
is-prop-is-pullback-Large-Precategory =
is-prop-Π³ (λ w' p' q' → is-prop-function-type is-property-is-contr)

is-pullback-prop-Large-Precategory :
Prop (α l1 ⊔ β l1 l1 ⊔ β l1 l2 ⊔ β l1 l3 ⊔ β l1 l4)
pr1 is-pullback-prop-Large-Precategory =
is-pullback-Large-Precategory C x y z f g w p q ε
pr2 is-pullback-prop-Large-Precategory =
is-prop-is-pullback-Large-Precategory
```
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