Open EgbertRijke opened 9 months ago
I've already defined the closed modality. It's in orthogonal-factorization-systems.closed-modalities
.
- [X] Prepare the file
pullback-squares for real-world applications, and possibly rename it to
cartesian squares`. Alternatively, introduce a file for cartesian morphisms of arrows.
Cartesian morphisms of arrows were added in #979.
This issue seems to be missing any mention of orthogonal maps or fiberwise orthogonal maps. Are they not needed?
When you write
- [ ] Define base change of morphisms of arrows.
Are you looking for a file about the following construct?
base-change-arrow :
{l1 l2 : Level} (l3 l4 : Level) {A : UU l1} {B : UU l2} → (A → B) →
UU (l1 ⊔ l2 ⊔ lsuc l3 ⊔ lsuc l4)
base-change-arrow l3 l4 f =
Σ (UU l3) (λ C → Σ (UU l4) (λ D → Σ (C → D) (λ g → cartesian-hom-arrow g f)))
Otherwise, just using cartesian morphisms of arrows seems to be working fine with me for now. Although I would probably say the same about cartesian morphisms of arrows vs. pullback cones if we didn't already have cartesian morphisms of arrows.
EDIT: Oh, wait, you want a base change of morphisms of arrows.
The cd-excision project
Project description
A complete cd-structure consists of a class of distinguished squares of maps, i.e., distinguished morphisms in the arrow category, that is stable under base change. A complete cd-structure is said to be regular if the following three conditions hold:
A type
F
is said to be a sheaf with respect to a cd-structure ifF
is right orthogonal toinclusion-im (cogap d)
for any distinguished squared
.A type
F
is said to satisfy excision with respect to a cd-structure ifF
is right orthogonal tocogap d
for any distinguished squared
.Goal of the project: Show that
F
is a sheaf with respect to a regular cd-structure if and only if it satisfies excision.Current participants: Reid Barton, Egbert Rijke, Fredrik Bakke.
Outline and tasks
Descent
The closed modality
The arrow category
pullback-squares
for real-world applications, and possibly rename it tocartesian squares
. Alternatively, introduce a file for cartesian morphisms of arrows.Wärn's first lemma
Orthogonality
Π
, left orthogonal maps underΣ
f
-local types (what is called fiberwise orthogonal in the notes)Cd-structures
Resources