Light condensed objects

This file defines the category of light condensed objects in a category C, following the work of Clausen-Scholze (see https://www.youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO).

def LightCondensed (C : Type w) [CategoryTheory.Category.{v, w} C] : Type (max (max (max (u + 1) w) u) v)

LightCondensed.{u} C is the category of light condensed objects in a category C, which are defined as sheaves on LightProfinite.{u} with respect to the coherent Grothendieck topology.

Equations

• LightCondensed C = CategoryTheory.Sheaf (CategoryTheory.coherentTopology LightProfinite) C

instance instCategoryLightCondensed {C : Type w} [CategoryTheory.Category.{v, w} C] : CategoryTheory.Category.{max (u + 1) v, max (max (u + 1) w) v} (LightCondensed C)

Equations

• instCategoryLightCondensed = inferInstance

@[reducible, inline]

abbrev LightCondSet : Type (u + 1)

Light condensed sets. Because LightProfinite is an essentially small category, we don't need the same universe bump as in CondensedSet.

Equations

• LightCondSet = LightCondensed (Type ?u.3)

@[simp]

theorem LightCondensed.id_val {C : Type w} [CategoryTheory.Category.{v, w} C] (X : LightCondensed C) : (CategoryTheory.CategoryStruct.id X).val = CategoryTheory.CategoryStruct.id X.val

@[simp]

theorem LightCondensed.comp_val {C : Type w} [CategoryTheory.Category.{v, w} C] {X : LightCondensed C} {Y : LightCondensed C} {Z : LightCondensed C} (f : X ⟶ Y) (g : Y ⟶ Z) : (CategoryTheory.CategoryStruct.comp f g).val = CategoryTheory.CategoryStruct.comp f.val g.val

theorem LightCondensed.hom_ext_iff {C : Type w} [CategoryTheory.Category.{v, w} C] {X : LightCondensed C} {Y : LightCondensed C} {f : X ⟶ Y} {g : X ⟶ Y} : f = g ↔ ∀ (S : LightProfiniteᵒᵖ), f.val.app S = g.val.app S

theorem LightCondensed.hom_ext {C : Type w} [CategoryTheory.Category.{v, w} C] {X : LightCondensed C} {Y : LightCondensed C} (f : X ⟶ Y) (g : X ⟶ Y) (h : ∀ (S : LightProfiniteᵒᵖ), f.val.app S = g.val.app S) : f = g

@[simp]

theorem LightCondSet.hom_naturality_apply {X : LightCondSet} {Y : LightCondSet} (f : X ⟶ Y) {S : LightProfiniteᵒᵖ} {T : LightProfiniteᵒᵖ} (g : S ⟶ T) (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x)