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).

@[reducible, inline]

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

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

@[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.1)

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

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