Condensed Objects

This file defines the category of condensed objects in a category C, following the work of Clausen-Scholze and Barwick-Haine.

Implementation Details

We use the coherent Grothendieck topology on CompHaus, and define condensed objects in C to be C-valued sheaves, with respect to this Grothendieck topology.

Note: Our definition more closely resembles "Pyknotic objects" in the sense of Barwick-Haine, as we do not impose cardinality bounds, and manage universes carefully instead.

References

• [barwickhaine2019]: Pyknotic objects, I. Basic notions, 2019.
• [scholze2019condensed]: Lectures on Condensed Mathematics, 2019.

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

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

Equations

• Condensed C = CategoryTheory.Sheaf (CategoryTheory.coherentTopology CompHaus) C

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

Equations

• instCategoryCondensed = inferInstance

@[reducible, inline]

abbrev CondensedSet : Type (u + 2)

Condensed sets (types) with the appropriate universe levels, i.e. Type (u+1)-valued sheaves on CompHaus.{u}.

Equations

• CondensedSet = Condensed (Type (?u.3 + 1))

@[simp]

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

@[simp]

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

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

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

@[simp]

theorem CondensedSet.hom_naturality_apply {X : CondensedSet} {Y : CondensedSet} (f : X ⟶ Y) {S : CompHausᵒᵖ} {T : CompHausᵒᵖ} (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)