Documentation

Mathlib.Condensed.Basic

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 #

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
Instances For
    Equations
    • instCategoryCondensed = let_fun this := inferInstance; this
    @[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
    Instances For
      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)