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.

Instances For
    • 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}.

    Instances For