Light condensed R-modules

This files defines light condensed modules over a ring R.

Main results

• Light condensed R-modules form an abelian category.

• The forgetful functor from light condensed R-modules to light condensed sets has a left adjoint, sending a light condensed set to the corresponding free light condensed R-module.

@[reducible, inline]

abbrev LightCondMod (R : Type u) [Ring R] : Type (u + 1)

The category of condensed R-modules, defined as sheaves of R-modules over CompHaus with respect to the coherent Grothendieck topology.

noncomputable instance instAbelianLightCondMod (R : Type u) [Ring R] : CategoryTheory.Abelian (LightCondMod R)

def LightCondensed.forget (R : Type u) [Ring R] : CategoryTheory.Functor (LightCondMod R) LightCondSet

The forgetful functor from condensed R-modules to condensed sets.

noncomputable def LightCondensed.free (R : Type u) [Ring R] : CategoryTheory.Functor LightCondSet (LightCondMod R)

The left adjoint to the forgetful functor. The free condensed R-module on a condensed set.

noncomputable def LightCondensed.freeForgetAdjunction (R : Type u) [Ring R] : free R ⊣ forget R

The condensed version of the free-forgetful adjunction.

@[reducible, inline]

abbrev LightCondAb : Type 1

The category of condensed abelian groups, defined as sheaves of abelian groups over CompHaus with respect to the coherent Grothendieck topology.

theorem LightCondMod.hom_naturality_apply (R : Type u) [Ring R] {X Y : LightCondMod R} (f : X ⟶ Y) {S T : LightProfiniteᵒᵖ} (g : S ⟶ T) (x : ↑(X.val.obj S)) : (CategoryTheory.ConcreteCategory.hom (f.val.app T)) ((CategoryTheory.ConcreteCategory.hom (X.val.map g)) x) = (CategoryTheory.ConcreteCategory.hom (Y.val.map g)) ((CategoryTheory.ConcreteCategory.hom (f.val.app S)) x)