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Mathlib.Algebra.Category.ModuleCat.FilteredColimits

The forgetful functor from R-modules preserves filtered colimits. #

Forgetful functors from algebraic categories usually don't preserve colimits. However, they tend to preserve filtered colimits.

In this file, we start with a ring R, a small filtered category J and a functor F : J ⥤ ModuleCat R. We show that the colimit of F ⋙ forget₂ (ModuleCat R) AddCommGroupCat (in AddCommGroupCat) carries the structure of an R-module, thereby showing that the forgetful functor forget₂ (ModuleCat R) AddCommGroupCat preserves filtered colimits. In particular, this implies that forget (ModuleCat R) preserves filtered colimits.

@[reducible, inline]

The colimit of F ⋙ forget₂ (ModuleCat R) AddCommGroupCat in the category AddCommGroupCat. In the following, we will show that this has the structure of an R-module.

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    @[reducible, inline]

    The canonical projection into the colimit, as a quotient type.

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      theorem ModuleCat.FilteredColimits.M.mk_eq {R : Type u} [Ring R] {J : Type v} [CategoryTheory.SmallCategory J] [CategoryTheory.IsFiltered J] (F : CategoryTheory.Functor J (ModuleCatMax R)) (x : (j : J) × (F.obj j)) (y : (j : J) × (F.obj j)) (h : ∃ (k : J) (f : x.fst k) (g : y.fst k), (F.map f) x.snd = (F.map g) y.snd) :

      The "unlifted" version of scalar multiplication in the colimit.

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        The bundled R-module giving the filtered colimit of a diagram.

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          The linear map from a given R-module in the diagram to the colimit module.

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            Given a cocone t of F, the induced monoid linear map from the colimit to the cocone point. We already know that this is a morphism between additive groups. The only thing left to see is that it is a linear map, i.e. preserves scalar multiplication.

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