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Mathlib/CategoryTheory/Abelian/GrothendieckCategory/ColimCoyoneda.lean
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… Grothendieck abelian categories (#20014) Let `C` be a Grothendieck abelian category. Let `X : C`. Assume that `κ` is a regular cardinal such that `Subobject X` is of cardinality `< κ`. Let `Y : J ⥤ C` be a functor from a `κ`-filtered category. We consider the map `colim_j (X ⟶ Y_j) → (X ⟶ colim Y)`. We show that it is injective, and under the additional assumption that all maps `Y_j ⟶ Y_j'` are monomorphisms, it is bijective. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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… Grothendieck abelian categories (#20014) Let `C` be a Grothendieck abelian category. Let `X : C`. Assume that `κ` is a regular cardinal such that `Subobject X` is of cardinality `< κ`. Let `Y : J ⥤ C` be a functor from a `κ`-filtered category. We consider the map `colim_j (X ⟶ Y_j) → (X ⟶ colim Y)`. We show that it is injective, and under the additional assumption that all maps `Y_j ⟶ Y_j'` are monomorphisms, it is bijective. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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Let
Cbe a Grothendieck abelian category. LetX : C. Assume thatκis a regular cardinal such thatSubobject Xis of cardinality< κ. LetY : J ⥤ Cbe a functor from aκ-filtered category. We consider the mapcolim_j (X ⟶ Y_j) → (X ⟶ colim Y). We show that it is injective, and under the additional assumption that all mapsY_j ⟶ Y_j'are monomorphisms, it is bijective.Arrow Ais finite iffAis a finite category #19945κ-filtered categories #20005