colimitPointwiseProductToProductColimit

English

If K is filtered and J finite, colimitLimitToLimitColimit F is an isomorphism.

Русский

Если K фильтрован, а J конечен, то colimitLimitToLimitColimit F является изоморфизмом.

LaTeX

$$IsIso (colimitLimitToLimitColimit F)$$

Lean4

/-- The natural morphism `colim_k (∏ᶜ s ↦ (F s).obj (k s)) ⟶ ∏ᶜ s ↦ colim_k (F s).obj (k s)`.
We will say that a category has the `IPC` property if this morphism is an isomorphism as long
as the indexing categories are filtered. -/
noncomputable def colimitPointwiseProductToProductColimit :
    colimit (pointwiseProduct F) ⟶ ∏ᶜ fun (s : α) => colimit (F s) :=
  colimit.desc (pointwiseProduct F) (coconePointwiseProduct F)

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