inclusionCreatesFiniteColimits — Mathlib

English

If each X i is nonempty, then the finite product over i is nonempty.

Русский

Если каждый X_i непуст, то конечное произведение по i непусто.

LaTeX

$$$\\forall i, X(i) \\neq \\varnothing \\Rightarrow (\\prod^{\\mathrm{c}}_{i} X(i)) \\neq \\varnothing$$$

Lean4

noncomputable instance inclusionCreatesFiniteColimits {J : Type} [SmallCategory J] [FinCategory J] :
    CreatesColimitsOfShape J FintypeCat.incl.{u} where
  CreatesColimit
    {K} :=
    createsColimitOfFullyFaithfulOfIso (FintypeCat.of <| colimit <| K ⋙ FintypeCat.incl)
      (Iso.refl _)
        /- Help typeclass inference to infer creation of finite colimits for the forgtful functor. -/

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