cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts

English

If F is full and faithful and has a left adjoint L which preserves binary products, then F is Cartesian closed.

Русский

Если F полно и полноffic, имеет левый сомножитель L, сохраняющий бинарные произведения, то F является картизианно замкнутым функтором.

LaTeX

$$$\\text{If } F \\\\text{ is full and faithful, and has a left adjoint } L \\text{ which preserves binary products, then } F \\text{ is Cartesian Closed.}$$$

Lean4

/-- If `F` is full and faithful, and has a left adjoint which preserves binary products, then it is
Cartesian closed.

TODO: Show the converse, that if `F` is Cartesian closed and its left adjoint preserves binary
products, then it is full and faithful.
-/
theorem cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [F.Full] [F.Faithful]
    [PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F where
  comparison_iso _ := expComparison_iso_of_frobeniusMorphism_iso F h _

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