English
If F is representably flat, then the Lan functor F.op.lan preserves finite limits; in particular, evaluation commutes with limits in the Lan construction.
Русский
Если F представимо плоский, то Lan F.op.lan сохраняет конечные пределы; в частности, выEvaluation и пределы в конструировании Lan согласованы.
LaTeX
$$PreservesFiniteLimits (F.op.lan)$$
Lean4
/-- Representably flat functors preserve finite limits. -/
theorem preservesFiniteLimits_of_flat (F : C ⥤ D) [RepresentablyFlat F] : PreservesFiniteLimits F :=
by
apply preservesFiniteLimits_of_preservesFiniteLimitsOfSize
intro J _ _; constructor
intro K; constructor
intro c hc
constructor
exact
{ lift := PreservesFiniteLimitsOfFlat.lift F hc
fac := PreservesFiniteLimitsOfFlat.fac F hc
uniq := fun s m h => by
apply PreservesFiniteLimitsOfFlat.uniq F hc
· exact h
· exact PreservesFiniteLimitsOfFlat.fac F hc s }
