English
If G preserves equalizers and finite products, it preserves finite limits.
Русский
Если G сохраняет равноселители и конечные произведения, то он сохраняет конечные пределы.
LaTeX
$$PreservesFiniteLimits G$$
Lean4
/-- If a functor creates equalizers and the appropriate products, it creates limits.
We additionally require the rather strong condition that the functor reflects isomorphisms. It is
unclear whether the statement remains true without this condition. There are various definitions of
"creating limits" in the literature, and whether or not the condition can be dropped seems to depend
on the specific definition that is used. -/
noncomputable def createsLimitsOfShapeOfCreatesEqualizersAndProducts : CreatesLimitsOfShape J G where
CreatesLimit
{K} :=
have : HasLimitsOfShape (Discrete J) C := hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape G
have : HasLimitsOfShape (Discrete (Σ p : J × J, p.1 ⟶ p.2)) C :=
hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape G
have : HasEqualizers C := hasLimitsOfShape_of_hasLimitsOfShape_createsLimitsOfShape G
have : HasLimit K := hasLimit_of_equalizer_and_product K
createsLimitOfReflectsIsomorphismsOfPreserves
