Limits over essentially small indexing categories #
If C has limits of size w and J is w-essentially small, then C has limits of shape J.
See also the file FinallySmall.lean for more general results.
theorem
CategoryTheory.Limits.hasLimitsOfShape_of_essentiallySmall
(J : Type uā)
[Category.{vā, uā} J]
(C : Type uā)
[Category.{vā, uā} C]
[EssentiallySmall.{wā, vā, uā} J]
[HasLimitsOfSize.{wā, wā, vā, uā} C]
:
HasLimitsOfShape J C
theorem
CategoryTheory.Limits.hasColimitsOfShape_of_essentiallySmall
(J : Type uā)
[Category.{vā, uā} J]
(C : Type uā)
[Category.{vā, uā} C]
[EssentiallySmall.{wā, vā, uā} J]
[HasColimitsOfSize.{wā, wā, vā, uā} C]
:
theorem
CategoryTheory.Limits.hasProductsOfShape_of_small
(C : Type uā)
[Category.{vā, uā} C]
(β : Type wā)
[Small.{wā, wā} β]
[HasProducts C]
:
HasProductsOfShape β C
theorem
CategoryTheory.Limits.hasCoproductsOfShape_of_small
(C : Type uā)
[Category.{vā, uā} C]
(β : Type wā)
[Small.{wā, wā} β]
[HasCoproducts C]
:
HasCoproductsOfShape β C