There is an implication "Grothendieck Topos -> exact filtered colimits". This also automatically settles a property of Sh(X) currently listed as unknown. To wit, any Grothendieck Topos arises as a left-exact localization of a presheaf category (https://ncatlab.org/nlab/show/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes). First, Set satisfies "exact filtered colimits", hence so does any presheaf category (since (co-)limits are point-wise). Then, we conclude by noting that "exact filtered colimits" is stable under left-exact localization, because colimits in the localization are computed by applying the reflector to the colimit as computed in the larger category.
This issue has been created by Thorgott via the submission form on https://catdat.app/category-property/exact_filtered_colimits
There is an implication "Grothendieck Topos -> exact filtered colimits". This also automatically settles a property of Sh(X) currently listed as unknown. To wit, any Grothendieck Topos arises as a left-exact localization of a presheaf category (https://ncatlab.org/nlab/show/sheaf+toposes+are+equivalently+the+left+exact+reflective+subcategories+of+presheaf+toposes). First, Set satisfies "exact filtered colimits", hence so does any presheaf category (since (co-)limits are point-wise). Then, we conclude by noting that "exact filtered colimits" is stable under left-exact localization, because colimits in the localization are computed by applying the reflector to the colimit as computed in the larger category.
This issue has been created by Thorgott via the submission form on https://catdat.app/category-property/exact_filtered_colimits