Absolutely Cocomplete Category

A category 𝒞 𝒞 \mathcal{C} is absolutely cocomplete if the Yoneda Embedding y : 𝒞 𝒞 ^ : 𝑦 𝒞 ^ 𝒞 y:\mathcal{C}\to\hat{\mathcal{C}} has a Left Adjoint. This induces some very strong Cocompleteness conditions on 𝒞 𝒞 \mathcal{C} , as Left Adjoints Preserve Colimits and Presheaf Categories are Cocomplete.

References