nLab completely distributive category


For locally small category 𝒦\mathcal{K} denote by 𝒫𝒦\mathcal{PK} the full subcategory of [𝒦 op,Set][\mathcal{K}^{op},Set] on small colimits of representables. The category 𝒦\mathcal{K} is cocomplete iff the Yoneda embedding Y:𝒦𝒫𝒦Y \colon \mathcal{K}\to\mathcal{PK} has a left adjoint colim:𝒫𝒦𝒦colim \colon \mathcal{PK}\to\mathcal{K}. If colimcolim has a further left adjoint W:𝒦𝒫𝒦W \colon \mathcal{K}\to\mathcal{PK} and 𝒦\mathcal{K} is moreover complete, then it is called completely distributive.



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