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Written by Zhixuan Yang on 2024-11-24 at 01:25

Let ρ : C^op × D → Set be a profunctor. The presheaf category over the collage of ρ is precisely the Artin gluing of Pr C and Pr D along the functor (A : Pr C) ↦ (d : Dºᵖ) ↦ Hom(ρ(-, d), A). How could I miss this simple description this long?!

(The collage of ρ is the category whose objects are the disjoint sum of ObC and ObD and

Hom(c',c) = C(c',c) Hom(d, d') = D(d, d'),

Hom(c,d) = ρ(c,d) Hom(d, c) = ∅.)

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Descendants