We study (vertically) normal lax double functors valued in the weak double category Cat of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for the theory of double categories they play a similar role as 2-functors valued in Cat for 2-categories. We first introduce representable lax double presheaves and establish a Yoneda lemma. Then we build a Grothendieck construction which gives a 2-equivalence between lax double presheaves and discrete double fibrations over a fixed double category. Finally, we prove a representation theorem showing that a lax double presheaf is represented by an object if and only if its Grothendieck construction has a double terminal object.
Keywords: Double categories, presheaves, Yoneda lemma, Grothendieck construction, representation theorem
2020 MSC: 18N10, 18D30, 18D60, 18D40
Theory and Applications of Categories, Vol. 41, 2024, No. 49, pp 1698-1782.
Published 2024-11-01.
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