Multivariable adjunction

Definition. An (n,m)-multivariable adjunction $P:({\mathbb{A}}_0,\dots ,{\mathbb{A}}_n)\to ({\mathbb{B}}_0,\dots ,{\mathbb{B}}_m)$ is a profunctor $P:{\mathbb{A}}_0^{op}\times \cdots \times {\mathbb{A}}_n^{op}\times {\mathbb{B}}_0\times \cdots \times {\mathbb{B}}_m\to \mathbf{Set}$ that is representable on each variable. This is to say that there exist representing functors, $F_j:{\mathbb{A}}_0\times \cdots \times {\mathbb{A}}_n\times {\mathbb{B}}_0^{op}\times \stackrel{\cancel{{\mathbb{B}}_j}}{\dots }\times {\mathbb{B}}_m^{op}\to {\mathbb{B}}_j,$ $G_i:{\mathbb{A}}_0^{op}\times \stackrel{\cancel{{\mathbb{A}}_i}}{\dots }\times {\mathbb{A}}_n^{op}\times {\mathbb{B}}_0\times \cdots \times {\mathbb{B}}_m\to {\mathbb{A}}_i,$ such that $P(A_0,\dots ,B_m)⟺{\mathbb{B}}_j(F_j(A_0,\dots ,B_m);B_j)⟺{\mathbb{A}}_i(A_i;G_i(A_0,\dots ,B_m)).$

Example. A (1,1)-adjunction is an ordinary adjunction. A (0,1)-adjunction is a representable copresheaf, and so it is the same as an object. A (2,1)-adjunction ${\mathbb{A}}_0,{\mathbb{A}}_1\to \mathbb{B}$ is a triple of functors $\mathbb{B}(F(A_0,A_1);B)\cong {\mathbb{A}}_0(A_1;G_0(A_0,B))\cong {\mathbb{A}}_1(A_0;G_1(A_1,B)).$ This is called a triple adjoint and it is a situation that occurs in monoidal closed categories. $C(X\otimes Y;Z)\cong C(X;Y⊸Z)\cong C(Y;X⊸Z).$

Tags: The 2-Chu-Dialectica Construction and the Polycategory of Multivariable Adjunctions (Shulman, 2020), profunctor, adjunction

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