Multivariable adjunction
Definition. An (n,m)-multivariable adjunction $P \colon (๐ธ_0,\dots,๐ธ_n) \to (๐น_0,\dots,๐น_m)$ is a profunctor $P \colon ๐ธ_0^{op} ร \dots ร ๐ธ_n^{op} ร ๐น_0 ร \dots ร ๐น_m \to \mathbf{Set}$ that is representable on each variable. This is to say that there exist representing functors, $$F_j \colon ๐ธ_0 \times \dots \times ๐ธ_n \times ๐น_0^{op} \times \overset{\cancel{ ๐น_j}}\dots \times ๐น_m^{op} \to ๐น_j,$$ $$G_i \colon ๐ธ_0^{op} \times \overset{\cancel{๐ธ_i}}\dots \times ๐ธ_n^{op} \times ๐น_0 \times \dots \times ๐น_m \to ๐ธ_i,$$ such that $$P(A_0,\dots,B_m) \iff ๐น_j(F_j(A_0,\dots,B_m); B_j) \iff ๐ธ_i(A_i;G_i(A_0,\dots,B_m)).$$
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 $๐ธ_0, ๐ธ_1 \to ๐น$ is a triple of functors $$ ๐น(F(A_0,A_1);B) \cong ๐ธ_0(A_1;G_0(A_0,B)) \cong ๐ธ_1(A_0;G_1(A_1,B)).$$ This is called a triple adjoint and it is a situation that occurs in monoidal closed categories. $$ โ(X โ Y; Z) โ โ(X ; Y โธ Z) โ โ(Y ; X โธ Z).$$
Tags: The 2-Chu-Dialectica Construction and the Polycategory of Multivariable Adjunctions (Shulman, 2020), profunctor, adjunction
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