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Multivariable adjunction

Last updated Oct 14, 2024

Definition. An (n,m)-multivariable adjunction $P \colon (๐”ธ_0,\dots,๐”ธ_n) \to (๐”น_0,\dots,๐”น_m)$ is a profunctor $P \colon ๐”ธ_0^{op} ร— \dots ร— ๐”ธ_n^{op} ร— ๐”น_0 ร— \dots ร— ๐”น_m \to \mathbf{Set}$ that is representable on each variable. This is to say that there exist representing functors, $$F_j \colon ๐”ธ_0 \times \dots \times ๐”ธ_n \times ๐”น_0^{op} \times \overset{\cancel{ ๐”น_j}}\dots \times ๐”น_m^{op} \to ๐”น_j,$$ $$G_i \colon ๐”ธ_0^{op} \times \overset{\cancel{๐”ธ_i}}\dots \times ๐”ธ_n^{op} \times ๐”น_0 \times \dots \times ๐”น_m \to ๐”ธ_i,$$ such that $$P(A_0,\dots,B_m) \iff ๐”น_j(F_j(A_0,\dots,B_m); B_j) \iff ๐”ธ_i(A_i;G_i(A_0,\dots,B_m)).$$

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 $๐”ธ_0, ๐”ธ_1 \to ๐”น$ is a triple of functors $$ ๐”น(F(A_0,A_1);B) \cong ๐”ธ_0(A_1;G_0(A_0,B)) \cong ๐”ธ_1(A_0;G_1(A_1,B)).$$ This is called a triple adjoint and it is a situation that occurs in monoidal closed categories. $$ โ„‚(X โŠ— Y; Z) โ‰… โ„‚(X ; Y โŠธ Z) โ‰… โ„‚(Y ; X โŠธ Z).$$

multivariable-adjunction

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

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