Multivariable adjunction

Last updated Oct 14, 2024

Definition. An (n,m)-multivariable adjunction $P \colon (𝔸_0,\dots,𝔸_n) \to (𝔹_0,\dots,𝔹_m)$ is a profunctor $P \colon 𝔸_0^{op} × \dots × 𝔸_n^{op} × 𝔹_0 × \dots × 𝔹_m \to \mathbf{Set}$ that is representable on each variable. This is to say that there exist representing functors, $$F_j \colon 𝔸_0 \times \dots \times 𝔸_n \times 𝔹_0^{op} \times \overset{\cancel{ 𝔹_j}}\dots \times 𝔹_m^{op} \to 𝔹_j,$$ $$G_i \colon 𝔸_0^{op} \times \overset{\cancel{𝔸_i}}\dots \times 𝔸_n^{op} \times 𝔹_0 \times \dots \times 𝔹_m \to 𝔸_i,$$ such that $$P(A_0,\dots,B_m) \iff 𝔹_j(F_j(A_0,\dots,B_m); B_j) \iff 𝔸_i(A_i;G_i(A_0,\dots,B_m)).$$

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 $𝔸_0, 𝔸_1 \to 𝔹$ is a triple of functors $$ 𝔹(F(A_0,A_1);B) \cong 𝔸_0(A_1;G_0(A_0,B)) \cong 𝔸_1(A_0;G_1(A_1,B)).$$ This is called a triple adjoint and it is a situation that occurs in monoidal closed categories. $$ ℂ(X ⊗ Y; Z) ≅ ℂ(X ; Y ⊸ Z) ≅ ℂ(Y ; X ⊸ Z).$$

multivariable-adjunction

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

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