nLab double Chu construction

Double Chu and 2-Chu constructions

Definition

The Chu construction can be expressed as a functor from the category of co- subunary (symmetric) polycategories to the category of symmetric polycategories, which is right adjoint to the forgetful functor. Since it is a right adjoint, it preserves limits, and therefore also internal categories.

Any 2-polycategory $C$ has a double polycategory $\mathbb{Q}(C)$ of quintets, in which the horizontal arrows are those of $C$ and the vertical arrows are the unary co-unary arrows in $C$, with the 2-cells filled by “square” 2-cells in $C$. (Of course, an ordinary polycategory can be regarded as a locally discrete 2-polycategory, and in this case the 2-cells in $\mathbb{Q}(C)$ are commutative squares in $C$.) If $C$ is a co-subunary 2-polycategory, then $Chu(\mathbb{Q}(C))$ is a double polycategory called its double Chu construction $\mathbb{C}hu(C)$.

Discarding the nonidentity vertical arrows in $\mathbb{C}hu(C)$ yields a 2-polycategory called the 2-Chu construction.

Related pages

• double gluing

References

• Mike Shulman, The 2-Chu-Dialectica construction and the polycategory of multivariable adjunctions. arxiv:1806.06082