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double Chu construction

Double Chu and 2-Chu constructions

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

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

References

Created on October 14, 2019 at 19:00:42. See the history of this page for a list of all contributions to it.