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I finally got around to having a look at the Hyland-Schalk paper. In trying to make abstract sense of it, I realized that (unless I’m mistaken) their input datum for a double gluing construction — a lax symmetric monoidal functor $L:C\to E$ and a functor $K:C\to E^{op}$ together with “contractions” $L(R) \otimes K(R\otimes S) \to K(S)$ (section 4.2.1) — is just a lax symmetric monoidal functor $C \to Chu(E,1)$. This explains their observation in section 4.3.1 that when $C$ is $\ast$-autonomous any lax symmetric monoidal $L:C\to E$ extends canonically to a $K$ with contraction, by Pavlovic’s observation that $Chu$ is right adjoint to the forgetful functor from $\ast$-autonomous categories to closed symmetric monoidal categories equipped with a chosen object (and morphisms that colaxly preserve the object, which is automatic when the chosen object in the codomain is terminal).
However, so far I haven’t been able to reformulate the actual construction of the double-glued category nicely in terms of this functor $C\to Chu(E,1)$ as input, which is very unsatisfying. Any ideas?
In particular, the ordinary gluing of $(L,K) : C\to Chu(E,1)$ would have, as objects, tuples consisting of an object $R\in C$, an object $(U,X)\in Chu(E,1)$ — which is to say two objects $U,X\in E$ — and a morphism $(U,X) \to (L,K)(R)$ in $Chu(E,1)$ — which is to say morphisms $U\to L(R)$ and $K(R) \to X$ in $E$. This looks almost like the double gluing except that the morphism $K(R) \to X$ goes the wrong way! Unless I am confused somehow.
Ah, I got it! It’s another comma double category. Regarding $C$ and $Chu(E,1)$ as vertically discrete double categories, we can map them both into the double Chu construction $\mathbb{C}hu(E,1)$ (the inclusion of $Chu(E,1)$ being an isomorphism onto the horizontal category). Then the double gluing category is the horizontal category of the comma object in double categories (and vertical transformations) of the cospan $C \to \mathbb{C}hu(E,1) \leftarrow Chu(E,1)$. And to encode the monoidal structure, we can use double polycategories (or multicategories) instead.
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