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\begin{document}

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\section*{double gluing}

\hypertarget{double_gluing}{}\section*{{Double gluing}}\label{double_gluing}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{double_gluing_along_homsets_with_orthogonality}{Double gluing along hom-sets with orthogonality}\dotfill \pageref*{double_gluing_along_homsets_with_orthogonality} \linebreak
\noindent\hyperlink{coherence_spaces_via_double_gluing}{Coherence spaces via double gluing}\dotfill \pageref*{coherence_spaces_via_double_gluing} \linebreak
\noindent\hyperlink{general_definition_of_double_gluing_along_homsets_with_tight_orthogonality}{General definition of double gluing along hom-sets with tight orthogonality}\dotfill \pageref*{general_definition_of_double_gluing_along_homsets_with_tight_orthogonality} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{general_double_gluing}{General double gluing}\dotfill \pageref*{general_double_gluing} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Double gluing}, coupled with orthogonality, is a method for creating new models of [[linear logic]] such as [[star-polycategories]] and [[star-autonomous categories]]. It is a variation of gluing methods (such as [[Artin gluing]]) that works for classical rather than intuitionistic (linear) logic. It is a general method that encompases many existing spaces such as [[phase spaces]], [[coherence spaces]], [[finiteness spaces]], [[totality spaces]], [[probabilistic coherence spaces]].

Although the name presumably refers to gluing along ``two functors at once'', it turns out to also be closely connected to [[double categories]], and in particular to [[comma double categories]].

\hypertarget{double_gluing_along_homsets_with_orthogonality}{}\subsection*{{Double gluing along hom-sets with orthogonality}}\label{double_gluing_along_homsets_with_orthogonality}

\hypertarget{coherence_spaces_via_double_gluing}{}\subsubsection*{{Coherence spaces via double gluing}}\label{coherence_spaces_via_double_gluing}

We begin with the definition of [[coherence spaces]], presented in a way that is amenable to generalization.

If $u\subset X$ and $v\subseteq X$, write $u\perp v$ if $|u\cap v|\leq 1$. Given $U\subseteq P(X)$, let ${U}^\perp=\{v \subseteq X \mid \forall u \in U. u \perp v\}$. Now a [[coherence space]] can be given by a set $X$ together with a set $U\subseteq P(X)$ of cliques and a set $V\subseteq P(X)$ of co-cliques that are orthogonal in the sense that $U=V^\perp$ and $V=U^\perp$.

In this presentation of coherence spaces, we can regard $u$ as a relation $1\to X$, and $v$ as a relation $X\to 1$. The relation $\perp$ is then a family of subsets $\perp_X\subseteq [1\to X]\times [X\to 1]$.

\hypertarget{general_definition_of_double_gluing_along_homsets_with_tight_orthogonality}{}\subsubsection*{{General definition of double gluing along hom-sets with tight orthogonality}}\label{general_definition_of_double_gluing_along_homsets_with_tight_orthogonality}

Let $C$ be a [[symmetric monoidal category]] with an object $I^*$, and let a family $\perp_X\subseteq C(I,X)\times C(X,I^*)$ of relations be given. (For coherence spaces, let $C=Rel$ and $I=I^*$ and $\perp$ be as above.

Let $X$ be an object of $C$. The relation $\perp$ defines a [[Galois connection]] between $C(I,X)$ and $C(X,I^*)$ in the usual way: for ${U}\subseteq C(I,X)$ we have ${U}^\perp = \{ v \in C(X,I^*) \mid \forall u\in {U}. u\perp v \}$, and likewise for ${V}\subseteq C(X,J)$ we have ${V}^\perp = \{ u \in C(I,X) \mid \forall v\in {V}. u\perp v \}$.

We define a \textbf{tight orthogonality space} $(X,U,V)$ to be an object $X\in C$ together with ${U}\subseteq C(I,X)$ and ${V}\subseteq C(X,I^*)$ that is a fixed point of this Galois connection, ${U}^\perp = {V}$ and ${V}^\perp= U$.\newline A \textbf{morphism} between tight orthogonality spaces $(X_1,{U_1},{V_1})\to (X_2,{U_2},{V_2})$ is a morphism $f\in C(X_1,X_2)$ such that

\begin{enumerate}%
\item if $u\in {U_1}$, then $f\circ u \in {U_2}$;
\item if $v\in{V_2}$ then $v\circ f\in {V_1}$.

\end{enumerate}
\textbf{Theorem:} (Hyland-Schalk): If an orthogonality $\bot_X\subseteq C(I,X)\times C(X,I^*)$ on a star-autonomous category satisfies four conditions for orthogonalities, a symmetry condition, and a precision condition, then the tight orthogonality spaces form a [[star-polycategory]].

One source of good orthogonality relations $\bot_X\subseteq C(I,X)\times C(X,I^*)$ are the focuses $F\subseteq C(I,I^*)$, from which we can let $u\bot_X v$ if $(v\circ u)\in F$. However, not all the useful orthogonalities arise from focuses.

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[coherence spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|\leq 1$.
\item [[totality spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|=1$.
\item [[finiteness spaces]]: Let $C=Rel$, and let $u\bot_X v$ if $|u\cap v|\lt\omega$.
\item [[phase semantics]]: Let $M$ be a commutative monoid, and let $F\subseteq M$ be given. Regarding $M$ as a one-object category, with object $I$, we put $u\bot_I v$ if $vu\in F$. Then an object of the orthogonality category $(I,U,V)$ is determined by a ``fact'' $U\subseteq M$. However, to get the poset version of phase semantics we restrict the morphisms of the orthogonality category to the ones that come from the identity $I\to I$.
\item probabilistic coherence spaces: Let $C$ be the category of weighted relations, i.e. countable sets and $[0,\infty]$-valued matrices between them. Let $u\bot_X v$ if $(v\circ u)\in [0,1]$. In other words, the focus $F\subseteq C(I,I^*)$ comprises the scalars in $[0,1]$. One then cuts down further to impose a bounded completeness condition.
\item quantum causal structure: Let $C=CP$, the category of finite dimensional Hilbert spaces and completely positive maps between them (see [[quantum operation]], although here we do not begin by insisting that they are trace preserving). Let $u\bot_X v$ if $v\circ u=1$. In other words, the focus $F\subseteq CP(\mathbb{C},\mathbb{C})$ is the singleton map $\{1\}$. Now, for any finite dimensional Hilbert space $X$, $\{trace_X\}^\perp$ comprises ``states'' of $X$, and a morphism $(X,\{trace_X\}^\perp,\{trace_X\}^{\perp\perp})\to (Y,\{trace_Y\}^\perp,\{trace_Y\}^{\perp\perp})$ is the same thing as a [[quantum operation]], i.e. a completely positive trace preserving map.
\item [[Frölicher spaces]]: Let $C=Set$, let $I=I^*=\mathbb{R}$, and let $u\bot_X v$ if $(v\circ u):\mathbb{R}\to\mathbb{R}$ is smooth. In other words, the focus $F\subseteq C(\mathbb{R},\mathbb{R})$ comprises the smooth maps.

\end{itemize}
\hypertarget{general_double_gluing}{}\subsection*{{General double gluing}}\label{general_double_gluing}

Let $C$ be any (symmetric) [[polycategory]] and $E$ a co-subunary polycategory (i.e. all morphisms have codomain arity 0 or 1), and let $L:C\to Chu(E)$ be a polycategory functor. We consider $C$ as a vertically discrete [[double polycategory]] and $L$ as a functor $C\to \mathbb{C}hu(E)$ into the [[double Chu construction]]. Similarly, consider $Chu(E)$ as a vertically discrete double polycategory, including into $\mathbb{C}hu(E)$ as the horizontal polycategory.

The \textbf{double gluing} polycategory is then the [[comma double category|comma double]] polycategory (i.e. the [[comma object]] in the 2-category of polycategories) $\backslash$begin\{tikzcd\} Gl(L) $\backslash$ard $\backslash$arr $\backslash$ardr,phantom,``$\backslash$Downarrow'' \& Chu(E) $\backslash$ard $\backslash$ C $\backslash$arr,``L''' \& $\backslash$mathbb\{C\}hu(E) $\backslash$end\{tikzcd\}

Note that:

\begin{enumerate}%
\item If $C$ is a multicategory (i.e. a co-unary polycategory), then so is $Gl(L)$.


\item If $E$ has a counit that is terminal so that $Chu(E) = Chu(E,1)$, $C$ is a representable multicategory (i.e. a symmetric monoidal category), and $E$ is representable and closed with products so that $Chu(E,1)$ is a $\ast$-autonomous category, then polycategory functors $C\to Chu(E)$ are equivalent to pairs of a lax symmetric monoidal functor $L:C\to E$ and a functor $K:C\to E^{op}$ together with a ``contraction'' $L(R) \otimes K(R\otimes S) \to K(S)$ satisfying a few axioms. This is how the definition is phrased in \hyperlink{HylandSchalk}{Hyland and Schalk}.


\item If $C$ is a $\ast$-polycategory, then $\ast$-polycategory functors $L:C\to Chu(E)$ are equivalent to functors $U_{\le 1} C\to E$ where $U_{\le 1} C$ is the underlying co-subunary polycategory of $C$. If $E$ has a counit that is terminal so that $Chu(E) = Chu(E,1)$, then these are equivalent to multicategory functors $U_{=1} C \to E$. In particular, we can double-glue along any lax symmetric monoidal functor with $\ast$-autonomous domain.



\end{enumerate}
If $C$ and $E$ are closed and representable with sufficient limits and colimits, then one can show (similarly to the conditions for a Chu construction to be representable) that $Gl(L)$ is also representable (as a multicategory or a polycategory, respectively) and closed. Thus, double gluing can produce closed symmetric monoidal and $\ast$-autonomous categories.

The case of double gluing along hom-functors, discussed above, is the case when $E=Set$ so that $Chu(E,1)$ is $\ast$-autonomous, as above, and $L(X) =C(I,X)$ with $K(X) = C(X,I^*)$, together with a restriction that the gluing morphisms be monic (and the additional ``orthogonality'' restriction.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Martin Hyland]] and Andrea Schalk, \emph{Glueing and orthogonality for models of linear logic}, Theoretical Computer Science 294 (2003) 183–231 (\href{https://core.ac.uk/download/pdf/21173316.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Aleks Kissinger]] and Sander Uijlen, \emph{A categorical semantics for causal structure}. Arxiv \href{https://arxiv.org/abs/1701.04732}{1701.04732}.

\end{itemize}


\end{document}