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Poly-FMCS-2024.tex
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Poly-FMCS-2024.tex
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\documentclass[aspectratio=169]{beamer}
\input{FMCS-preamble}
\definecolor{darkMagenta}{HTML}{ff9900}
\definecolor{violet}{HTML}{ff9900}
\definecolor{teal}{HTML}{1260cc}
\usepackage{tabularx}
\usepackage{xcolor,colortbl}
\usepackage{ cmll }
\usepackage{panqueques}
\makeatletter
\def\blfootnote{\gdef\@thefnmark{}\@footnotetext}
\makeatother
\newcommand{\purple}[1]{\textcolor{purple}{#1}}
\newcommand{\tcolor}[1]{\textcolor{magenta}{#1}}
\title{What kind of linearly distributive categories \\ do polynomial functors form?}
\author{Priyaa Varshinee Srinivasan}
\date{\today}
\begin{document}
\begin{frame}[noframenumbering,plain]
\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=2cm]current page.north west);
\end{tikzpicture}
\begingroup
%\flushright
\centering
{\fontfamily{qag}\selectfont\large\bfseries\color{black}\MyTitle}
\par \vspace{1em} \MyAuthor
\par \par \vspace{1em} \includegraphics[scale=0.5]{pics/Topos.png}
\par \vspace{-1em} \small{Topos Institute, Berkeley}
\par \vspace{2em} \small{FMCS 2024, Kananaskis}
\par \vspace{-0.5em}
\small{\MyDate}\par
\endgroup
\end{frame}
\begin{frame}[noframenumbering, plain]
\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=2cm]current page.north west);
\end{tikzpicture}
\begingroup
{\centering
This is joint work with \textcolor{magenta}{David Spivak}.
\[ \includegraphics[scale=0.35]{pics/hqdefault.jpg} \]
Our paper is available in \textcolor{magenta}{arXiv: 2407.01849} (July 2024)
}
\endgroup
\end{frame}
\begin{frame}[noframenumbering,plain]
\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=2cm]current page.north west);
\end{tikzpicture}
\begingroup
\flushleft
{\fontfamily{qag}\selectfont\hspace{2 cm}\Large\bfseries\color{black}{Linearly distributive categories}}\vspace{1em}\\
\hspace{2 cm} Robin Cockett, and Robert Seely. {\em Weakly distributive categories} (1997) \\
\endgroup
\end{frame}
\iffalse
\begin{frame}{Linear logic}
In 1987, Girard introduced linear logic as a logic for resources manipulation.
\vspace{0.5em}
Classical logic treats statements as truth values; linear logic treats statements as resources which cannot be duplicated or destroyed.
\vspace{-1em}
\begin{align*}
\textcolor{magenta}{p} &: \text{to spend a dollar} \\
\textcolor{magenta}{q} &: \text{to buy an apple}
\end{align*}
``$p \Rightarrow q$" has the meaning that if a dollar is spent then an apple
can be bought.
\vspace{0.5em}
A person can either have \textcolor{magenta}{a dollar} or \textcolor{magenta}{an apple} at a given time but not both.
\vspace{0.5em}
The word ``linear" refers to this resource sensitivity of the
logic.
\end{frame}
\begin{frame}{Categorical semantics of multiplicative linear logic}
{\small
\begin{table}[h]
\centering
\begin{tabular}{ | l | l | }
\hline
{\bf Linear logic fragment} & {\bf Connectives} \\
\hline
\hline
Negation & $A^\perp$ \\
\hline
\cellcolor{red!20}Multiplicative & \cellcolor{red!20} $(\ox, 1)$ and $(\parr, \bot)$ \\
\hline
Additive & $(\with, \top)$ and $(\oplus, 0)$ \\
\hline
Exponentials & $!$ and $?$ \\
\hline
\end{tabular}
\end{table}
}
{\bf Negation}:
\textcolor{magenta}{p}: to spend a dollar \qquad \qquad
\textcolor{magenta}{$p^\perp$}: to receive a dollar
\vspace{0.25em}
{\bf Multiplicative fragment}:
$(\ox, 1)$: an apple $\ox$ an orange ($p \Rightarrow q \ox r$ means access to resources at the same time; conjunction from classical logic)
$(\parr, \bot)$ = De Morgan's Rule: $(A^\perp \ox B^\perp)^\perp$ (means don't have access to either resource, so someone else owns it; disjunction from classical logic)
\end{frame}
\begin{frame}{Categorical semantics of multiplicative linear logic}
{\small
\begin{table}[h]
\centering
\begin{tabular}{ | l | l | }
\hline
{\bf Linear logic fragment} & {\bf Connectives} \\
\hline
\hline
Negation & $A^\perp$ \\
\hline
\cellcolor{red!20}Multiplicative & \cellcolor{red!20} $(\ox, 1)$ and $(\parr, \bot)$ \\
\hline
Additive & $(\with, \top)$ and $(\oplus, 0)$ \\
\hline
Exponentials & $!$ and $?$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{ | l | l | }
\hline
\cellcolor{violet!20}{\bf Linear logic fragment} & \cellcolor{violet!20}{\bf Categorical proof theory} \\
\hline
\hline
\cellcolor{red!20}{MLL} & \cellcolor{red!20}{Linearly distributive categories\footnote{Cockett and Seely (1997) ``Weakly Distributive Categories"} } \\
\cellcolor{red!20}{ $(\ox, 1)$ and $(\parr, \bot)$ }& \cellcolor{red!20}{$(\X, \ox, \top, \oa, \bot)$}\\
\hline
MLL with negation & $*$-autonomous categories\footnote{Barr (1991) ``$*$-autonomous categories and linear logic"} \\
\hline
\cellcolor{red!20} Compact MLL & \cellcolor{red!20} Monoidal categories \\
\cellcolor{red!20}($\otimes = \parr$, $1 = \bot$) & \cellcolor{red!20} $(\X, \ox, I)$ \\
\hline
Compact closed categories\footnote{Kelly and Laplaza (1980) ``Coherences for compact closed categories"} & Compact MLL with negation \\
\hline
\end{tabular}
\label{Table: MLL}
\end{table}
}
\end{frame}
\fi
\begin{frame}{Linearly distributive categories}
{\bf Linearly distributive categories (LDCs)\footnote{Cockett and Seely (1997) ``Weakly distributive categories"}:}
\[
(\X, \ox, \top, a_\ox, u_\ox^L, u_\ox^R) ~~~~~~~~~ (\X, \oa, \bot, a_\oa, u_\oa^L, u_\oa^R)
\] linked by {linear distributors}:
\[\partial^L: A \ox (B \oa C) \rightarrow (A \ox B) \oa C \]
\[ \textcolor{gray} {A \times (B + C) \simeq (A \times B) + (A \times C)} \]
\[ \partial^R: (B \oa C) \ox A \rightarrow B \oa (C \ox A) \]
\textbf{Intuition:} At a restaurant the waiter, A, can choose to address either person at the table, B or C. Once assigned to B, A cannot choose C.
\textit{The distributor is not an equality or isomorphism in general!}
{\bf Monoidal categories:} LDCs in which $\ox = \oa; \top = \bot$
\vspace{1em}
\end{frame}
\begin{frame}{Symmetry and $\ox$-symmetry}
A \tcolor{symmetric LDC} is an LDC in which $\ox$ and the $\oa$ products are symmetric and the diagram commutes.
\[ \begin{tikzcd}[ampersand replacement=\&]
(A \oa B) \ox C \ar[d, "\tcolor{\partial^R}"'] \ar[r, "\sigma_\ox" ]
\& C \ox (A \oa B) \ar[r, "\sigma_\oa"]
\& C \ox (B \oa A) \ar[d, "\tcolor{\partial^L}"] \\
A \oa (B \ox C)
\& A \oa (C \ox B) \ar[l, "\sigma_\oa"]
\& A \oa (B \ox C) \ar[l, "\sigma_\ox"]
\end{tikzcd}\]
\vspace{1em}
A \tcolor{$\ox$-symmetric LDC} is an LDC in which only the $\ox$ product is symmetric.
\end{frame}
\begin{frame}{Spectrum of LDCs}
\vspace{0.5em}
\[ \begin{tikzpicture}[scale=2]
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2, color=magenta, fill=magenta] (0) at (-5.75, 2.75) {};
\node [style=circle, scale=2, color=black!80, fill=black!80] (1) at (-3.5, 2.75) {};
\node [style=circle, scale=2, color=black!60, fill=black!60] (2) at (-1, 2.75) {};
\node [none, color=white] (99) at (-3.5, 2.75) {?};
\node [style=circle, scale=2, color=black!40, fill=black!40] (3) at (1.75, 2.75) {};
\node[style=none, color=white] (99) at (1.75, 2.75) {?};
\node[style=none, color=white] (99) at (-1, 2.75) {?};
\node [style=none] (4) at (-7.75, 2.75) {};
\node [style=circle, scale=2, color=magenta!20, fill=magenta!20] (5) at (4, 2.75) {};
\node [style=none] (6) at (6, 2.75) {};
\node [style=none] (7) at (-5.75, 2) {{\large \bf LDC}};
\node [style=none] (8) at (-3.5, 4.35) {};
\node [style=none] (9) at (-3.5, 3.85) {};
\node [style=none] (10) at (-1, 2) {};
\node [style=none] (11) at (1.75, 4.1) {};
\node [style=none] (12) at (1.75, 3.5) {};
\node [style=none] (13) at (4, 2) {};
\node [style=none] (14) at (-1, 1.4) {};
\node [style=none] (15) at (4, 1.5) {};
\node [style=none] (16) at (-5.75, 1.5) {$(\X, \ox, \top, \oa, \bot)$};
\node [style=none] (17) at (-3.5, 3.35) { };
\node [style=none] (13) at (4, 2) {{\bf \large Monoidal} category};
\node [style=none] (15) at (4, 1.5) {$(\X, \ox, I)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted] (4.center) to (0);
\draw (0) to (1);
\draw (1) to (2);
\draw (2) to (3);
\draw (3) to (5);
\draw [dotted] (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}\]
\end{frame}
\begin{frame}{Mix LDCs}
{\bf Mix category} \footnote{Richard Blute, Robin Cockett, and Robert Seely (2000). "Feedback for linearly distributive categories: traces and fixpoints."}: LDC with $m: \bot \to \top$ called the {\bf mix map} with
\begin{center}
$\indep_{A,B}: A \ox B \to A \oa B :=$
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=ox] (0) at (0, 0.2500001) {};
\node [style=circ] (1) at (0.5000001, -0.2500001) {};
\node [style=circ] (2) at (0, -1) {$\bot$};
\node [style=map] (3) at (0, -1.75) {m};
\node [style=circ] (4) at (0, -2.5) {$\top$};
\node [style=circ] (5) at (-0.5000001, -3.25) {};
\node [style=oa] (6) at (0, -3.75) {};
\node [style=nothing] (7) at (0, 0.7499999) {};
\node [style=nothing] (8) at (0, -4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7) to (0);
\draw (0) to (1);
\draw [in=45, out=-60, looseness=1.00] (1) to (6);
\draw [in=120, out=-135, looseness=1.00] (0) to (5);
\draw (5) to (6);
\draw (6) to (8);
\draw [densely dotted, in=-90, out=45, looseness=1.00] (5) to (4);
\draw (4) to (3);
\draw (3) to (2);
\draw [densely dotted, in=-135, out=90, looseness=1.00] (2) to (1);
\end{pgfonlayer}
\end{tikzpicture}
=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circ] (0) at (-0.5000001, -0.2500001) {};
\node [style=circ] (1) at (0, -1) {$\bot$};
\node [style=map] (2) at (0, -1.75) {m};
\node [style=circ] (3) at (0, -2.5) {$\top$};
\node [style=circ] (4) at (0.5000001, -3.25) {};
\node [style=nothing] (5) at (0, 0.7499999) {};
\node [style=nothing] (6) at (0, -4.25) {};
\node [style=oa] (7) at (0, -3.75) {};
\node [style=ox] (8) at (0, 0.2500001) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [densely dotted, in=-90, out=150, looseness=1.00] (4) to (3);
\draw (3) to (2);
\draw (2) to (1);
\draw [densely dotted, in=-45, out=90, looseness=1.00] (1) to (0);
\draw (8) to (5);
\draw (8) to (0);
\draw [in=135, out=-120, looseness=1.00] (0) to (7);
\draw (7) to (6);
\draw (7) to (4);
\draw [in=-45, out=60, looseness=1.00] (4) to (8);
\end{pgfonlayer}
\end{tikzpicture}
\[ (1 \oa (u_\oa^L)^{-1}) (1 \ox (\m \oa 1)) \delta^L(u_\ox^R \oa 1)\]
\end{center}
The \tcolor{indep}\footnote{Also referred to as the mixor} must be natural in $A$ and $B$.
\vspace{0.5em}
\end{frame}
\begin{frame}[noframenumbering]{Spectrum of LDCs}
\[
\begin{tikzpicture}[scale=2]
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2, color=magenta, fill=magenta] (0) at (-5.75, 2.75) {};
\node [style=circle, scale=2, color=magenta!80, fill=magenta!80] (1) at (-3.5, 2.75) {};
\node [style=circle, scale=2, color=black!60, fill=black!60] (2) at (-1, 2.75) {};
\node [style=circle, scale=2, color=black!40, fill=black!40] (3) at (1.75, 2.75) {};
\node [style=none] (4) at (-7.75, 2.75) {};
\node [style=circle, scale=2, color=magenta!20, fill=magenta!20] (5) at (4, 2.75) {};
\node [style=none] (6) at (6, 2.75) {};
\node [style=none] (7) at (-5.75, 2) {{\large \bf LDC}};
\node [style=none] (8) at (-3.5, 4.35) {{\bf \large Mix} category};
\node [style=none] (9) at (-3.5, 3.85) {$\m: \bot \to \top$};
\node [style=none] (10) at (-1, 2) {};
\node [style=none] (11) at (1.75, 4.1) {};
\node [style=none] (12) at (1.75, 3.5) {};
\node [style=none] (13) at (4, 2) {};
\node [style=none] (14) at (-1, 1.4) {};
\node [style=none] (15) at (4, 1.5) {};
\node [style=none] (16) at (-5.75, 1.5) {$(\X, \ox, \top, \oa, \bot)$};
\node [style=none] (17) at (-3.5, 3.35) {$\mx: A \ox B \to A \oa B$};
\node [style=none] (13) at (4, 2) {{\bf \large Monoidal} category};
\node [style=none] (15) at (4, 1.5) {$(\X, \ox, I)$};
\node[style=none, color=white] (99) at (1.75, 2.75) {?};
\node[style=none, color=white] (99) at (-1, 2.75) {?};
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\draw (2) to (3);
\draw (3) to (5);
\draw [dotted] (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\]
\end{frame}
\begin{frame}{Isomix LDCs }
\vspace{0.5em}
It is an {\bf isomix} category if $m$ is an isomorphism.
\vspace{1em}
$m$ being an isomorphism does not make the indep an isomorphism.
\vspace{1em}
\[ \begin{tikzpicture}[scale=2]
\begin{pgfonlayer}{nodelayer}
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%\node [style=none] (15) at (4, 1.5) {$\m = 1$, $\mx=1$};
\node [style=none] (16) at (-5.75, 1.5) {$(\X, \ox, \top, \oa, \bot)$};
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\end{tikzpicture} \]
\end{frame}
\begin{frame}{Compact LDCs}
A {\bf compact LDC} is an LDC in which every indep map is an isomorphism.
\[ \indep_{A,B}: A \ox B \to^{\simeq} A \oa B \]
\vspace{1.5em}
Compact LDCs $(\X, \ox, \top, \oa, \bot)$ are linearly equivalent to the underlying
monoidal categories $(\X, \ox, \top)$ and $(\X, \oa, \bot)$.
\end{frame}
\begin{frame}{Spectrum of LDCs}
\[ \begin{tikzpicture}[scale=2]
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\end{frame}
\begin{frame}{Typical examples of LDCs}
\vspace{2em}
Every monoidal category is an LDC.
\vspace{0.75em}
A bounded distributive lattice regarded as a category is an LDC.
\vspace{0.75em}
All $*$-autonomous categories are LDCs: \\
Ehrhard’s finiteness spaces, Girard's coherence spaces, Chu spaces.
\vspace{1.5em}
\begin{center}
\textcolor{purple}{\Large Can we have a simple yet a structurally rich example of LDCs?}
\vspace{0.5em}
{\Large Yes!!}
\end{center}
\end{frame}
\begin{frame}{This talk is about ... }
\vspace{1em}
\textcolor{purple}{Poly} the category of polynomial functors and transformations as an example of isomix LDCs.
\vspace{1em}
We will find non-trivial examples of various {structures of LDCs} in Poly.
\vspace{1em}
We will also {discover new properties} of LDCs by studying Poly.
\vspace{2em}
\begin{center} \textcolor{magenta}{Poly seems to be a golden goose for LDCs!!!} \end{center}
\end{frame}
\begin{frame}{Acknowledgement}
\centering
Thanks to Harrison Grodin and Reed Mullanix for their conjecture, that normal duoidal categories are linear distributive categories, which spawned this work.
\end{frame}
\begin{frame}{ Recap: Polynomial functors and natural transformations}
\vspace{1em}
A {\bf polynomial functor} is a functor that is isomorphic to a coproduct of representables.
\[ p \cong \sum_{P : p(1)} y^{p[P]} : {\sf Set \to Set} \]
\vspace{0.5em}
A schematic of a polynomial functor map, $\varphi: (y^3 + y^2) \to (y + y^2)$.
\[ \includegraphics[scale=0.09]{pics/corolla-map.jpeg} \]
\end{frame}
\begin{frame}{ Recap: Tensor product in Poly}
The tensor product $\ox$ is given by the Day convolution of Cartesian product $\times$ in $\Set$.
\[ p \ox q = \sum_{P: p(1)} y^{p[P]} \ox \sum_{Q: q(1)} y^{q[Q]} := \sum_{\textcolor{purple}{(P,Q): p(1) \times q(1)}} y^{\textcolor{purple}{p[P] \times q[Q]}} \]
Tensor product is symmetric.
\end{frame}
\begin{frame}{Recap: Tri product in Poly}
The substitution product $\tri$ is given by functor composition.
\[ p \tri q = \sum_{P: p(1)} y^{p[P]} \tri \sum_{Q: q(1)} y^{q[Q]} :=
\sum_{P: p(1)} \Big(\sum_{Q: q(1)} y^{q[Q]} \Big)^{p[P]}
= \sum_{P:p(1)} \sum_{\color{purple}{f:p[P] \to q(1)}} \prod_{d:p[P]} \prod_{\color{purple}{e:q[f(d)]}} y \]
Read $p \tri q$ as \textcolor{purple}{q then p}.
\vspace{1em}
The tri product $\tri$ is non-symmetric.
\[ \includegraphics[scale=0.08]{pics/tri.jpeg} \]
\end{frame}
\begin{frame}[noframenumbering,plain]
\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=4cm]current page.north west);
\end{tikzpicture}
\begingroup
\flushleft
{\fontfamily{qag}\selectfont\hspace{2 cm}\Large\bfseries\color{black}{Part I: The category Poly is a $\ox$-symmetric isomix LDC}} \vspace{1em}
\endgroup
\end{frame}
\begin{frame}{Duoidal categories}
A \tcolor{duoidal category}\footnote{Marcelo Aguiar and Swapneel Arvind Mahajan. Monoidal functors, species and Hopf algebras (2010)} is a category $\X$ with two monoidal structures $(\X, \ox, \top)$ and $(\X, \oa, \bot)$ along with a natural transformation:
\[ {\duo: ~(A \oa B) \ox (C \oa D) \to (A \ox C) \oa (B \ox D)} \]
called the \tcolor{interchange law}, and morphisms:
\[ e_\top: ~\top \to \top \oa \top \quad \quad e_\bot: ~\bot \otimes \bot \to \bot \]
such that the functors $\oa$ and $\bot$ are $\ox$-lax monoidal, and the assosciativity and unitor natural isomorphisms of $(\oa, \bot)$ are $\ox$-monoidal natural transformations.
\vspace{1em}
\end{frame}
\begin{frame}{Normal duoidal category}
In a duoidal category, we have a map $\tcolor{k: \top \to \bot}$
\[ \top \to^{\cong} \top \ox \top \to^{\cong} (\top \oa \bot) \ox (\bot \oa \top) \to^{\tcolor{\duo}} (\top \ox \bot) \oa (\top \ox \bot) \to^{\cong} \bot \oa \bot \to^{\cong} \bot \]
\vspace{0.5 em}
A duoidal category is \tcolor{normal} if the above composite is an isomorphism.
\end{frame}
\begin{frame}{Bilax duoidal functor}
\tcolor{Bilax duoidal functor} $F: (\X, \ox, \top, \tri, \bot) \to (\Y, \ox, \top, \tri, \bot)$ consists of a functor $F: \X \to \Y$ that is
$\ox$-lax monoidal and $\tri$-colax monoidal satisfying,
\[ \xymatrix{
F((a \tri b) \otimes (c \tri d)) \ar[d]_{F(\duo)} &F(a \tri b) \otimes F(c \tri d) \ar[r]^{\hspace{-20pt}n_\tri \otimes n_\tri} \ar[l]_{m_\otimes} & (F(a) \tri F(b)) \otimes (F(c) \tri F(d)) \ar[d]^{\duo} \\
F((a \otimes c) \tri (b \otimes d)) \ar[r]_{n_\tri} &F(a \otimes c) \tri F(b \otimes d) & (F(a) \otimes F(c)) \tri (F(b) \otimes F(d)) \ar[l]^{\hspace{-20pt}m_\otimes \tri m_\otimes}
}
\]
and 3 more coherences (two for unit laxors the duoidal map and one for the unit laxors with the $k$ map.
\end{frame}
\begin{frame}{Normal duoidal is also isomix}
{\bf normalDuo} category of normal duoidal categories and bilax duoidal functors
\vspace{0.5em}
{\bf Isomix} category of isomix LDCs and isomix functors
\vspace{1em}
\textcolor{teal}{Theorem:} There is a faithful functor from \tcolor{normalDuo} to \tcolor{Isomix}.
\textcolor{teal}{Proof sketch:} Define the left distributor as follows:
\[
\begin{tikzcd}[ampersand replacement=\&]
A \ox (B \oa C) \ar[r, "\color{magenta}\cong"] \& (A \oa I) \ox (B \oa C) \ar[r, "\color{magenta}\duo"] \& (A \ox B) \oa (I \ox C) \ar[r, "\color{magenta}\cong"] \& (A \ox B) \oa C
\end{tikzcd}
\]
\vspace{0.75em}
\textcolor{teal}{Lemma}: The category \tcolor{$($Poly$, \otimes, \tri, y )$ is normal duoidal} hence an isomix LDC. Additionally, Poly is $\ox$-symmetric.
\vspace{0.5em}
\color{violet}{\bf Attention:} \color{black} Since in this tutorial we will mostly be in a $\ox$-symmetric setting, we will \\ \tcolor{use $\tri$ instead of $\oa$}.
\end{frame}
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\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=4cm]current page.north west);
\end{tikzpicture}
\begingroup
\flushleft
{\fontfamily{qag}\selectfont\hspace{2 cm}\Large\bfseries\color{black}{Part II: Meet the linear duals and biclosed LDCs}}\vspace{1em}\\
\endgroup
\end{frame}
\begin{frame}{What is a linear dual?}
In an LDC, an object {\bf $B$ is left dual to $A$} if there exist\footnote{Robin Cockett, Jurgen Koslowski and Robert Seely. Introduction to Linear bicategories (1999)}:
\[ \eta: \top \to A \tri B ~~~~~~~~ \epsilon: B \ox A \to \bot \]
such that:
\[ \begin{tikzpicture}[scale=1.2]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (6) at (1, 0) {};
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\node [style=none, scale=1.2] (13) at (2.5, 0.1) {$\epsilon$};
\node [style=none] (14) at (0.75, 0.25) {$A$};
\node [style=none] (15) at (3.25, 1.75) {$A$};
\end{pgfonlayer}
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\draw (10.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}[scale=1.2]
\draw (0,2.5) -- (0,0);
\end{tikzpicture} ~~~~~~~~~~
\begin{tikzpicture}[scale=1.2]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (6) at (3, 0) {};
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\begin{pgfonlayer}{edgelayer}
\draw (6.center) to (7.center);
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\draw (10.center) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}[scale=0.9]
\draw (0,2.5) -- (0,0);
\end{tikzpicture} \]
A symmetric {\bf $*$-autonomous category} is an LDC in which every object has a chosen dual object.
\end{frame}
\begin{frame}{Biclosed LDCs}
An LDC is:
\tcolor{$\ox$-closed} if for all $A:\X$, the functor $ - \otimes A : \X \to \X$ has a right adjoint.
\[ \X(A \ox B, C) \cong \X(B, A \lollipop C) \quad \quad \color{teal}{\eval: A \ox (A \lollipop B) \to B} \]
\tcolor{$\tri$-coclosed} if for all $A: \X$, the functor $ - \tri A: \X \to \X$ has a left adjoint.
\[ \X \left(A \slash C, B \right) \cong \X(A, B \tri C) \quad \quad \color{teal}{\coeval: A \to (A \slash B) \tri B} \]
\tcolor{biclosed} if $\X$ is both $\ox$-closed and $\tri$-coclosed.
\end{frame}
\begin{frame}{Poly is biclosed}
The category \textcolor{magenta}{$\Poly$ is a biclosed isomix LDC} where, for any two polynomials $p, q$:
\[ p \lollipop q := [p,q] = \prod_{P : p(1)} \sum_{\color{magenta} Q : q(1)} \prod_{d : q[Q]} \sum_{\color{magenta}d' : p[P]} y \]
\[ p \slash q := \coclose{q}{p} = \sum_{P : p(1)} y^{ q \tri p[P]} = \sum_{P : p(1)} \prod_{\color{magenta} Q: q(1)} \prod_{d:q[Q]\to p[P]} y \]
\end{frame}
\begin{frame}{Double closure maps (1)}
For any object $A:\X$ in a biclosed isomix LDC, there is a sort of a ``double closure'' map:
\[ \begin{tikzcd}[ampersand replacement=\&]
A \ar[r, "{\color{magenta}\coeval}"] \ar[rrr, bend right = 15pt, "\Phi_A"'] \& [10pt] \left(\coclose{[A, y]}{y} \tri [A,y] \right) \ox A \ar[r, "\color{magenta}\partial^R"] \& \coclose{[A, y]}{y} \tri ([A,y]) \ox A) \ar[r, "\color{magenta}\eval"] \& \coclose{[A,y]}{y}
\end{tikzcd} \]
\textcolor{teal}{Theorem:} If $A$ is right dual to $[A,y]$ with
\[ \epsilon = \eval: A \ox [A,y] \to y \]
then $\Phi_A$ has a retraction $\chi_A$,
\[ \begin{tikzcd}[ampersand replacement=\&]
A \ar[r, "\Phi_A"] \ar[rr, bend right, equals] \& \coclose{[A,y]}{y} \ar[r, "\chi_A"] \& A
\end{tikzcd} \]
For the other direction, we need an extra condition: $ \coeval \then \partial^R \then \chi_A \then \eval = Id_{[A, y]}$.
\end{frame}
\begin{frame}{Double closure maps (2)}
For any object $A:\X$ in a biclosed isomix LDC there is a sort of a ``double closure'' map:
\[ \begin{tikzcd}[ampersand replacement=\&]
\left[ \coclose{A}{ y }, y \right] \ar[r, "\color{magenta}\coeval"] \ar[rrr, bend right = 15pt, "\Psi_A"']
\& \left[ \coclose{A}{ y }, y \right] \ox \left( \coclose{A}{ y } \tri A \right) \ar[r, "\color{magenta}\partial^L"]
\& \left( \left[ \coclose{A}{ y }, y \right] \ox \coclose{A}{ y } \right) \tri A \ar[r, "\color{magenta}\eval"]
\& A
\end{tikzcd} \]
\textcolor{teal}{Theorem:} If $A$ is left dual to $\coclose{A}{y}$ with
\[ \eta = \coeval: y \to \coclose{A}{y} \tri A \]
then $\Psi_A$ has a section $\Omega_a$:
\[ \begin{tikzcd}[ampersand replacement=\&]
A \ar[ r, "\Omega_A"] \ar[rr, bend right, equals] \& \left[ \coclose{A}{y}, y \right] \ar[r, "\Psi_A"] \& A
\end{tikzcd}
\]
\end{frame}
\begin{frame}{Double closure maps in Poly }
%\[ [Ay, y] \cong y^A \quad \quad \quad \quad \quad \quad \quad [y^A, y] \cong Ay \]
%\vspace{1em}
\textcolor{teal}{Theorem:} A polynomial \textcolor{magenta}{$p = y^A$} for $A: \text{Set}$ if and only if \tcolor{$\Phi_p$ is the identity}:
\[ \id_p = \Phi_p : p \to \coclose{[p,y]}{y} \]
\textcolor{teal}{Corollary:} For any set $A$, \tcolor{$Ay$ is left dual to $y^A$} with $\eta = \coeval$ and $\epsilon = \eval$.
\[ \includegraphics[scale=0.08]{pics/dual2.jpeg}\]
\textcolor{teal}{Theorem:} A polynomial \textcolor{magenta}{$q = Ay$} for $A: \text{Set}$ if and only if \tcolor{$\Psi_q$ is the identity}.
\[ \id_q = \Psi_q : \left[ \coclose{q}{y}, y \right] \to q \]
We get the same corollary again!
\end{frame}
\begin{frame}{What are the linear duals in Poly?}
\textcolor{teal}{Theorem:} If a polynomial $p$ is left dual to $q$, then $p = Ay$ and $q = y^A$ for some $A : {\sf Set}$.
\[ \includegraphics[scale=0.08]{pics/dual3.jpeg}\]
Thus, the only polynomials with duals are linear polynomials and representables!
\vspace{1em}
\textcolor{teal}{Theorem}: If $q \dual p$ and $p \dual q$, then $p = q = y$.
\vspace{0.25em}
Any polynomial which is both a left and a right dual of the same polynomial is trivial.
\end{frame}
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\begin{tikzpicture}[remember picture, overlay]
\fill[black] (current page.south west) rectangle ([xshift=4cm]current page.north west);
\end{tikzpicture}
\begingroup
\flushleft
{\fontfamily{qag}\selectfont\hspace{2 cm}\Large\bfseries\color{black}{Part III: The core of mix LDC}}\vspace{1em}\\
\endgroup
\end{frame}
\begin{frame}{The core of a mix LDC}
The {\bf core}\footnote{Richard Blute, Robin Cockett, Robert Seely, "Feedback for linearly dis- tributive categories: traces and fixpoints" (2000)} of a mix LDC is the full subcategory determined by objects $A$ for which the natural transformation is also an isomorphism:
\[ \indep_{A,-}: A \ox - \to A\tri - \quad \quad \quad \indep_{-, A}: - \ox A \to - \tri A \]
\vspace{0.25em}
The core of a \textcolor{purple}{mix category} is closed to $\otimes$ and $\oplus$.
\vspace{0.5em}
The core of an \textcolor{purple}{isomix} LDC contains the monoidal units $\top$ and $\bot$.
\vspace{0.5em}
The core of an isomix LDC is linearly equivalent to its underlying monoidal categories.
\end{frame}
\begin{frame}{Left and right core of mix LDCs}
For a { mix ($\ox$-symmetric) LDC} $(\X, \otimes, \top, \tri, \bot)$,
\tcolor{Left core:} Full subcategory of objects $A: \X$ such that
\[ \color{magenta}{\indep_{A,-}: A \ox - \to A \tri -} \]
\tcolor{Right core:} Full subcategory of objects $B:\X$ such that
\[ \color{magenta}{\indep_{-, B}: - \ox B \to - \tri B} \]
\textcolor{teal}{Lemma:} For an isomix LDC, the unit object is both in the left and the right cores.
In an isomix LDC, the left and the right core are compact LDCs.
\end{frame}
\begin{frame}{Opposing cores}
\tcolor{Opposing cores:} A mix LDC is said to have opposing cores if
\[ \star: \rCore(\X)^{\op} \to^{\cong} \lCore(\X) \]
\textcolor{teal}{Examples:} Compact closed categories and Poly
\vspace{0.75em}
\textcolor{teal}{Lemma:} In Poly, a polynomial \tcolor{$p$ in the left core} if and only if $p \cong Ay$ for some $A:\Set$.
\textcolor{teal}{Lemma:} In Poly, a polynomial \tcolor{$q$ in the right core} if and only if $q \cong y^B$ for some $B:\Set$.
\[ \includegraphics[scale=0.1]{pics/boxes1.jpeg} \]
\end{frame}
\begin{frame}{Opposing cores of Poly }
\textcolor{teal}{Corollary:} The category $\Poly$ has opposing cores.
\textcolor{teal}{Proof:} For any $A: \Set$, \[ \left(y^A \right)^\star := Ay \]
For any $A,B: \Set$, and map $\varphi: y^B \to y^A$,
\[ \varphi^\star: Ay \to By ; \quad (\varphi^\star)_1 ~:=~ \varphi^\sharp \]
\textcolor{teal}{Corollary:} In Poly, a polynomial \tcolor{$p$ is left dual to $q$} if and only if \tcolor{$p$ is from the left core} and \tcolor{$q$ is from the right core}.
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{\fontfamily{qag}\selectfont\hspace{2 cm}\Large\bfseries\color{black}{Part IV: Linear monoids and linear comonoids}}\vspace{1em}\\
\hspace{2 cm} Robin Cockett, Jurgen Koslowski and Robert Seely. \\
\hspace{2.5 cm} {\em Introduction to Linear bicategories} (1999) \\
\hspace{2 cm} Priyaa Varshinee Srinivasan. (PhD Thesis) \\
\hspace{2.5 cm} {\em Dagger linear logic for categorical quantum mechanics} (2021) \\
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\begin{frame}{Linear monoids in LDCs}
In an LDC, a {\bf linear monoid}\footnote{Robin Cockett, Jurgen Koslowski and Robert Seely.{Introduction to Linear bicategories} (1999)}, $A \linmonw B$, contains a:
\vspace{0.5em}
- a $\ox$-monoid $(A, \mulmap{1.35}{white}: A \ox A \to A, ~\unitmap{1.35}{white}: \top \to A)$
\vspace{0.5 em}
- cyclic duals, $A \dual B$ and $B \dual A$
\vspace{0.5em}
together producing a $\tri$-comonoid $(B, \comulmap{1.5}{white}: B \to B \tri B, \counitmap{1.5}{white}: B \to \bot)$
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