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30 changes: 15 additions & 15 deletions P1-Polynomials.tex
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Expand Up @@ -949,7 +949,11 @@ \section{Sums and products of functors $\smset\to\smset$} \label{sec.poly.rep-se
%-------- Section --------%
\section{Summary and further reading}

% TODO: fill in summary
In this chapter, we reviewed the construction of a representable functor on the category of $\smset$ as well as the proof of the Yoneda lemma, a foundational result characterizing natural transformations out of these representables.
We then reviewed other categorical constructions in $\smset$.
Many of the properties of $\smset$ will carry over to the polynomial functors we will introduce in the next chapter.

For other introductions to the Yoneda lemma, the category of sets, or both, take your pick of \cite{Pierce:1991,Borceux:1994a,MacLane:1998a,Leinster:2014a,Riehl:2017a,fong2019seven,cheng_2022}.

%-------- Section --------%
\section{Exercise solutions}
Expand Down Expand Up @@ -1731,11 +1735,12 @@ \section{Polyboxes}
%-------- Section --------%
\section{Summary and further reading}

In this chapter we explained the mathematics behind our main objects of study in this book, polynomial functors. A polynomial $p=\sum_{i\in I}\yon^{p[i]}$ can be considered as
In this chapter we explained the mathematics behind our main objects of study in this book, polynomial functors. A polynomial $p\iso\sum_{i\in I}\yon^{p[i]}$ can be considered as
\begin{enumerate}
\item combinatorial data: an indexed family of sets $(p[i])_{i\in I}$;
\item a picture: for each $i\in I$, a corolla with $p[i]$-many leaves;
\item a functor $\smset\to\smset$: for each $X:\smset$, a new set $\sum_{i\in I}X^{p[i]}$.
\item a functor $\smset\to\smset$, assigning each $X\in\smset$ to a new set $\sum_{i\in I}X^{p[i]}$;
\item an indexed family of sets $(p[i])_{i\in I}$;
\item a forest of corollas with $|p[i]|$-many leaves for each $i\in I$;
\item a polybox picture resembling two stacked cells in a spreadsheet, to be filled in with an element of $i$ below and an element of $p[i]$ above.
\end{enumerate}

There are many fine sources on polynomial functors. Some of the computer science literature is more relaxed about what a polynomial is. For example, the ``coalgebra community'' often defines a polynomial to include finite power sets (see e.g.\ \cite{jacobs2017introduction}). Other computer science communities use the same definition of polynomial, but refer to it as a \emph{container} and use different words for its positions (they call them ``shapes'') and directions (they call them, rather unfortunately, ``positions''). See e.g.\ \cite{abbot2003categoriesthesis,abbott2005containers}.
Expand Down Expand Up @@ -4213,13 +4218,10 @@ \subsection{The parallel product} \label{subsec.poly.cat.monoidal.par}
%-------- Section --------%
\section{Summary and further reading}

In this chapter we explained the mathematics behind our main object of study in this book, the category $\poly$ of polynomial functors. A polynomial $p=\sum_{i\in I}\yon^{p[i]}$ can be considered as
\begin{enumerate}
\item combinatorial data: an indexed family of sets $(p[i])_{i\in I}$;
\item a picture: for each $i\in I$, a corolla with $p[i]$-many leaves;
\item a functor $\smset\to\smset$: for each $X:\smset$, a new set $\sum_{i\in I}X^{p[i]}$.
\end{enumerate}
The last of these ties anchors us to the rest of category theory, so that rather than needing to ask for morphisms between combinatorial objects or between pictures, we can ask for morphisms between functors $\smset\to\smset$, and that's well established territory. Once we said that morphisms $p\to q$ between polynomial functors should just be natural transformations, we translated that back to the combinatorial and pictorial settings and saw the ``on-positions, on-directions'' description. We want to emphasize that we will be referring to each morphism (natural transformation) $p\to q$ between polynomials as a \emph{lens}.
In this chapter we introduced and examined morphisms between our polynomials.
First, we viewed our polynomials as functors and introduced the morphisms between them as natural transformations.
Next, we interpreted these morphisms in terms of positions and directions and viewed them pictorially.
We want to emphasize that we will be referring to each morphism (natural transformation) $p\to q$ between polynomials as a \emph{lens}.

Once $\poly$ was defined, we considered various properties it has, e.g.\ that it has all products and coproducts, and that these distribute: $\prod\sum\to\sum\prod$.
\begin{align*}
Expand All @@ -4238,9 +4240,7 @@ \section{Summary and further reading}
p_1\otimes p_2\coloneqq\sum_{i_1,i_2)\in p_1(1)\times p_2(1)}\yon^{p_1[i_1]\times p_2[i_2]}
\]

There are many fine sources on polynomial functors. Some of the computer science literature is more relaxed about what a polynomial is. For example, the ``coalgebra community'' often defines a polynomial to include finite power sets (see e.g.\ \cite{jacobs2017introduction}). Other computer science communities use the same definition of polynomial, but refer to it as a \emph{container} and use different words for its positions (they call them ``shapes'') and directions (they call them, rather unfortunately, ``positions''). See e.g.\ \cite{abbot2003categoriesthesis,abbott2005containers}.

But the notion of polynomial functors seems to have originated from Andr\'{e} Joyal. A good introduction to polynomial functors can be found in \cite{kock2012polynomial}; in particular the related work section on page~3 provides a nice survey of the field. A reader may also be interested in the Workshops on Polynomial Functors organized by the Topos Institute: \url{https://topos.site/p-func-workshop/}.
Variants of lenses are studied in compositional game theory \cite{hedges2016compositionality,hedges2017coherence,hedges2018limits,hedges2018morphisms}, in categorical database theory \cite{johnson2012lenses}, in functional programming and programming language theory \cite{bohannon2006relational,oconnor2011functor,abou2016reflections}, and in more generalized categorical settings \cite{gibbons2012relating,spivak2019generalized}.

%-------- Section --------%
\section{Exercise solutions}
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32 changes: 17 additions & 15 deletions refs.bib
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@@ -1,6 +1,6 @@
%1
@misc{1lab-poly, title={Cat.instances.poly}, url={https://1lab.dev/Cat.Instances.Poly.html}, journal={1Lab}, author={Liao, Amélia}, year={2022}, month={08}}
@misc{1lab-poly, title={Cat.instances.poly}, url={https://1lab.dev/Cat.Instances.Poly.html}, journal={1Lab}, author={Liao, Amélia}, year={2022}, month={08}}


%A
Expand Down Expand Up @@ -47,7 +47,7 @@ @article{abbott2005containers
}

@phdthesis{abbot2003categoriesthesis,
author = {Michael Gordon Abbott},
author = {Michael Gordon Abbott},
title = {Categories of Containers},
school = {University of Leicester},
year = 2003,
Expand Down Expand Up @@ -170,10 +170,10 @@ @book{Abiteboul:1995a
Date = {1995},
%Isbn = = {0-201-53771-0}
}


@phdthesis{Adam:2017a,
author = {Elie M.\ Adam},
author = {Elie M.\ Adam},
title = {Systems, Generativity and Interactional Effects},
school = {Massachusetts Institute of Technology},
year = 2017,
Expand Down Expand Up @@ -708,7 +708,7 @@ @inbook{Bourn.Gran:2004a
}

@phdthesis{brown1961some,
author = {Ronnie Brown},
author = {Ronnie Brown},
title = {Some problems of algebraic topology: a study of function spaces, function complexes, and FD-complexes},
school = {University of Oxford},
year = 1961,
Expand Down Expand Up @@ -773,7 +773,7 @@ @inproceedings{Canny.Kaltofen.Yagati:1989a
}

@misc{capucci2022diegetic,
url = {https://arxiv.org/abs/2206.12338},
url = {https://arxiv.org/abs/2206.12338},
author = {Capucci, Matteo},
title = {Diegetic representation of feedback in open games},
publisher = {arXiv},
Expand Down Expand Up @@ -826,7 +826,7 @@ @article{Censi:2016a
year = {2016},
%ignorepdf ={http://purl.org/censi/research/201510-monotone/1-MonotoneCodesign.pdf},
}

@article{censi:2017a,
author = "Censi, Andrea",
title = "Uncertainty in Monotone Co-Design Problems",
Expand All @@ -836,7 +836,7 @@ @article{censi:2017a
year = "2017",
%pdf = "https://arxiv.org/pdf/1609.03103"
}

@inproceedings{chandra1977optimal,
title={Optimal implementation of conjunctive queries in relational data bases},
author={Chandra, Ashok K and Merlin, Philip M},
Expand All @@ -846,7 +846,7 @@ @inproceedings{chandra1977optimal
organization={ACM}
}


@misc{cheng2004mathematics,
title={Mathematics, morally},
author={Cheng, Eugenia},
Expand All @@ -855,6 +855,8 @@ @misc{cheng2004mathematics
URL = {\url{http://www.cheng.staff.shef.ac.uk/morality/morality.pdf}}
}

@book{cheng_2022, place={Cambridge}, title={The Joy of Abstraction: An Exploration of Math, Category Theory, and Life}, DOI={10.1017/9781108769389}, publisher={Cambridge University Press}, author={Cheng, Eugenia}, year={2022}}

@article{Choy.Srinivasan.Cheu:2006a,
title={Neural networks for continuous online learning and control},
author={Choy, Min Chee and Srinivasan, Dipti and Cheu, Ruey Long},
Expand Down Expand Up @@ -1342,7 +1344,7 @@ @misc{fong2019abelian
}

@book{fong2019seven,
title={An Invitation to Applied Category Theory: Seven Sketches in Compositionality},
title={Seven Sketches in Compositionality: An Invitation to Applied Category Theory},
author={Fong, Brendan and Spivak, David I.},
year={2019},
publisher={Cambridge University Press},
Expand Down Expand Up @@ -2636,7 +2638,7 @@ @book{Nielson:1999:PPA:555142
%ISBN = = {3540654100},
publisher = {Springer-Verlag New York, Inc.},
address = {Secaucus, NJ, USA},
}
}

@book{nipkow2002isabelle,
title={Isabelle/HOL: a proof assistant for higher-order logic},
Expand Down Expand Up @@ -2689,7 +2691,7 @@ @misc{Nlab:nerve-and-realization


@misc{Nlab:symmeric-monoidal-category,
author = "nLab, Contributors To",
author = "nLab, Contributors To",
Title = {Symmetric monoidal category},
year=2018,
URL = "https://ncatlab.org/nlab/revision/symmetric+monoidal+category/30"
Expand Down Expand Up @@ -3284,7 +3286,7 @@ @misc{spivak2020dirichlet
}

@misc{spivak2022polynomial,
title={Polynomial functors and Shannon entropy},
title={Polynomial functors and Shannon entropy},
author={David I. Spivak},
year={2022},
eprint={2201.12878},
Expand All @@ -3305,7 +3307,7 @@ @article{Steenrod:1967
url = {http://dx.doi.org/10.1307/mmj/1028999711},
volume = {14},
year = {1967}
}
}

@article{Stewart.Parker:2007a,
title={Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations},
Expand Down

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