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1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
4 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
4 \def\mmpar#1#2#3{\smallskip\noindent{\bf #1} (#2). {\it #3} \smallskip} |
5 |
5 |
6 \section{$n$-categories and their modules} |
6 \section{\texorpdfstring{$n$}{n}-categories and their modules} |
7 \label{sec:ncats} |
7 \label{sec:ncats} |
8 |
8 |
9 \subsection{Definition of $n$-categories} |
9 \subsection{Definition of \texorpdfstring{$n$}{n}-categories} |
10 \label{ss:n-cat-def} |
10 \label{ss:n-cat-def} |
11 |
11 |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
658 \end{align*} |
658 \end{align*} |
659 This $n$-category can be thought of as the local part of the fields. |
659 This $n$-category can be thought of as the local part of the fields. |
660 Conversely, given a topological $n$-category we can construct a system of fields via |
660 Conversely, given a topological $n$-category we can construct a system of fields via |
661 a colimit construction; see \S \ref{ss:ncat_fields} below. |
661 a colimit construction; see \S \ref{ss:ncat_fields} below. |
662 |
662 |
663 \subsection{Examples of $n$-categories} |
663 \subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
664 \label{ss:ncat-examples} |
664 \label{ss:ncat-examples} |
665 |
665 |
666 |
666 |
667 We now describe several classes of examples of $n$-categories satisfying our axioms. |
667 We now describe several classes of examples of $n$-categories satisfying our axioms. |
668 We typically specify only the morphisms; the rest of the data for the category |
668 We typically specify only the morphisms; the rest of the data for the category |
1513 on the choice of 1-ball $J$. |
1513 on the choice of 1-ball $J$. |
1514 |
1514 |
1515 We will define a more general self tensor product (categorified coend) below. |
1515 We will define a more general self tensor product (categorified coend) below. |
1516 |
1516 |
1517 |
1517 |
1518 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1518 \subsection{Morphisms of \texorpdfstring{$A_\infty$}{A-infinity} 1-category modules} |
1519 \label{ss:module-morphisms} |
1519 \label{ss:module-morphisms} |
1520 |
1520 |
1521 In order to state and prove our version of the higher dimensional Deligne conjecture |
1521 In order to state and prove our version of the higher dimensional Deligne conjecture |
1522 (\S\ref{sec:deligne}), |
1522 (\S\ref{sec:deligne}), |
1523 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1523 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1783 %certainly the simple kind (strictly commute with gluing) arise in nature.} |
1783 %certainly the simple kind (strictly commute with gluing) arise in nature.} |
1784 |
1784 |
1785 |
1785 |
1786 |
1786 |
1787 |
1787 |
1788 \subsection{The $n{+}1$-category of sphere modules} |
1788 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1789 \label{ssec:spherecat} |
1789 \label{ssec:spherecat} |
1790 |
1790 |
1791 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules" |
1791 In this subsection we define $n{+}1$-categories $\cS$ of ``sphere modules" |
1792 whose objects are $n$-categories. |
1792 whose objects are $n$-categories. |
1793 With future applications in mind, we treat simultaneously the big category |
1793 With future applications in mind, we treat simultaneously the big category |