2 |
2 |
3 \section{Comparing \texorpdfstring{$n$}{n}-category definitions} |
3 \section{Comparing \texorpdfstring{$n$}{n}-category definitions} |
4 \label{sec:comparing-defs} |
4 \label{sec:comparing-defs} |
5 |
5 |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
6 In \S\ref{sec:example:traditional-n-categories(fields)} we showed how to construct |
7 a disk-like $n$-category from a traditional $n$-category; the morphisms of the |
7 a topological $n$-category from a traditional $n$-category; the morphisms of the |
8 disk-like $n$-category are string diagrams labeled by the traditional $n$-category. |
8 topological $n$-category are string diagrams labeled by the traditional $n$-category. |
9 In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
9 In this appendix we sketch how to go the other direction, for $n=1$ and 2. |
10 The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms |
10 The basic recipe, given a disk-like $n$-category $\cC$, is to define the $k$-morphisms |
11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
11 of the corresponding traditional $n$-category to be $\cC(B^k)$, where |
12 $B^k$ is the {\it standard} $k$-ball. |
12 $B^k$ is the {\it standard} $k$-ball. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
13 One must then show that the axioms of \S\ref{ss:n-cat-def} imply the traditional $n$-category axioms. |
573 |
573 |
574 |
574 |
575 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
575 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
576 \label{sec:comparing-A-infty} |
576 \label{sec:comparing-A-infty} |
577 In this section, we make contact between the usual definition of an $A_\infty$ category |
577 In this section, we make contact between the usual definition of an $A_\infty$ category |
578 and our definition of an $A_\infty$ disk-like $1$-category, from \S \ref{ss:n-cat-def}. |
578 and our definition of a disk-like $A_\infty$ $1$-category, from \S \ref{ss:n-cat-def}. |
579 |
579 |
580 \medskip |
580 \medskip |
581 |
581 |
582 Given an $A_\infty$ disk-like $1$-category $\cC$, we define an ``$m_k$-style" |
582 Given a disk-like $A_\infty$ $1$-category $\cC$, we define an ``$m_k$-style" |
583 $A_\infty$ $1$-category $A$ as follows. |
583 $A_\infty$ $1$-category $A$ as follows. |
584 The objects of $A$ are $\cC(pt)$. |
584 The objects of $A$ are $\cC(pt)$. |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
619 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
619 Corresponding to this decomposition the operad action gives a map $\mu: A\ot A\to A$. |
620 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
620 Define the gluing map to send $(f_1, a_1)\ot (f_2, a_2)$ to $(g, \mu(a_1\ot a_2))$. |
621 Operad associativity for $A$ implies that this gluing map is independent of the choice of |
621 Operad associativity for $A$ implies that this gluing map is independent of the choice of |
622 $g$ and the choice of representative $(f_i, a_i)$. |
622 $g$ and the choice of representative $(f_i, a_i)$. |
623 |
623 |
624 It is straightforward to verify the remaining axioms for a $A_\infty$ disk-like 1-category. |
624 It is straightforward to verify the remaining axioms for a disk-like $A_\infty$ 1-category. |
625 |
625 |
626 |
626 |
627 |
627 |
628 |
628 |
629 |
629 |