570 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
569 \draw[->, thick, orange!50!brown] (1.45,-1.1)-- node[left, black] {$f$} +(0,.7); |
571 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
570 \draw[->, thick, orange!50!brown] (4.35,.4)-- node[left, black] {$g$} +(0,.7); |
572 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
571 \draw[->, thick, blue!75!yellow] (1.5,.78) node[black, above] {$(b\cdot c)\times I$} -- (2.5,0); |
573 \end{tikzpicture}} |
572 \end{tikzpicture}} |
574 \end{equation*} |
573 \end{equation*} |
|
574 \begin{equation*} |
|
575 \mathfig{0.6}{triangle/triangle3b} |
|
576 \end{equation*} |
575 \caption{Horizontal composition of 2-morphisms} |
577 \caption{Horizontal composition of 2-morphisms} |
576 \label{fzo5} |
578 \label{fzo5} |
|
579 \end{figure} |
|
580 \begin{figure}[t] |
|
581 $$ |
|
582 \mathfig{0.6}{triangle/triangle3c} |
|
583 $$ |
|
584 \caption{Part of the proof that the four different horizontal compositions of 2-morphisms are equal.} |
|
585 \label{fig:horizontal-compositions-equal} |
577 \end{figure} |
586 \end{figure} |
578 |
587 |
579 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
588 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
580 as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. |
589 as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. |
581 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
590 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
|
591 \begin{figure}[t] |
|
592 $$ |
|
593 \mathfig{0.4}{triangle/triangle4a} |
|
594 $$ |
|
595 \nn{remember to change the labels} |
|
596 \caption{An associator.} |
|
597 \label{fig:associator} |
|
598 \end{figure} |
582 |
599 |
583 Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
600 Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
584 We have already defined above |
601 We have already defined above |
585 structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
602 structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
586 $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
603 $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
587 Figure \nn{triangle 4.b, 4.c and 4.a; note change from ``assoc" to ``$\alpha$"}. |
604 Figure \ref{fig:ingredients-triangle-axiom}. |
588 (See also Figures \ref{fzo2} and \nn{previous associator fig}.) |
605 (See also Figures \ref{fzo2} and \ref{fig:associator}.) |
589 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
606 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
590 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
607 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
591 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
608 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
592 |
609 \begin{figure}[t] |
593 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \nn{triangle 4.d and 4.e} |
610 \begin{align*} |
|
611 \mathfig{0.4}{triangle/triangle4a} \\ |
|
612 \mathfig{0.4}{triangle/triangle4b} \\ |
|
613 \mathfig{0.4}{triangle/triangle4c} |
|
614 \end{align*} |
|
615 \nn{remember to change `assoc' to $\alpha$} |
|
616 \caption{Ingredients for the triangle axiom.} |
|
617 \label{fig:ingredients-triangle-axiom} |
|
618 \end{figure} |
|
619 |
|
620 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition} |
594 (see also Figure \ref{fzo5}). |
621 (see also Figure \ref{fzo5}). |
595 The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \nn{triangle 4.f}. |
622 The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \ref{fig:vertical-composition}. |
596 Figure \nn{triangle 5} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
623 Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
597 Figure \nn{ref to 4.f above} by an isotopy rel boundary. |
624 Figure \ref{fig:vertical-composition} by an isotopy rel boundary. |
598 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
625 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
599 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
626 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
600 |
627 \begin{figure}[t] |
601 |
628 \begin{align*} |
602 |
629 \mathfig{0.4}{triangle/triangle4d} |
603 |
630 \mathfig{0.4}{triangle/triangle4e} |
604 |
631 \end{align*} |
605 |
632 \caption{Horizontal compositions in the triangle axiom.} |
606 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
633 \label{fig:horizontal-composition} |
607 |
634 \end{figure} |
608 |
635 \begin{figure}[t] |
609 |
636 \begin{align*} |
610 |
637 \mathfig{0.4}{triangle/triangle4f} |
611 |
638 \end{align*} |
612 |
639 \caption{Vertical composition in the triangle axiom.} |
|
640 \label{fig:vertical-composition} |
|
641 \end{figure} |
|
642 \begin{figure}[t] |
|
643 \begin{align*} |
|
644 \mathfig{0.4}{triangle/triangle5} |
|
645 \end{align*} |
|
646 \caption{Adding a collar in the proof of the triangle axiom.} |
|
647 \label{fig:adding-a-collar} |
|
648 \end{figure} |
613 |
649 |
614 |
650 |
615 |
651 |
616 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
652 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
617 \label{sec:comparing-A-infty} |
653 \label{sec:comparing-A-infty} |