578 |
578 |
579 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
579 Given 1-morphisms $a$, $b$ and $c$ of $D$, we define the associator from $(a\bullet b)\bullet c$ to $a\bullet(b\bullet c)$ |
580 as in Figure \nn{like triangle 4.a, but more general; use three colors as in that fig}. |
580 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 $\cC$. |
581 This is just a reparameterization of the pinched product $(a\bullet b\bullet c)\times I$ of $\cC$. |
582 |
582 |
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583 Let $x,y,z$ be objects of $D$ and let $a:x\to y$ and $b:y\to z$ be 1-morphisms of $D$. |
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584 We have already defined above |
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585 structure maps $u:a\bullet \id_y\to a$ and $v:\id_y\bullet b\to b$, as well as an associator |
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586 $\alpha: (a\bullet \id_y)\bullet b\to a\bullet(\id_y\bullet b)$, as shown in |
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587 Figure \nn{triangle 4.b, 4.c and 4.a; note change from ``assoc" to ``$\alpha$"}. |
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588 (See also Figures \ref{fzo2} and \nn{previous associator fig}.) |
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589 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ |
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590 is equal to the composition of $\alpha$ and $\id_a\bullet v$. |
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591 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.) |
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592 |
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593 The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \nn{triangle 4.d and 4.e} |
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594 (see also Figure \ref{fzo5}). |
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595 The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \nn{triangle 4.f}. |
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596 Figure \nn{triangle 5} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from |
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597 Figure \nn{ref to 4.f above} by an isotopy rel boundary. |
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598 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) |
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599 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1). |
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602 |
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583 |
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584 |
605 |
585 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
606 %\nn{need to find a list of axioms for pivotal 2-cats to check} |
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607 |
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608 |
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610 |
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611 |
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612 |
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613 |
586 |
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587 |
615 |
588 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
616 \subsection{\texorpdfstring{$A_\infty$}{A-infinity} 1-categories} |
589 \label{sec:comparing-A-infty} |
617 \label{sec:comparing-A-infty} |
590 In this section, we make contact between the usual definition of an $A_\infty$ category |
618 In this section, we make contact between the usual definition of an $A_\infty$ category |