32 

    32 

    33 \medskip

    33 \medskip

    34 

    34 

    35 The axioms for an $n$-category are spread throughout this section.

    35 The axioms for an $n$-category are spread throughout this section.

    36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, 

    36 Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, 

    38 For an enriched $n$-category we add Axiom \ref{axiom:enriched}.

    38 For an enriched $n$-category we add Axiom \ref{axiom:enriched}.

    39 For an $A_\infty$ $n$-category, we replace 

    39 For an $A_\infty$ $n$-category, we replace 

    40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.

    40 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}.

    41 

    41 

    42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms 

    42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms 

   670 In addition, collar maps act trivially on $\cC(X)$.

   670 In addition, collar maps act trivially on $\cC(X)$.

   671 \end{axiom}

   671 \end{axiom}

   672 

   672 

   673 \medskip

   673 \medskip

   674 

   674 

   679 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.

   700 If $P$ is a poset let $P\times I$ denote the product poset, where $I = \{0, 1\}$ with ordering $0\le 1$.

   680 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.

   701 Let $\Cone(P)$ denote $P$ adjoined an additional object $v$ (the vertex of the cone) with $p\le v$ for all objects $p$ of $P$.

   681 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.

   702 Finally, let $\vcone(P)$ denote $P\times I \cup \Cone(P)$, where we identify $P\times \{0\}$ with the base of the cone.

   682 We call $P\times \{1\}$ the base of $\vcone(P)$.

   703 We call $P\times \{1\}$ the base of $\vcone(P)$.

   683 (See Figure \ref{vcone-fig}.)

   704 (See Figure \ref{vcone-fig}.)

   784 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}

   805 \caption{(a) $P$, (b) $P\times I$, (c) $\Cone(P)$, (d) $\vcone(P)$}

   785 \label{vcone-fig}

   806 \label{vcone-fig}

   786 \end{figure}

   807 \end{figure}

   787 

   808 

   788 

   809 

   789 \begin{axiom}[Splittings]

   942 \begin{axiom}[Splittings]

   790 \label{axiom:vcones}

   943 \label{axiom:vcones}

   791 Let $c\in \cC_k(X)$, with $0\le k < n$, and

   944 Let $c\in \cC_k(X)$, with $0\le k < n$, and

   792 let $P$ be a finite poset of splittings of $c$.

   945 let $P$ be a finite poset of splittings of $c$.

   793 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   946 Then we can embed $\vcone(P)$ into the splittings of $c$, with $P$ corresponding to the base of $\vcone(P)$.

   809 poset of splittings of $c$ is connected.

   962 poset of splittings of $c$ is connected.

   810 Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).

   963 Note that we do not require that any two splittings of $c$ have a common refinement (i.e.\ replace the letter W with the letter V).

   811 Two decompositions of $X$ might intersect in a very messy way, but one can always find a third

   964 Two decompositions of $X$ might intersect in a very messy way, but one can always find a third

   812 decomposition which has common refinements with each of the original two decompositions.

   965 decomposition which has common refinements with each of the original two decompositions.

   813 

   966 

   814 

   968 

   815 \medskip

   969 \medskip

   816 

   970 

   817 This completes the definition of an $n$-category.

   971 This completes the definition of an $n$-category.

   818 Next we define enriched $n$-categories.

   972 Next we define enriched $n$-categories.

  1022 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}).

  1176 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}).

  1023 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}).

  1177 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}).

  1024 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}).

  1178 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}).

  1025 \item If enriching in an auxiliary category, all of the data should be compatible 

  1179 \item If enriching in an auxiliary category, all of the data should be compatible 

  1026 with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}).

  1180 with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}).

  1028 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies 

  1182 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies 

  1029 and collar maps (Axiom \ref{axiom:extended-isotopies}).

  1183 and collar maps (Axiom \ref{axiom:extended-isotopies}).

  1030 \end{itemize}

  1184 \end{itemize}

  1031 

  1185 

  1032 

  1186 

  1462 We want to define $y_1\bullet y_2 \in \cl\cC(W)$.

  1616 We want to define $y_1\bullet y_2 \in \cl\cC(W)$.

  1463 Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements 

  1617 Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements 

  1464 $y_{ia} \in \cC(X_{ia})$ representing $y_i$.

  1618 $y_{ia} \in \cC(X_{ia})$ representing $y_i$.

  1465 It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$,

  1619 It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$,

  1466 since intersections of the pieces with $\bd W$ might not be well-behaved.

  1620 since intersections of the pieces with $\bd W$ might not be well-behaved.

  1468 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement

  1622 we can choose the decomposition $\du_{a} X_{ia}$ so that its restriction to $\bd W_i$ is a refinement

  1469 of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$

  1623 of the splitting along $\bd Y$, and this implies that the combined decomposition $\du_{ia} X_{ia}$

  1470 is permissible.

  1624 is permissible.

  1472 shows that this is independent of the choices of representatives of $y_i$.

  1626 shows that this is independent of the choices of representatives of $y_i$.

  1473 

  1627 

  1474 

  1628 

  1475 \medskip

  1629 \medskip

  1476 

  1630