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}, |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and |
37 \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product}, \ref{axiom:vcones} and |
38 \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace |
38 \ref{axiom:extended-isotopies}. |
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39 For an enriched $n$-category we add \ref{axiom:enriched}. |
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40 For an $A_\infty$ $n$-category, we replace |
39 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
41 Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. |
40 \nn{need to revise this after we're done rearranging the a-inf and enriched stuff} |
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41 |
42 |
42 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
43 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
43 for $k{-}1$-morphisms. |
44 for $k{-}1$-morphisms. |
44 Readers who prefer things to be presented in a strictly logical order should read this |
45 Readers who prefer things to be presented in a strictly logical order should read this |
45 subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
46 subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. |
982 a colimit construction; see \S \ref{ss:ncat_fields} below. |
983 a colimit construction; see \S \ref{ss:ncat_fields} below. |
983 |
984 |
984 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
985 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
985 Here's a summary of the definition which segregates the data from the properties. |
986 Here's a summary of the definition which segregates the data from the properties. |
986 |
987 |
987 An $n$-category consists of the following data: \nn{need to revise this list} |
988 An $n$-category consists of the following data: |
988 \begin{itemize} |
989 \begin{itemize} |
989 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
990 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
990 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
991 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
991 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
992 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
992 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
993 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
993 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$; |
994 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
994 \item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
995 %\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
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996 \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
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997 and collar maps (Axiom \ref{axiom:families}). |
995 \end{itemize} |
998 \end{itemize} |
996 The above data must satisfy the following conditions: |
999 The above data must satisfy the following conditions: |
997 \begin{itemize} |
1000 \begin{itemize} |
998 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
1001 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
999 restrictions (Axiom \ref{axiom:composition}). |
1002 restrictions (Axiom \ref{axiom:composition}). |
1000 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
1003 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
1001 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
1004 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
1002 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). |
1005 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). |
1003 \item If enriching in an auxiliary category, all of the data should be compatible |
1006 \item If enriching in an auxiliary category, all of the data should be compatible |
1004 with the auxiliary category structure on $\cC_n(X; c)$. |
1007 with the auxiliary category structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}). |
1005 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies (Axiom \ref{axiom:extended-isotopies}). |
1008 \item The possible splittings of a morphism satisfy various conditions (Axiom \ref{axiom:vcones}). |
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1009 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies |
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1010 and collar maps (Axiom \ref{axiom:extended-isotopies}). |
1006 \end{itemize} |
1011 \end{itemize} |
1007 |
1012 |
1008 |
1013 |
1009 \subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
1014 \subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
1010 \label{ss:ncat-examples} |
1015 \label{ss:ncat-examples} |