668 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
668 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
669 \end{align*} |
669 \end{align*} |
670 This $n$-category can be thought of as the local part of the fields. |
670 This $n$-category can be thought of as the local part of the fields. |
671 Conversely, given a topological $n$-category we can construct a system of fields via |
671 Conversely, given a topological $n$-category we can construct a system of fields via |
672 a colimit construction; see \S \ref{ss:ncat_fields} below. |
672 a colimit construction; see \S \ref{ss:ncat_fields} below. |
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673 |
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674 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
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675 Here's a summary of the definition which segregates the data from the properties. |
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676 |
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677 An $n$-category consists of the following data: |
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678 \begin{itemize} |
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679 \item Functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}) |
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680 \item Boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}) |
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681 \item Composition/gluing maps $\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B_1\cup_Y B_2)_E$ (Axiom \ref{axiom:composition}) |
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682 \item Product/identity maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}) |
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683 \item If enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ |
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684 \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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685 \end{itemize} |
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686 The above data must satisfy the following conditions: |
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687 \begin{itemize} |
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688 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
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689 restrictions (Axiom \ref{axiom:composition}). |
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690 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
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691 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
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692 \item The product maps are associative and also compatible with homeomorphism actions, gluing and restriction (Axiom \ref{axiom:product}). |
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693 \item If enriching in an auxiliary category, all of the data should be compatible |
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694 with the auxiliary category structure on $\cC_n(X; c)$. |
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695 \item For ordinary categories, invariance of $n$-morphisms under extended isotopies (Axiom \ref{axiom:extended-isotopies}). |
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696 \end{itemize} |
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697 |
673 |
698 |
674 \subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
699 \subsection{Examples of \texorpdfstring{$n$}{n}-categories} |
675 \label{ss:ncat-examples} |
700 \label{ss:ncat-examples} |
676 |
701 |
677 |
702 |