43 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
43 One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
44 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
44 This is described in more detail in \S \ref{sec:example:traditional-n-categories(fields)}. |
45 \end{example} |
45 \end{example} |
46 |
46 |
47 Now for the rest of the definition of system of fields. |
47 Now for the rest of the definition of system of fields. |
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48 (Readers desiring a more precise definition should refer to Subsection \ref{ss:n-cat-def} |
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49 and replace $k$-balls with $k$-manifolds.) |
48 \begin{enumerate} |
50 \begin{enumerate} |
49 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
51 \item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
50 and these maps are a natural |
52 and these maps comprise a natural |
51 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
53 transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
52 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
54 For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
53 $\cC(X)$ which restricts to $c$. |
55 $\cC(X)$ which restricts to $c$. |
54 In this context, we will call $c$ a boundary condition. |
56 In this context, we will call $c$ a boundary condition. |
55 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
57 \item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
56 \item $\cC_k$ is compatible with the symmetric monoidal |
58 \item $\cC_k$ is compatible with the symmetric monoidal |
57 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
59 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
58 compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
60 compatibly with homeomorphisms and restriction to boundary. |
59 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
61 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
60 restriction maps. |
62 restriction maps. |
61 \item Gluing without corners. |
63 \item Gluing without corners. |
62 Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
64 Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
63 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
65 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
64 Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
66 Using the boundary restriction and disjoint union |
65 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
67 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
66 copies of $Y$ in $\bd X$. |
68 copies of $Y$ in $\bd X$. |
67 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
69 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
68 Then (here's the axiom/definition part) there is an injective ``gluing" map |
70 Then (here's the axiom/definition part) there is an injective ``gluing" map |
69 \[ |
71 \[ |
70 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
72 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
71 \] |
73 \] |
72 and this gluing map is compatible with all of the above structure (actions |
74 and this gluing map is compatible with all of the above structure (actions |
73 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
75 of homeomorphisms, boundary restrictions, disjoint union). |
74 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
76 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
75 the gluing map is surjective. |
77 the gluing map is surjective. |
76 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
78 We say that fields on $X\sgl$ in the image of the gluing map |
77 gluing surface, we say that fields in the image of the gluing map |
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78 are transverse to $Y$ or splittable along $Y$. |
79 are transverse to $Y$ or splittable along $Y$. |
79 \item Gluing with corners. |
80 \item Gluing with corners. |
80 Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
81 Let $\bd X = Y \cup Y \cup W$, where the two copies of $Y$ and |
81 Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
82 $W$ might intersect along their boundaries. |
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83 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
82 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
84 Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
83 (without corners) along two copies of $\bd Y$. |
85 (without corners) along two copies of $\bd Y$. |
84 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
86 Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a splittable field on $W\sgl$ and let |
85 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
87 $c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
86 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
88 Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
95 \] |
97 \] |
96 and this gluing map is compatible with all of the above structure (actions |
98 and this gluing map is compatible with all of the above structure (actions |
97 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
99 of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
98 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
100 Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
99 the gluing map is surjective. |
101 the gluing map is surjective. |
100 From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
102 We say that fields in the image of the gluing map |
101 gluing surface, we say that fields in the image of the gluing map |
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102 are transverse to $Y$ or splittable along $Y$. |
103 are transverse to $Y$ or splittable along $Y$. |
103 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
104 \item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
104 $c \mapsto c\times I$. |
105 $c \mapsto c\times I$. |
105 These maps comprise a natural transformation of functors, and commute appropriately |
106 These maps comprise a natural transformation of functors, and commute appropriately |
106 with all the structure maps above (disjoint union, boundary restriction, etc.). |
107 with all the structure maps above (disjoint union, boundary restriction, etc.). |
186 \label{sec:example:traditional-n-categories(fields)} |
187 \label{sec:example:traditional-n-categories(fields)} |
187 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled |
188 We now describe in more detail Example \ref{ex:traditional-n-categories(fields)}, systems of fields coming from sub-cell-complexes labeled |
188 by $n$-category morphisms. |
189 by $n$-category morphisms. |
189 |
190 |
190 Given an $n$-category $C$ with the right sort of duality |
191 Given an $n$-category $C$ with the right sort of duality |
191 (e.g. a pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
192 (e.g. a pivotal 2-category, 1-category with duals, star 1-category), |
192 we can construct a system of fields as follows. |
193 we can construct a system of fields as follows. |
193 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
194 Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
194 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
195 with codimension $i$ cells labeled by $i$-morphisms of $C$. |
195 We'll spell this out for $n=1,2$ and then describe the general case. |
196 We'll spell this out for $n=1,2$ and then describe the general case. |
196 |
197 |