equal
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93 Instead, we will combine the domain and range into a single entity which we call the |
93 Instead, we will combine the domain and range into a single entity which we call the |
94 boundary of a morphism. |
94 boundary of a morphism. |
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
97 $1\le k \le n$. |
97 $1\le k \le n$. |
98 At first might seem that we need another axiom for this, but in fact once we have |
98 At first it might seem that we need another axiom for this, but in fact once we have |
99 all the axioms in the subsection for $0$ through $k-1$ we can use a coend |
99 all the axioms in the subsection for $0$ through $k-1$ we can use a coend |
100 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
100 construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ |
101 to spheres (and any other manifolds): |
101 to spheres (and any other manifolds): |
102 |
102 |
103 \begin{prop} |
103 \begin{prop} |
105 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
105 For each $1 \le k \le n$, we have a functor $\cC_{k-1}$ from |
106 the category of $k{-}1$-spheres and |
106 the category of $k{-}1$-spheres and |
107 homeomorphisms to the category of sets and bijections. |
107 homeomorphisms to the category of sets and bijections. |
108 \end{prop} |
108 \end{prop} |
109 |
109 |
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110 We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in other Axioms at lower levels. |
110 |
111 |
111 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
112 %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. |
112 |
113 |
113 \begin{axiom}[Boundaries]\label{nca-boundary} |
114 \begin{axiom}[Boundaries]\label{nca-boundary} |
114 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
115 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cC_{k-1}(\bd X)$. |
513 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
514 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo |
514 homotopies fixed on $\bd X$. |
515 homotopies fixed on $\bd X$. |
515 (Note that homotopy invariance implies isotopy invariance.) |
516 (Note that homotopy invariance implies isotopy invariance.) |
516 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
517 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
517 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
518 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
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519 |
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520 Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. |
518 \end{example} |
521 \end{example} |
519 |
522 |
520 \begin{example}[Maps to a space, with a fiber] |
523 \begin{example}[Maps to a space, with a fiber] |
521 \rm |
524 \rm |
522 \label{ex:maps-to-a-space-with-a-fiber}% |
525 \label{ex:maps-to-a-space-with-a-fiber}% |
554 Define $\cC(X)$, for $\dim(X) < n$, |
557 Define $\cC(X)$, for $\dim(X) < n$, |
555 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
558 to be the set of all $C$-labeled sub cell complexes of $X\times F$. |
556 Define $\cC(X; c)$, for $X$ an $n$-ball, |
559 Define $\cC(X; c)$, for $X$ an $n$-ball, |
557 to be the dual Hilbert space $A(X\times F; c)$. |
560 to be the dual Hilbert space $A(X\times F; c)$. |
558 \nn{refer elsewhere for details?} |
561 \nn{refer elsewhere for details?} |
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562 |
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563 |
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564 Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. |
559 \end{example} |
565 \end{example} |
560 |
566 |
561 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
567 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
562 |
568 |
563 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
569 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |