3 \section{TQFTs via fields} |
3 \section{TQFTs via fields} |
4 \label{sec:fields} |
4 \label{sec:fields} |
5 \label{sec:tqftsviafields} |
5 \label{sec:tqftsviafields} |
6 |
6 |
7 In this section we review the notion of a ``system of fields and local relations". |
7 In this section we review the notion of a ``system of fields and local relations". |
8 For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-TQFT}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
8 For more details see \cite{kw:tqft}. From a system of fields and local relations we can readily construct TQFT invariants of manifolds. This is described in \S \ref{sec:constructing-a-tqft}. A system of fields is very closely related to an $n$-category. In Example \ref{ex:traditional-n-categories(fields)}, which runs throughout this section, we sketch the construction of a system of fields from an $n$-category. We make this more precise for $n=1$ or $2$ in \S \ref{sec:example:traditional-n-categories(fields)}, and much later, after we've have given our own definition of a `topological $n$-category' in \S \ref{sec:ncats}, we explain precisely how to go back and forth between a topological $n$-category and a system of fields and local relations. |
9 |
9 |
10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
10 We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
11 submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
12 $\overline{X \setmin Y}$. |
12 $\overline{X \setmin Y}$. |
13 |
13 |
19 $k$ and morphisms homeomorphisms. |
19 $k$ and morphisms homeomorphisms. |
20 (We could equally well work with a different category of manifolds --- |
20 (We could equally well work with a different category of manifolds --- |
21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
21 oriented, topological, smooth, spin, etc. --- but for definiteness we |
22 will stick with unoriented PL.) |
22 will stick with unoriented PL.) |
23 |
23 |
24 Fix a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. |
24 Fix a symmetric monoidal category $\cS$. While reading the definition, you should just think about the cases $\cS = \Set$ or $\cS = \Vect$. The presentation here requires that the objects of $\cS$ have an underlying set, but this could probably be avoided if desired. |
25 |
25 |
26 A $n$-dimensional {\it system of fields} in $\cS$ |
26 A $n$-dimensional {\it system of fields} in $\cS$ |
27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
28 together with some additional data and satisfying some additional conditions, all specified below. |
28 together with some additional data and satisfying some additional conditions, all specified below. |
29 |
29 |
324 |
324 |
325 In more detail, let $X$ be an $n$-manifold. |
325 In more detail, let $X$ be an $n$-manifold. |
326 %To harmonize notation with the next section, |
326 %To harmonize notation with the next section, |
327 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
327 %let $\bc_0(X)$ be the vector space of finite linear combinations of fields on $X$, so |
328 %$\bc_0(X) = \lf(X)$. |
328 %$\bc_0(X) = \lf(X)$. |
329 Define $U(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
329 \begin{defn} |
330 $U(X)$ is generated by things of the form $u\bullet r$, where |
330 \label{defn:TQFT-invariant} |
331 $u\in U(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
331 The TQFT invariant of $X$ associated to a system of fields $\cF$ and local relations $\cU$ is |
332 Define |
332 $$A(X) \deq \lf(X) / U(X),$$ |
333 \[ |
333 where $\cU(X) \sub \lf(X)$ to be the space of local relations in $\lf(X)$; |
334 A(X) \deq \lf(X) / U(X) . |
334 $\cU(X)$ is generated by things of the form $u\bullet r$, where |
335 \] |
335 $u\in \cU(B)$ for some embedded $n$-ball $B\sub X$ and $r\in \cC(X\setmin B)$. |
|
336 \end{defn} |
336 (The blob complex, defined in the next section, |
337 (The blob complex, defined in the next section, |
337 is in some sense the derived version of $A(X)$.) |
338 is in some sense the derived version of $A(X)$.) |
338 If $X$ has boundary we can similarly define $A(X; c)$ for each |
339 If $X$ has boundary we can similarly define $A(X; c)$ for each |
339 boundary condition $c\in\cC(\bd X)$. |
340 boundary condition $c\in\cC(\bd X)$. |
340 |
341 |