143 Note that if $S$ is an interval, we can interpret the string diagram as a recipe for a morphism in $\cC$, at least after we fix one boundary point as `incoming' and the other `outgoing'. There's a (half-)$1$-cell adjacent to the incoming boundary point, and another adjacent to the outgoing boundary point. These will be the source and target of the morphism we build. Flip all the transverse orientations of the $0$-cells so they are compatible with the overall orientation of the interval. Now we simply compose all the morphisms living on the $0$-cells. |
143 Note that if $S$ is an interval, we can interpret the string diagram as a recipe for a morphism in $\cC$, at least after we fix one boundary point as `incoming' and the other `outgoing'. There's a (half-)$1$-cell adjacent to the incoming boundary point, and another adjacent to the outgoing boundary point. These will be the source and target of the morphism we build. Flip all the transverse orientations of the $0$-cells so they are compatible with the overall orientation of the interval. Now we simply compose all the morphisms living on the $0$-cells. |
144 |
144 |
145 If $\cC$ were a $*$-algebra (i.e., it has only one $0$-morphism) we could forget the labels on the $1$-cells, and a string diagram would just consist of a finite collection of oriented points in the interior, labelled by elements of the algebra, up to flipping an orientation and taking $*$ of the corresponding element. |
145 If $\cC$ were a $*$-algebra (i.e., it has only one $0$-morphism) we could forget the labels on the $1$-cells, and a string diagram would just consist of a finite collection of oriented points in the interior, labelled by elements of the algebra, up to flipping an orientation and taking $*$ of the corresponding element. |
146 |
146 |
147 \subsubsection{$n=2$} |
147 \subsubsection{$n=2$} |
148 Now suppose $\cC$ is a pivotal $2$-category. (The usual definition in the literature is for a pivotal monoidal category; by a pivotal $2$-category we mean to take the axioms for a pivotal monoidal category, think of a monoidal category as a $2$-category with only one object, then forget that restriction. There is an unfortunate other use of the phrase `pivotal $2$-category' in the literature, which actually refers to a $3$-category, but that's their fault.) |
148 Now suppose $\cC$ is a (strict) pivotal $*$-$2$-category. (The usual definition in the literature is for a pivotal monoidal category; by a pivotal $2$-category we mean to take the axioms for a pivotal monoidal category, think of a monoidal category as a $2$-category with only one object, then forget that restriction. There is an unfortunate other use of the phrase `pivotal $2$-category' in the literature, which actually refers to a $3$-category, but that's their fault.) The $*$ here means that in addition to being able to rotate $2$-morphisms via the pivotal structure, we can also reflect them. |
149 |
149 |
150 A string diagram on a $0$-manifold is a labeling of each point by an object (a.k.a. a $0$-morphism) of $\cC$. A string diagram on a $1$-manifold is exactly as in the $n=1$ case, with labels taken from the $0$- and $1$-morphisms of $\cC$. |
150 A string diagram on a $0$-manifold is a labeling of each point by an object (a.k.a. a $0$-morphism) of $\cC$. A string diagram on a $1$-manifold is exactly as in the $n=1$ case, with labels taken from the $0$- and $1$-morphisms of $\cC$. |
151 |
151 |
152 A string diagram on a $2$-manifold $Y$ consists of |
152 A string diagram on a $2$-manifold $Y$ consists of |
153 \begin{itemize} |
153 \begin{itemize} |
154 \item a cell decomposition of $Y$: the $1$-skeleton is a graph embedded in $Y$, but the $2$-cells don't need to be balls. |
154 \item a `generalized cell decomposition' of $Y$: the $1$-skeleton is a graph embedded in $Y$, but the $2$-cells don't need to be balls. |
155 \item a $0$-morphism of $\cC$ on each $2$-cell; |
155 \item a $0$-morphism of $\cC$ on each $2$-cell; |
156 \item a transverse orientation of each $1$-cell; |
156 \item a transverse orientation of each $1$-cell; |
157 \item a $1$-morphism of $\cC$ on each $1$-cell, with source and target given by the labels on the $2$-cells on the incoming and outgoing sides; |
157 \item a $1$-morphism of $\cC$ on each $1$-cell, with source and target given by the labels on the $2$-cells on the incoming and outgoing sides; |
158 \item for each $0$-cell, a homeomorphism of its link to $S^1$ (this is `the boundary of the standard $2$-bihedron') such that none of the intersections of $1$-cells with the link are sent to $\pm 1$ (this is the `general position' requirement; the points $\pm 1$ are special, as part of the structure of a standard bihedron); |
158 \item for each $0$-cell, a homeomorphism of its link to $S^1$ (this is `the boundary of the standard $2$-bihedron') such that none of the intersections of $1$-cells with the link are sent to $\pm 1$ (this is the `general position' requirement; the points $\pm 1$ are special, as part of the structure of a standard bihedron); |
159 \item a $2$-morphism of $\cC$ for each $0$-cell, with source and target given by the labels of the $1$-cells crossing the incoming and outgoing faces of the bihedron. |
159 \item a $2$-morphism of $\cC$ for each $0$-cell, with source and target given by the labels of the $1$-cells crossing the incoming and outgoing faces of the bihedron. |
160 \end{itemize} |
160 \end{itemize} |
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161 |
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162 You can see here why we can't insist on an actual cell decomposition: asking that the $2$-cells are balls is a non-local condition, so we wouldn't be able to glue fields together. |
161 |
163 |
162 Let's spell out this stuff about bihedra. Suppose the neighborhood of a $0$-cell looks like the following. |
164 Let's spell out this stuff about bihedra. Suppose the neighborhood of a $0$-cell looks like the following. |
163 $$ |
165 $$ |
164 \begin{tikzpicture} |
166 \begin{tikzpicture} |
165 \draw[fill] (0,0) circle (0.5mm) node[anchor=north west] (x) {$x$}; |
167 \draw[fill] (0,0) circle (0.5mm) node[anchor=north west] (x) {$x$}; |
191 Finally, when $Y$ is a ball, how do we interpret a string diagram on $Y$ as a $2$-morphism in $\cC$? First choose a parametrization of $Y$ as a standard bihedron; now `sweep out' the interior of $Y$. We'll build a $2$-morphism from the tensor product of the $1$-morphisms labeling the $1$-cells meeting the lower boundary to the tensor product of the $1$-morphisms labelling the upper boundary. As we pass critical points in the $1$-cells, apply a pairing or copairing map from the category. As we pass $0$-cells, modify the parametrization to match the direction we're sweeping out, and compose with the label of the $0$-cell, acting on the appropriate tensor factors. |
193 Finally, when $Y$ is a ball, how do we interpret a string diagram on $Y$ as a $2$-morphism in $\cC$? First choose a parametrization of $Y$ as a standard bihedron; now `sweep out' the interior of $Y$. We'll build a $2$-morphism from the tensor product of the $1$-morphisms labeling the $1$-cells meeting the lower boundary to the tensor product of the $1$-morphisms labelling the upper boundary. As we pass critical points in the $1$-cells, apply a pairing or copairing map from the category. As we pass $0$-cells, modify the parametrization to match the direction we're sweeping out, and compose with the label of the $0$-cell, acting on the appropriate tensor factors. |
192 |
194 |
193 As usual for fields based on string diagrams, the corresponding local relations are exactly the kernel of this `evaluation' map. |
195 As usual for fields based on string diagrams, the corresponding local relations are exactly the kernel of this `evaluation' map. |
194 |
196 |
195 \subsection{Conic stratifications} |
197 \subsection{Conic stratifications} |
196 Ugh. Here's my attempt to make ``looks locally like a cell decomposition'' sensible. A conic stratification of $M$ is a stratification $$M_0 \subset M_1 \subset \cdots \subset M_n = M$$ |
198 Here's my attempt to make ``looks locally like a cell decomposition'' sensible. A conic stratification of $M$ is a stratification $$M_0 \subset M_1 \subset \cdots \subset M_n = M$$ |
197 (so $M_k \setminus M_{k-1}$ is a $k$-manifold, the connected components of which we'll still call $k$-cells, even though they need not be balls), which has a certain local model. |
199 (so $M_k \setminus M_{k-1}$ is a $k$-manifold, the connected components of which we'll still call $k$-cells, even though they need not be balls), which has a certain local model. |
198 |
200 |
199 Any point on $k$-cell has a neighborhood $U$ which is homeomorphic to $B^k \times \Cone(X)$, where $X$ is some conic stratification of $S^{n-k-1}$, and this homeomorphism preserves strata. (In $B^k \times \Cone(X)$, there are no strata below level $k$, the cone points are the $k$-strata, and the points over the $i$-strata of $X$ form the $i+k+1$ strata.) |
201 Any point on $k$-cell has a neighborhood $U$ which is homeomorphic to $B^k \times \Cone(X)$, where $X$ is some conic stratification of $S^{n-k-1}$, and this homeomorphism preserves strata. (In $B^k \times \Cone(X)$, there are no strata below level $k$, the cone points are the $k$-strata, and the points over the $i$-strata of $X$ form the $i+k+1$ strata.) |
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202 |
200 |
203 |
201 \section{Axioms for fields} |
204 \section{Axioms for fields} |
202 A $n$-dimensional system of fields and local relations $(\cF, \cU)$ enriched in a symmetric monoidal category $\cS$ consists of the following data: |
205 A $n$-dimensional system of fields and local relations $(\cF, \cU)$ enriched in a symmetric monoidal category $\cS$ consists of the following data: |
203 \begin{description} |
206 \begin{description} |
204 \item[fields] functors $\cF_k$ from $k$-manifolds (and homeomorphisms) to sets; |
207 \item[fields] functors $\cF_k$ from $k$-manifolds (and homeomorphisms) to sets; |
205 \item[boundaries] natural transformations $\bdy : \cF_k \to (\cF_{k-1} \circ \bdy)$; |
208 \item[boundaries] natural transformations $\bdy : \cF_k \to (\cF_{k-1} \circ \bdy)$; |
206 \item[structure] the structure of an object of $\cS$ on each set $\cF_n(X; c)$, and below, appropriate compatibility at the level of morphisms; |
209 \item[structure] the structure of an object of $\cS$ on each set $\cF_n(X; c)$, and below, appropriate compatibility at the level of morphisms; |
207 \item[gluing] when $\bdy X = (Y \sqcup Y) \cup Z$, there is an injective map $$\cF_k(X; y \bullet y \bullet z) \into \cF_k(X \bigcup_Y \selfarrow; z)$$ for each $y \in \cF_{k-1}(Y), z \in \cF_{k-1}(Z)$; |
210 \item[gluing] when $\bdy X = (Y \sqcup Y) \cup Z$, there is an injective map $$\cF_k(X; y \bullet y \bullet z) \into \cF_k(X \bigcup_Y \selfarrow; z)$$ for each $y \in \cF_{k-1}(Y), z \in \cF_{k-1}(Z)$; |
208 \item[identities] natural transformations $\times I: \cF_k \to (\cF_{k+1} \circ \times I)$; |
211 \item[identities] natural transformations $\times I: \cF_k \to (\cF_{k+1} \circ \times I)$; |
209 \item[local relations] a functor $\cU$ from $n$-balls (and homeomorphisms) to sets, so $\cU \subset \cF$; |
212 \item[local relations] a functor $\cU$ from $n$-balls (and homeomorphisms) to sets, so $\cU \subset \cF$. |
210 \end{description} |
213 \end{description} |
211 and these data satisfy the following properties: |
214 and these data satisfy the following properties: |
212 \begin{itemize} |
215 \begin{itemize} |
213 \item everything respects the symmetric monoidal structures on $k$-manifolds (disjoint union), sets, and $\cS$ $$\cF_k(A \sqcup B) = \cF_k(A) \times \cF_k(B);$$ |
216 \item everything respects the symmetric monoidal structures on $k$-manifolds (disjoint union), sets, and $\cS$ $$\cF_k(A \sqcup B) = \cF_k(A) \times \cF_k(B);$$ |
214 \item gluing is compatible with action of homeomorphisms; |
217 \item gluing is compatible with action of homeomorphisms; |
215 \item the local relations form an ideal under gluing; |
218 \item the local relations form an ideal under gluing; |
216 \item ... gluing is surjective up to isotopy (collaring?) ... |
219 \item gluing is surjective up to isotopy; |
217 \item identities are compatible on the nose with everything in sight... |
220 \item identities are compatible on the nose with everything in sight. |
218 \end{itemize} |
221 \end{itemize} |
219 |
222 |
220 Actually in the `gluing' axiom above, the field $z$ on the right hand side actually needs to be interpreted as the image of $z$ under a gluing map one dimensional down, because it's now meant to be a field on $Z \bigcup_{\bdy Y} \selfarrow$. |
223 Actually in the `gluing' axiom above, the field $z$ on the right hand side actually needs to be interpreted as the image of $z$ under a gluing map one dimensional down, because it's now meant to be a field on $Z \bigcup_{\bdy Y} \selfarrow$. |
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224 |
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225 It's admittedly a little peculiar looking that we insist that gluing is surjective up to isotopy, but it's a feature of the examples and turns out to be useful (see the proof of the gluing formula below). We may actually want to relax this axiom even further: we didn't talk about this, but for systems of fields based on pasting diagrams (as opposed to string diagrams) for $n$-categories, we need to be able to `insert identities', as well as isotope, before gluing becomes surjective. `Inserting an identity' means cutting open a field somewhere that it is splittable, gluing on an identity morphism, then using a collaring morphism before gluing the field up again. Essentially the difference is that string diagrams `have identities everywhere', so they are always splittable after a small isotopy. |
221 |
226 |
222 \section{TQFT from fields} |
227 \section{TQFT from fields} |
223 Given a system of fields and local relations $\cF, \cU$, we define the corresponding vector space valued invariant of $n$-manifolds $A$ as follows. For $X$ an $n$-manifold, write $\cU(X)$ for the subspace of $\cF(X)$ consisting of the span of the images of a gluing map $\cU(B; c) \tensor \cF(X \setminus B; c)$ for any embedded $n$-ball $B \subset X$, and boundary field $c \in \cF(\bdy B)$. We then define |
228 Given a system of fields and local relations $\cF, \cU$, we define the corresponding vector space valued invariant of $n$-manifolds $A$ as follows. For $X$ an $n$-manifold, write $\cU(X)$ for the subspace of $\cF(X)$ consisting of the span of the images of a gluing map $\cU(B; c) \tensor \cF(X \setminus B; c)$ for any embedded $n$-ball $B \subset X$, and boundary field $c \in \cF(\bdy B)$. We then define |
224 $$A(X) = \cF(X) / \cU(X).$$ |
229 $$A(X) = \cF(X) / \cU(X).$$ |
225 It's clear that homeomorphisms of $X$ act on this space. Actually, this collapses to an action of the mapping class group: |
230 It's clear that homeomorphisms of $X$ act on this space. Actually, this collapses to an action of the mapping class group: |