| changeset 703 | 2313b05f4906 |
| parent 702 | 062dc08cdefd |
| 702:062dc08cdefd | 703:2313b05f4906 |
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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 (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. |
148 Now suppose $\cC$ is a (strict) pivotal $*$-$2$-category. (The usual definition in the literature is for a pivotal tensor category; by a pivotal $2$-category we mean to take the axioms for a pivotal tensor category, think of a tensor 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} |
198 Here's my attempt to make ``looks locally like a cell decomposition'' sensible. A conic stratification of $M$ is a stratification $$M_0 \subset M_1 \subset \cdots \subset M_n = M$$ |
198 Here's my attempt to make ``looks locally like a cell decomposition'' sensible. A conic stratification of $M$ is a stratification $$M_0 \subset M_1 \subset \cdots \subset M_n = M$$ |
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. |
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. |
200 |
200 |
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.) |
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.) |
202 |
202 |
203 It's interesting to think about the details of this definition in dimensions $3$ and maybe even $4$, but in practice we have so few examples of such higher categories that particular axiomatizations of `string diagrams' are not deeply important. |
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203 |
204 |
204 \section{Axioms for fields} |
205 \section{Axioms for fields} |
205 A $n$-dimensional system of fields and local relations $(\cF, \cU)$ enriched in a symmetric monoidal category $\cS$ consists of the following data: |
206 A $n$-dimensional system of fields and local relations $(\cF, \cU)$ enriched in a symmetric monoidal category $\cS$ consists of the following data: |
206 \begin{description} |
207 \begin{description} |
207 \item[fields] functors $\cF_k$ from $k$-manifolds (and homeomorphisms) to sets; |
208 \item[fields] functors $\cF_k$ from $k$-manifolds (and homeomorphisms) to sets; |
211 \item[identities] natural transformations $\times I: \cF_k \to (\cF_{k+1} \circ \times I)$; |
212 \item[identities] natural transformations $\times I: \cF_k \to (\cF_{k+1} \circ \times I)$; |
212 \item[local relations] a functor $\cU$ from $n$-balls (and homeomorphisms) to sets, so $\cU \subset \cF$. |
213 \item[local relations] a functor $\cU$ from $n$-balls (and homeomorphisms) to sets, so $\cU \subset \cF$. |
213 \end{description} |
214 \end{description} |
214 and these data satisfy the following properties: |
215 and these data satisfy the following properties: |
215 \begin{itemize} |
216 \begin{itemize} |
216 \item everything respects the symmetric monoidal structures on $k$-manifolds (disjoint union), sets, and $\cS$ $$\cF_k(A \sqcup B) = \cF_k(A) \times \cF_k(B);$$ |
217 \item everything respects the symmetric monoidal structures on $k$-manifolds (disjoint union), sets, and $\cS$: in particular, $$\cF_k(A \sqcup B) = \cF_k(A) \times \cF_k(B);$$ |
217 \item gluing is compatible with action of homeomorphisms; |
218 \item gluing is compatible with action of homeomorphisms; |
218 \item the local relations form an ideal under gluing; |
219 \item the local relations form an ideal under gluing; |
219 \item gluing is surjective up to isotopy; |
220 \item gluing is surjective up to isotopy; |
220 \item identities are compatible on the nose with everything in sight. |
221 \item identities are compatible on the nose with everything in sight. |
221 \end{itemize} |
222 \end{itemize} |
399 There is a map the other way, too. There isn't quite a map $\cF(X \bigcup_Y \selfarrow) \to \cF(X)$, since a field on $X \bigcup_Y \selfarrow$ need not be splittable along $Y$. Nevertheless, every field is isotopic to one that is splittable along $Y$, and combining this with the lemma above we obtain a map $\cF(X \bigcup_Y \selfarrow) / (\text{isotopy}) \to A(X) \Tensor_{A(Y)} \selfarrow$. We now need to show that this descends to a map from $A(X \bigcup_Y \selfarrow)$. Consider an field of the form $u \bullet f$, for some ball $B$ embedded in $X \bigcup_Y \selfarrow$ and $u \in \cU(B), f \in \cF(X \bigcup_Y \selfarrow \setminus B)$. Now $B$ might cross $Y$, but we can choose an isotopy of $X \bigcup_Y \selfarrow$ so that it doesn't. Thus $u \bullet f$ is sent to a field in $\cU(X)$, and is zero in $A(X) \Tensor_{A(Y)} \selfarrow$. |
400 There is a map the other way, too. There isn't quite a map $\cF(X \bigcup_Y \selfarrow) \to \cF(X)$, since a field on $X \bigcup_Y \selfarrow$ need not be splittable along $Y$. Nevertheless, every field is isotopic to one that is splittable along $Y$, and combining this with the lemma above we obtain a map $\cF(X \bigcup_Y \selfarrow) / (\text{isotopy}) \to A(X) \Tensor_{A(Y)} \selfarrow$. We now need to show that this descends to a map from $A(X \bigcup_Y \selfarrow)$. Consider an field of the form $u \bullet f$, for some ball $B$ embedded in $X \bigcup_Y \selfarrow$ and $u \in \cU(B), f \in \cF(X \bigcup_Y \selfarrow \setminus B)$. Now $B$ might cross $Y$, but we can choose an isotopy of $X \bigcup_Y \selfarrow$ so that it doesn't. Thus $u \bullet f$ is sent to a field in $\cU(X)$, and is zero in $A(X) \Tensor_{A(Y)} \selfarrow$. |
400 |
401 |
401 It's not too hard to see that these maps are mutual inverses. |
402 It's not too hard to see that these maps are mutual inverses. |
402 \end{proof} |
403 \end{proof} |
403 |
404 |
404 \subsubsection{Codimension 2 gluing} |
405 We can also state a codimension $2$ gluing formula, but even just defining what modules and tensor products over $2$-categories mean is painful. (Maybe I'll expand these notes in the unlikely event that I still have time in the second talk.) Our eventual notion of $n$-category will significantly alleviate this problem, but we still shy away from stating a nice gluing formula in all codimensions simply because the blob complex paper never defines a notion of equivalence of $k$-categories. We're pretty sure we're on the right track with this, however, and the statements are all relatively easy. |
405 |
406 |
406 \section{$n$-categories and fields} |
407 \section{$n$-categories and fields} |
407 Roughly, the data of a system of fields and local relations and the data of a disklike $n$-category (from \S 6) are intended to be equivalent. |
408 Roughly, the data of a system of fields and local relations and the data of a disklike $n$-category (from \S 6) are intended to be equivalent. |
408 |
409 |
409 You essentially recover the axioms for a disklike $n$-category by just remembering everything about $\cF(X) / \cU(X)$ for $X$ a ball. |
410 You essentially recover the axioms for a disklike $n$-category by just remembering everything about $\cF(X) / \cU(X)$ for $X$ a ball. |