77 There are two main examples which will motivate the precise definitions, so we'll go and understand these in some detail first. |
80 There are two main examples which will motivate the precise definitions, so we'll go and understand these in some detail first. |
78 |
81 |
79 \subsection{Maps to a target space} |
82 \subsection{Maps to a target space} |
80 Fixing a target space $T$, we can define a system of fields $\Maps(- \to T)$. Actually, it's best to modify this a bit, just in the top dimension, where we'll linearize in the following way: define $\Maps(X^n \to T)$ on an $n$-manifold $X$ to be \emph{formal linear combinations} of maps to $T$, extending a \emph{fixed} linear map on $\bdy X$. (That is, arbitrary boundary conditions are allowed, but we can only take linear combinations of maps with the same boundary conditions.) This will be a common feature for all `linear' systems of fields: at the top dimension the set associated to an $n$-manifold will break up into a vector space for each possible boundary condition. |
83 Fixing a target space $T$, we can define a system of fields $\Maps(- \to T)$. Actually, it's best to modify this a bit, just in the top dimension, where we'll linearize in the following way: define $\Maps(X^n \to T)$ on an $n$-manifold $X$ to be \emph{formal linear combinations} of maps to $T$, extending a \emph{fixed} linear map on $\bdy X$. (That is, arbitrary boundary conditions are allowed, but we can only take linear combinations of maps with the same boundary conditions.) This will be a common feature for all `linear' systems of fields: at the top dimension the set associated to an $n$-manifold will break up into a vector space for each possible boundary condition. |
81 |
84 |
82 What then are the local relations? We define $U(B)$, the local relations on an $n$-ball $B$, to be the subspace of $\Maps(B \to T)$ spanned by differences $f-g$ of maps which are homotopic rel boundary. |
85 What then are the local relations? We define $\cU(B)$, the local relations on an $n$-ball $B$, to be the subspace of $\Maps(B \to T)$ spanned by differences $f-g$ of maps which are homotopic rel boundary. |
83 |
86 |
84 Let's identify some useful features of this system of fields and local relations; later these will inspire the axioms. |
87 Let's identify some useful features of this system of fields and local relations; later these will inspire the axioms. |
85 |
88 |
86 \begin{description} |
89 \begin{description} |
87 \item[Boundaries] We can restrict $f: X \to T$ to a map $\bdy f: \bdy X \to T$. |
90 \item[Boundaries] We can restrict $f: X \to T$ to a map $\bdy f: \bdy X \to T$. |
89 \item[Relations form an ideal] |
92 \item[Relations form an ideal] |
90 Suppose $X$ and $Y$ are $n$-balls, and we can glue them together to form another $n$-ball $X \cup_S Y$. |
93 Suppose $X$ and $Y$ are $n$-balls, and we can glue them together to form another $n$-ball $X \cup_S Y$. |
91 If $f, g: X \to T$ are homotopic maps, and $h: Y \to T$ is an arbitrary map, and all agree on the $(n-1)$-ball $S$, then $f \bullet_S h$ and $g \bullet_S h$ are again homotopic to each other. Said otherwise, $f-g$ was a local relation on $X$, and $(f-g) \bullet_S h$ is a local relation on $X \cup_S Y$. |
94 If $f, g: X \to T$ are homotopic maps, and $h: Y \to T$ is an arbitrary map, and all agree on the $(n-1)$-ball $S$, then $f \bullet_S h$ and $g \bullet_S h$ are again homotopic to each other. Said otherwise, $f-g$ was a local relation on $X$, and $(f-g) \bullet_S h$ is a local relation on $X \cup_S Y$. |
92 \end{description} |
95 \end{description} |
93 |
96 |
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97 Finally, a puzzle for you to think about if the next example gets bogged down in nitty-gritty: |
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98 \begin{puzzle} |
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99 Let $\cU(X)$ denote fields of the form $u \bullet f$, where $u \in \cU(B)$ for some ball $B$ in $X$, and $f$ is a map from $X \setminus B$ to $T$. Then $$\Maps(X \to T) / \cU(X) \iso \mathbb{C}[X \to T]$$ Why? |
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100 \end{puzzle} |
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101 (The difficulty is meant to be that we only mod out by `local' homotopies, not all homotopies.) |
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102 |
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103 |
94 \subsection{String diagrams} |
104 \subsection{String diagrams} |
95 This will be a more complicated example, and also a very important one. Essentially, it's a recipe for constructing a system of fields and local relations from a suitable $n$-category. As we haven't yet talked about a definition of an $n$-category, I'll be somewhat vague about what we actually require from one. I'll spell out the construction precisely in the cases $n=1$ and $n=2$, where there are familiar concrete definitions to work with. Later, in \S 6, when we introduce our notion of a `disklike $n$-category', you should think of the definition as being optimized to make the transition back and forth between $n$-categories and systems of fields as straightforward as possible. |
105 This will be a more complicated example, and also a very important one. Essentially, it's a recipe for constructing a system of fields and local relations from a suitable $n$-category. As we haven't yet talked about a definition of an $n$-category, I'll be somewhat vague about what we actually require from one. I'll spell out the construction precisely in the cases $n=1$ and $n=2$, where there are familiar concrete definitions to work with. Later, in \S 6, when we introduce our notion of a `disklike $n$-category', you should think of the definition as being optimized to make the transition back and forth between $n$-categories and systems of fields as straightforward as possible. |
96 |
106 |
97 The core idea is to fix a diagrammatic calculus which represents the algebraic operations in an $n$-category. The diagrams are drawn in $n$-balls. Each diagram is a recipe for composing some collection of morphisms. Modifying the diagram by an isotopy should not change the result of the corresponding composition (perhaps for some types of $n$-categories not all isotopies should be allowed, but we'll generally work in `most invariant' situation, which roughly corresponds to the $n$-categories have lots of nice duality properties). Moreover, the allowed diagrams should be specified by some `local rule': e.g. the diagrams are locally modeled on a certain collection of subdiagrams. Because the diagrams are specified in this way, we can then allow ourselves to draw the same diagrams on arbitrary manifolds, and these become our fields. When we restrict our attention to balls, the `local relations' are precisely those diagrams are a recipe for a composition which is zero in the $n$-category. |
107 The core idea is to fix a diagrammatic calculus which represents the algebraic operations in an $n$-category. The diagrams are drawn in $n$-balls. Each diagram is a recipe for composing some collection of morphisms. Modifying the diagram by an isotopy should not change the result of the corresponding composition (perhaps for some types of $n$-categories not all isotopies should be allowed, but we'll generally work in `most invariant' situation, which roughly corresponds to the $n$-categories have lots of nice duality properties). Moreover, the allowed diagrams should be specified by some `local rule': e.g. the diagrams are locally modeled on a certain collection of subdiagrams. Because the diagrams are specified in this way, we can then allow ourselves to draw the same diagrams on arbitrary manifolds, and these become our fields. When we restrict our attention to balls, the `local relations' are precisely those diagrams are a recipe for a composition which is zero in the $n$-category. |
98 |
108 |
99 There are several alternative schemes for realizing this idea. Two that may be familiar are `string diagrams' (which we'll discuss in detail below, beloved of quantum topologists) and `pasting diagrams' (familiar to category theorists). In fact, these are geometrically dual to each other (and one could look at them as limiting cases of diagrams based on handle decompositions, as the core or co-core diameter goes to zero). The use of string diagrams significantly predates the term (or indeed `quantum topology', and perhaps also `higher category'): Penrose was using them by the late '60s. |
109 There are several alternative schemes for realizing this idea. Two that may be familiar are `string diagrams' (which we'll discuss in detail below, beloved of quantum topologists) and `pasting diagrams' (familiar to category theorists). In fact, these are geometrically dual to each other (and one could look at them as limiting cases of diagrams based on handle decompositions, as the core or co-core diameter goes to zero). The use of string diagrams significantly predates the term (or indeed `quantum topology', and perhaps also `higher category'): Penrose was using them by the late '60s. |
100 |
110 |
101 Fix an $n$-category $\cC$, according to your favorite definition. Suppose that it has `the right sort of duality'. Let's state the general definition, but then to preserve sanity unwind it in dimensions $1$ and $2$. |
111 Fix an $n$-category $\cC$, according to your favorite definition. Suppose that it has `the right sort of duality'. Let's state the general definition, but then to preserve sanity unwind it in dimensions $1$ and $2$. |
102 A string diagram on a $k$-manifold $X$ consists of |
112 A string diagram on a $k$-manifold $X$ consists of |
103 \begin{itemize} |
113 \begin{itemize} |
104 \item a cell decomposition of X; |
114 \item a `conic stratification' (see below, think ``looks locally like a cell decomposition'') of X; |
105 \item a general position homeomorphism from the link of each $j$-cell to the boundary of the standard $(k-j)$-dimensional bihedron; and |
115 \item a general position homeomorphism from the link of each $j$-cell to the boundary of the standard $(k-j)$-dimensional bihedron; and |
106 \item a labelling of each $j$-cell by a $(k-j)$-dimensional morphism of $\cC$, with domain and range determined by the labelings of the link of the $j$-cell. |
116 \item a labelling of each $j$-cell by a $(k-j)$-dimensional morphism of $\cC$, with domain and range determined by the labelings of the link of the $j$-cell. |
107 \end{itemize} |
117 \end{itemize} |
108 Actually, this data is just a representative of a string diagram, and we consider this data up to a certain equivalence; we can modify the homeomorphism parametrizing the link of a $j$-cell, at the expense of replacing the corresponding $(k-j)$-morphism labelling that $j$-cell by the `appropriate dual'. |
118 Actually, this data is just a representative of a string diagram, and we consider this data up to a certain equivalence; we can modify the homeomorphism parametrizing the link of a $j$-cell, at the expense of replacing the corresponding $(k-j)$-morphism labelling that $j$-cell by the `appropriate dual'. |
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119 |
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120 What is a conic stratification? Actually, I just made up that name. In the blob complex paper we just say ``cell decomposition'' but this is wrong (and we'll fix it)! Really we want something that looks locally like a cell decomposition. Let's postpone this, as it's just a distraction for now. |
109 |
121 |
110 When $X$ has boundary, we ask that each cell meets the boundary transversely (so cells meeting the boundary are only half-cells). Note that this means that a string diagram on $X$ restricts to a string diagram on $\bdy X$. |
122 When $X$ has boundary, we ask that each cell meets the boundary transversely (so cells meeting the boundary are only half-cells). Note that this means that a string diagram on $X$ restricts to a string diagram on $\bdy X$. |
111 |
123 |
112 \subsubsection{$n=1$} |
124 \subsubsection{$n=1$} |
113 Now suppose $n=1$; here the right sort of duality means that we want $\cC$ to be a $*$-$1$-category. |
125 Now suppose $n=1$; here the right sort of duality means that we want $\cC$ to be a $*$-$1$-category. |
132 |
144 |
133 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 $\cC$. |
145 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 $\cC$. |
134 |
146 |
135 A string diagram on a $2$-manifold $Y$ consists of |
147 A string diagram on a $2$-manifold $Y$ consists of |
136 \begin{itemize} |
148 \begin{itemize} |
137 \item a cell decomposition of $Y$: the $1$-skeleton is a graph embedded in $Y$, and the $2$-cells ensure that each component of the complement of this graph is a disk); |
149 \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. |
138 \item a $0$-morphism of $\cC$ on each $2$-cell; |
150 \item a $0$-morphism of $\cC$ on each $2$-cell; |
139 \item a transverse orientation of each $1$-cell; |
151 \item a transverse orientation of each $1$-cell; |
140 \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; |
152 \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; |
141 \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); |
153 \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); |
142 \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. |
154 \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. |
173 |
185 |
174 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. |
186 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. |
175 |
187 |
176 As usual for fields based on string diagrams, the corresponding local relations are exactly the kernel of this `evaluation' map. |
188 As usual for fields based on string diagrams, the corresponding local relations are exactly the kernel of this `evaluation' map. |
177 |
189 |
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190 \subsection{Conic stratifications} |
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191 Ugh. Here's my attempt to make ``looks locally like a cell decomposition'' sensible. A conic stratification of $M$ is a stratification $$M_0 \subset M_1 \subset \cdots \subset M_n = M$$ |
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192 (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. |
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193 |
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194 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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195 |
178 \section{Axioms for fields} |
196 \section{Axioms for fields} |
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197 A $n$-dimensional system of fields and local relations $(\cF, \cU)$ enriched in a symmetric monoidal category $\cS$ consists of the following data: |
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198 \begin{description} |
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199 \item[fields] functors $\cF_k$ from $k$-manifolds (and homeomorphisms) to sets; |
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200 \item[boundaries] natural transformations $\bdy : \cF_k \to (\cF_{k-1} \circ \bdy)$; |
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201 \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; |
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202 \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 $y \in \cF_{k-1}(Y), z \in \cF_{k-1}(Z)$; |
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203 \item[identities] natural transformations $\times I: \cF_k \to (\cF_{k+1} \circ \times I)$; |
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204 \item[local relations] a functor $\cU$ from $n$-balls (and homeomorphisms) to sets, so $\cU \subset \cF$; |
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205 \end{description} |
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206 and these data satisfy the following properties: |
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207 \begin{itemize} |
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208 \item everything respects the symmetric monoidal structures on $k$-manifolds (disjoint union), sets, and $\cS$ $$\cF_k(A \sqcup B) = \cF_k(A) \times \cF_k(B);$$ |
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209 \item gluing is compatible with action of homeomorphisms; |
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210 \item the local relations form an ideal under gluing; |
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211 \item ... gluing is surjective up to isotopy (collaring?) ... |
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212 \item identities are compatible on the nose with everything in sight... |
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213 \end{itemize} |
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214 |
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215 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 $Z \bigcup_{\bdy Y} \selfarrow$. |
179 |
216 |
180 \section{TQFT from fields} |
217 \section{TQFT from fields} |
181 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 |
218 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 |
182 $$A(X) = \cF(X) / \cU(X).$$ |
219 $$A(X) = \cF(X) / \cU(X).$$ |
183 It's clear that homeomorphisms of $X$ act on this space. Actually, this collapses to an action of the mapping class group: |
220 It's clear that homeomorphisms of $X$ act on this space. Actually, this collapses to an action of the mapping class group: |