580 |
580 |
581 The next axiom says, roughly, that we have strict associativity in dimension $n$, |
581 The next axiom says, roughly, that we have strict associativity in dimension $n$, |
582 even when we reparametrize our $n$-balls. |
582 even when we reparametrize our $n$-balls. |
583 |
583 |
584 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
584 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
|
585 \label{axiom:isotopy-preliminary} |
585 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
586 Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which |
586 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
587 acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. |
587 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
588 (Keep in mind the important special case where $f$ restricted to $\bd X$ is the identity.) |
588 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act |
589 Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which act |
589 trivially on $\bd b$. |
590 trivially on $\bd b$. |
677 We use this axiom in the proofs of \ref{lem:d-a-acyclic}, \ref{lem:colim-injective} \nn{...}. |
678 We use this axiom in the proofs of \ref{lem:d-a-acyclic}, \ref{lem:colim-injective} \nn{...}. |
678 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
679 All of the examples of (disk-like) $n$-categories we consider in this paper satisfy the axiom, but |
679 nevertheless we feel that it is too strong. |
680 nevertheless we feel that it is too strong. |
680 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
681 In the future we would like to see this provisional version of the axiom replaced by something less restrictive. |
681 |
682 |
682 We give two alternate versions of he axiom, one better suited for smooth examples, and one better suited to PL examples. |
683 We give two alternate versions of the axiom, one better suited for smooth examples, and one better suited to PL examples. |
683 |
684 |
684 \begin{axiom}[Splittings] |
685 \begin{axiom}[Splittings] |
685 \label{axiom:splittings} |
686 \label{axiom:splittings} |
686 Let $c\in \cC_k(X)$, with $0\le k < n$. |
687 Let $c\in \cC_k(X)$, with $0\le k < n$. |
687 Let $X = \cup_i X_i$ be a splitting of $X$. |
688 Let $X = \cup_i X_i$ be a splitting of $X$. |
1001 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
1002 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
1002 we need a preliminary definition. |
1003 we need a preliminary definition. |
1003 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
1004 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
1004 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
1005 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
1005 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
1006 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
1006 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are |
1007 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are |
1007 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
1008 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
1008 %Let $\pi_0(\bbc)$ denote |
1009 %Let $\pi_0(\bbc)$ denote |
1009 |
1010 |
1010 \begin{axiom}[Enriched $n$-categories] |
1011 \begin{axiom}[Enriched $n$-categories] |
1011 \label{axiom:enriched} |
1012 \label{axiom:enriched} |
1045 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
1046 Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. |
1046 After stating it, we will briefly discuss ways in which it can be made more general. |
1047 After stating it, we will briefly discuss ways in which it can be made more general. |
1047 } |
1048 } |
1048 |
1049 |
1049 Recall the category $\bbc$ of balls with boundary conditions. |
1050 Recall the category $\bbc$ of balls with boundary conditions. |
1050 Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
1051 Note that the morphisms $\Homeo(X;c \to X'; c')$ from $(X, c)$ to $(X', c')$ form a topological space. |
1051 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
1052 Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) |
1052 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
1053 and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ |
1053 (e.g.\ the singular chain functor $C_*$). |
1054 (e.g.\ the singular chain functor $C_*$). |
1054 |
1055 |
1055 \begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] |
1056 \begin{axiom}[\textup{\textbf{[$A_\infty$ replacement for Axiom \ref{axiom:extended-isotopies}]}} Families of homeomorphisms act in dimension $n$.] |
1056 \label{axiom:families} |
1057 \label{axiom:families} |
1057 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
1058 For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism |
1058 \[ |
1059 \[ |
1059 \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') . |
1060 \cJ(\Homeo(X;c \to X'; c')) \ot \cC(X; c) \to \cC(X'; c') . |
1060 \] |
1061 \] |
1061 Similarly, we have an $\cS$-morphism |
1062 Similarly, we have an $\cS$-morphism |
1062 \[ |
1063 \[ |
1063 \cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c), |
1064 \cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c), |
1064 \] |
1065 \] |
1069 a diagram like the one in Theorem \ref{thm:CH} commutes. |
1070 a diagram like the one in Theorem \ref{thm:CH} commutes. |
1070 % say something about compatibility with product morphisms? |
1071 % say something about compatibility with product morphisms? |
1071 \end{axiom} |
1072 \end{axiom} |
1072 |
1073 |
1073 We now describe the topology on $\Coll(X; c)$. |
1074 We now describe the topology on $\Coll(X; c)$. |
1074 We retain notation from the above definition of collar map. |
1075 We retain notation from the above definition of collar map (after Axiom \ref{axiom:isotopy-preliminary}). |
1075 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
1076 Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to |
1076 (possibly length zero) embedded intervals in $X$ terminating at $p$. |
1077 (possibly length zero) embedded intervals in $X$ terminating at $p$. |
1077 If $p \in Y$ this interval is the image of $\{p\}\times J$. |
1078 If $p \in Y$ this interval is the image of $\{p\}\times J$. |
1078 If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. |
1079 If $p \notin Y$ then $p$ is assigned the length zero interval $\{p\}$. |
1079 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
1080 Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. |
1080 Note in particular that parts of the collar are allowed to shrink continuously to zero length. |
1081 Note in particular that parts of the collar are allowed to shrink continuously to zero length. |
1081 (This is the real content; if nothing shrinks to zero length then the action of families of collar |
1082 (This is the real content; if nothing shrinks to zero length then the action of families of collar |
1082 maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
1083 maps follows from the action of families of homeomorphisms and compatibility with gluing.) |
1083 |
1084 |
1084 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
1085 The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} |
1085 $\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
1086 $\Homeo(X;c\to X'; c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above |
1086 action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible). |
1087 action of $\cJ(\Homeo(X;c\to X'; c'))$ to be strictly associative as well (assuming the two actions are compatible). |
1087 In fact, compatibility implies less than this. |
1088 In fact, compatibility implies less than this. |
1088 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
1089 For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. |
1089 (This is the example most relevant to this paper.) |
1090 (This is the example most relevant to this paper.) |
1090 Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action |
1091 Then compatibility implies that the action of $C_*(\Homeo(X;c\to X'; c'))$ agrees with the action |
1091 of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
1092 of $C_0(\Homeo(X;c\to X'; c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. |
1092 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. |
1093 And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction (see Example \ref{ex:blob-complexes-of-balls} below). |
1093 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
1094 Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, |
1094 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
1095 such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. |
1095 |
1096 |
1096 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
1097 An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} |
1097 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
1098 supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a |
1140 (and their boundaries), while for fields we consider all manifolds. |
1141 (and their boundaries), while for fields we consider all manifolds. |
1141 Second, in the category definition we directly impose isotopy |
1142 Second, in the category definition we directly impose isotopy |
1142 invariance in dimension $n$, while in the fields definition we |
1143 invariance in dimension $n$, while in the fields definition we |
1143 instead remember a subspace of local relations which contain differences of isotopic fields. |
1144 instead remember a subspace of local relations which contain differences of isotopic fields. |
1144 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
1145 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
1145 Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
1146 Thus a \nn{lemma-ize} system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
1146 balls and, at level $n$, quotienting out by the local relations: |
1147 balls and, at level $n$, quotienting out by the local relations: |
1147 \begin{align*} |
1148 \begin{align*} |
1148 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
1149 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
1149 \end{align*} |
1150 \end{align*} |
1150 This $n$-category can be thought of as the local part of the fields. |
1151 This $n$-category can be thought of as the local part of the fields. |
1243 Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
1244 Let $\cF$ be a TQFT in the sense of \S\ref{sec:fields}: an $n$-dimensional |
1244 system of fields (also denoted $\cF$) and local relations. |
1245 system of fields (also denoted $\cF$) and local relations. |
1245 Let $W$ be an $n{-}j$-manifold. |
1246 Let $W$ be an $n{-}j$-manifold. |
1246 Define the $j$-category $\cF(W)$ as follows. |
1247 Define the $j$-category $\cF(W)$ as follows. |
1247 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
1248 If $X$ is a $k$-ball with $k<j$, let $\cF(W)(X) \deq \cF(W\times X)$. |
1248 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
1249 If $X$ is a $j$-ball and $c\in \cl{\cF(W)}(\bd X)$, |
1249 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
1250 let $\cF(W)(X; c) \deq A_\cF(W\times X; c)$. |
1250 \end{example} |
1251 \end{example} |
1251 |
1252 |
1252 The next example is only intended to be illustrative, as we don't specify |
1253 The next example is only intended to be illustrative, as we don't specify |
1253 which definition of a ``traditional $n$-category" we intend. |
1254 which definition of a ``traditional $n$-category" we intend. |
1996 The above operad-like structure is analogous to the swiss cheese operad |
1997 The above operad-like structure is analogous to the swiss cheese operad |
1997 \cite{MR1718089}. |
1998 \cite{MR1718089}. |
1998 |
1999 |
1999 \medskip |
2000 \medskip |
2000 |
2001 |
2001 We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the |
2002 We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the |
2002 plain ball case. |
2003 plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval. |
|
2004 \nn{figure, 2 examples} |
2003 Note that a marked pinched product can be decomposed into either |
2005 Note that a marked pinched product can be decomposed into either |
2004 two marked pinched products or a plain pinched product and a marked pinched product. |
2006 two marked pinched products or a plain pinched product and a marked pinched product. |
2005 %\nn{should maybe give figure} |
2007 \nn{should give figure} |
2006 |
2008 |
2007 \begin{module-axiom}[Product (identity) morphisms] |
2009 \begin{module-axiom}[Product (identity) morphisms] |
2008 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
2010 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
2009 $k{+}m$-ball ($m\ge 1$), |
2011 $k{+}m$-ball ($m\ge 1$), |
2010 there is a map $\pi^*:\cM(M)\to \cM(E)$. |
2012 there is a map $\pi^*:\cM(M)\to \cM(E)$. |