62 |
62 |
63 %% The AMS math files are commonly used to gain access to useful features |
63 %% The AMS math files are commonly used to gain access to useful features |
64 %% like extended math fonts and math commands. |
64 %% like extended math fonts and math commands. |
65 |
65 |
66 \usepackage{amssymb,amsfonts,amsmath,amsthm} |
66 \usepackage{amssymb,amsfonts,amsmath,amsthm} |
|
67 |
|
68 % fiddle with fonts |
|
69 |
|
70 \usepackage{microtype} |
|
71 |
|
72 \usepackage{ifxetex} |
|
73 \ifxetex |
|
74 \usepackage{xunicode,fontspec,xltxtra} |
|
75 \setmainfont[Ligatures={}]{Linux Libertine O} |
|
76 \usepackage{unicode-math} |
|
77 \setmathfont{Asana Math} |
|
78 \fi |
|
79 |
67 |
80 |
68 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
81 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
69 %% OPTIONAL MACRO FILES |
82 %% OPTIONAL MACRO FILES |
70 %% Insert self-defined macros here. |
83 %% Insert self-defined macros here. |
71 %% \newcommand definitions are recommended; \def definitions are supported |
84 %% \newcommand definitions are recommended; \def definitions are supported |
335 \maketitle |
350 \maketitle |
336 |
351 |
337 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
352 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
338 \begin{article} |
353 \begin{article} |
339 |
354 |
|
355 |
340 \begin{abstract} |
356 \begin{abstract} |
341 We summarize our axioms for higher categories, and describe the ``blob complex". |
357 We summarize our axioms for higher categories, and describe the ``blob complex". |
342 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
358 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$. |
343 The $0$-th homology of this chain complex recovers the usual topological quantum field theory invariants of $W$. |
359 The $0$-th homology of this chain complex recovers the usual topological quantum field theory invariants of $W$. |
344 The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide). |
360 The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when $W=S^1$ they coincide). |
492 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
508 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
493 of higher associativity relations. |
509 of higher associativity relations. |
494 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
510 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
495 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
511 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
496 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
512 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
497 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
513 {\it strictly associative} composition $\Omega_r \times \Omega_s \to \Omega_{r+s}$. |
498 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
514 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
499 We wish to imitate this strategy in higher categories. |
515 We wish to imitate this strategy in higher categories. |
500 Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
516 Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
501 a product of $k$ intervals (c.f.\ \cite{0909.2212}) but rather with any $k$-ball, that is, |
517 a product of $k$ intervals (c.f.\ \cite{0909.2212}) but rather with any $k$-ball, that is, |
502 % \cite{ulrike-tillmann-2008,0909.2212} |
518 % \cite{ulrike-tillmann-2008,0909.2212} |
904 \begin{equation*} |
920 \begin{equation*} |
905 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
921 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
906 \end{equation*} |
922 \end{equation*} |
907 \end{property} |
923 \end{property} |
908 |
924 |
|
925 |
909 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
926 If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
910 write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
927 write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
911 \begin{property}[Gluing map] |
928 \begin{property}[Gluing map] |
912 \label{property:gluing-map}% |
929 \label{property:gluing-map}% |
913 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
930 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
914 %\begin{equation*} |
931 %\begin{equation*} |
915 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
932 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
916 %\end{equation*} |
933 %\end{equation*} |
917 Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
934 Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
918 a map |
935 a map |
919 \[ |
936 $ |
920 \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
937 \bc_*(X) \to \bc_*\left(X \bigcup_{Y} \selfarrow\right), |
921 \] |
938 $ |
922 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
939 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
923 \end{property} |
940 \end{property} |
924 |
941 |
925 \begin{property}[Contractibility] |
942 \begin{property}[Contractibility] |
926 \label{property:contractibility}% |
943 \label{property:contractibility}% |