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    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

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    64 %% like extended math fonts and math commands.

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    66 \usepackage{amssymb,amsfonts,amsmath,amsthm}

    66 \usepackage{amssymb,amsfonts,amsmath,amsthm}

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    69 %% OPTIONAL MACRO FILES

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   157 \def\BD{BD}

   170 \def\BD{BD}

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   159 \def\spl{_\pitchfork}

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   162 % equations

   177 % equations

   163 \newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}}

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   335 \maketitle

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   337 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   352 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   338 \begin{article}

   353 \begin{article}

   339 

   354 

   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 

   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 

   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

   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}%