162 

   162 \todo{Check font size in an actual PNAS article: this looks a little big}

   163 \section{}

   180 \subsection{$n$-categories}

   179 \subsection{$n$-categories} \mbox{}

   181 \nn{

   180 

   182 Axioms for $n$-categories, examples (maps, string diagrams)

   181 \todo{This is just a copy and paste of the statements of the axioms. We need to rewrite this into something that's both compact and comprehensible! The first few at least aren't that terrifying, but we definitely don't want to derail the reader with the actual product axiom, for example.}

   183 }

   182 

   184 \nn{

   183 \begin{axiom}[Morphisms]

   184 \label{axiom:morphisms}

   185 For each $0 \le k \le n$, we have a functor $\cC_k$ from 

   186 the category of $k$-balls and 

   187 homeomorphisms to the category of sets and bijections.

   188 \end{axiom}

   189 \begin{lem}

   190 \label{lem:spheres}

   191 For each $1 \le k \le n$, we have a functor $\cl{\cC}_{k-1}$ from 

   192 the category of $k{-}1$-spheres and 

   193 homeomorphisms to the category of sets and bijections.

   194 \end{lem}

   195 \begin{axiom}[Boundaries]\label{nca-boundary}

   196 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.

   197 These maps, for various $X$, comprise a natural transformation of functors.

   198 \end{axiom}

   199 \begin{lem}[Boundary from domain and range]

   200 \label{lem:domain-and-range}

   201 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,

   202 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}).

   203 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the 

   204 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$.

   205 Then we have an injective map

   206 \[

   207 	\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S)

   208 \]

   209 which is natural with respect to the actions of homeomorphisms.

   210 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product

   211 becomes a normal product.)

   212 \end{lem}

   213 \begin{axiom}[Composition]

   214 \label{axiom:composition}

   215 Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)

   216 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}).

   217 Let $E = \bd Y$, which is a $k{-}2$-sphere.

   218 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.

   219 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.

   220 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. 

   221 We have a map

   222 \[

   223 	\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E

   224 \]

   225 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions

   226 to the intersection of the boundaries of $B$ and $B_i$.

   227 If $k < n$,

   228 or if $k=n$ and we are in the $A_\infty$ case, 

   229 we require that $\gl_Y$ is injective.

   230 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)

   231 \end{axiom}

   232 \begin{axiom}[Strict associativity] \label{nca-assoc}

   233 The composition (gluing) maps above are strictly associative.

   234 Given any splitting of a ball $B$ into smaller balls

   235 $$\bigsqcup B_i \to B,$$ 

   236 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.

   237 \end{axiom}

   238 \begin{axiom}[Product (identity) morphisms]

   239 \label{axiom:product}

   240 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),

   241 there is a map $\pi^*:\cC(X)\to \cC(E)$.

   242 These maps must satisfy the following conditions.

   243 \begin{enumerate}

   244 \item

   245 If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and

   246 if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram

   247 \[ \xymatrix{

   248 	E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\

   249 	X \ar[r]^{f} & X'

   250 } \]

   251 commutes, then we have 

   252 \[

   253 	\pi'^*\circ f = \tilde{f}\circ \pi^*.

   254 \]

   255 \item

   256 Product morphisms are compatible with gluing (composition).

   257 Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ 

   258 be pinched products with $E = E_1\cup E_2$.

   259 Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.

   260 Then 

   261 \[

   262 	\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .

   263 \]

   264 \item

   265 Product morphisms are associative.

   266 If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then

   267 \[

   268 	\rho^*\circ\pi^* = (\pi\circ\rho)^* .

   269 \]

   270 \item

   271 Product morphisms are compatible with restriction.

   272 If we have a commutative diagram

   273 \[ \xymatrix{

   274 	D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\

   275 	Y \ar@{^(->}[r] & X

   276 } \]

   277 such that $\rho$ and $\pi$ are pinched products, then

   278 \[

   279 	\res_D\circ\pi^* = \rho^*\circ\res_Y .

   280 \]

   281 \end{enumerate}

   282 \end{axiom}

   283 \begin{axiom}[\textup{\textbf{[plain  version]}} Extended isotopy invariance in dimension $n$.]

   284 \label{axiom:extended-isotopies}

   285 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts

   286 to the identity on $\bd X$ and isotopic (rel boundary) to the identity.

   287 Then $f$ acts trivially on $\cC(X)$.

   288 In addition, collar maps act trivially on $\cC(X)$.

   289 \end{axiom}

   290 

   291 \smallskip

   292 

   293 For $A_\infty$ $n$-categories, we replace

   294 isotopy invariance with the requirement that families of homeomorphisms act.

   295 For the moment, assume that our $n$-morphisms are enriched over chain complexes.

   296 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and

   297 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space.

   298 

   299 

   300 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]

   301 \label{axiom:families}

   302 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes

   303 \[

   304 	C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .

   305 \]

   306 These action maps are required to be associative up to homotopy,

   307 and also compatible with composition (gluing) in the sense that

   308 a diagram like the one in Theorem \ref{thm:CH} commutes.

   309 \end{axiom}

   310 

   311 

   312 \todo{

   228 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).

   360 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category.

   232 

   364 We will use the term `field on $W$' to refer to \nn{a point} of this functor,

   233 \nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls}

   365 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.

   366 

   367 \todo{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?}

   368 

   235 \nn{How can we extend an $n$-category from balls to arbitrary manifolds?}

   370 \nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?}

   236 

   371 

   237 \nn{In practice, this gives the old definition}

   372 We now define the blob complex $\bc_*(W; \cC)$ of an $n$-manifold $W$

   238 \subsubsection{}

   373 with coefficients in the $n$-category $\cC$ to be the homotopy colimit

   374 of the functor $\psi_{\cC; W}$ described above. 

   375 

   376 When $\cC$ is a topological $n$-category,

   377 the flexibility available in the construction of a homotopy colimit allows

   378 us to give a much more explicit description of the blob complex.

   379 

   380 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}

   381 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that

   382 each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions.

   383 

   384 The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of

   385 \begin{itemize}

   386 \item a permissible collection of $k$ embedded balls (called `blobs') in $W$,

   387 \item an ordering of the balls, and

   388 \item for each resulting piece of $W$, a field,

   389 \end{itemize}

   390 such that for any innermost blob $B$, the field on $B$ goes to zero under the composition map from $\cC$.

   391 

   392 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with signs given by the ordering.

   393 

   394 \todo{Say why this really is the homotopy colimit}

   395 \todo{Spell out $k=0, 1, 2$}

   396