245 Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$.

   245 For each $k$-ball $X$ and $m$-ball $D$, with $k+m \le n$, there is a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$. These maps must satisfy the following conditions.

   246 Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.

   246 \begin{enumerate}

   247 \item

   437 a colimit construction; see below.

   442 a colimit construction; see \S \ref{ss:ncat_fields} below.

   450 We know describe several classes of examples of $n$-categories satisfying our axioms.

   455 We now describe several classes of examples of $n$-categories satisfying our axioms.

   451 

   456 

   452 \begin{example}{Maps to a space}

   457 \begin{example}[Maps to a space]

   458 \rm

   454 Fix $F$ a closed $m$-manifold (keep in mind the case where $F$ is a point). Fix a `target space' $T$, any topological space.

   460 Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.

   455 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of 

   461 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of 

   456 all maps from $X\times F$ to $T$.

   462 all continuous maps from $X$ to $T$.

   457 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo

   463 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo

   464 \begin{example}{Linearized, twisted, maps to a space}

   470 \begin{example}[Maps to a space, with a fiber]

   471 \rm

   472 \label{ex:maps-to-a-space-with-a-fiber}%

   473 We can modify the example above, by fixing an $m$-manifold $F$, and defining $\pi^{\times F}_{\leq n}(T)(X) = \Maps(X \times F \to T)$, otherwise leaving the definition in Example \ref{ex:maps-to-a-space} unchanged. Taking $F$ to be a point recovers the previous case.

   474 \end{example}

   475 

   476 \begin{example}[Linearized, twisted, maps to a space]

   477 \rm

   466 We can linearize the above example as follows.

   479 We can linearize Examples \ref{ex:maps-to-a-space} and \ref{ex:maps-to-a-space-with-a-fiber} as follows.

   468 (e.g.\ the trivial cocycle).

   481 (have in mind the trivial cocycle).

   469 For $X$ of dimension less than $n$ define $\cC(X)$ as before.

   482 For $X$ of dimension less than $n$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X)$ as before, ignoring $\alpha$.

   470 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be

   483 For $X$ an $n$-ball and $c\in \Maps(\bdy X \times F \to T)$ define $\pi^{\alpha, \times F}_{\leq n}(T)(X; c)$ to be

   471 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$,

   484 the $R$-module of finite linear combinations of continuous maps from $X\times F$ to $T$,

   473 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$.

   486 $h: X\times F\times I \to T$, then $a = \alpha(h)b$.

   477 \begin{itemize}

   490 The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend. Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.

   478 

   491 \begin{example}[Traditional $n$-categories]

   479 \item \nn{Continue converting these into examples}

   492 \rm

   480 

   493 \label{ex:traditional-n-categories}

   481 \item Given a traditional $n$-category $C$ (with strong duality etc.),

   494 Given a `traditional $n$-category with strong duality' $C$

   482 define $\cC(X)$ (with $\dim(X) < n$) 

   495 define $\cC(X)$, for $X$ a $k$-ball or $k$-sphere with $k < n$,

   488 Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$,

   501 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$,

   489 and with the same labeling as $a$.

   502 with each cell labelled by the $m$-th iterated identity morphism of the corresponding cell for $a$.

   496 

   509 \end{example}

   510 

   511 Finally, we describe a version of the bordism $n$-category suitable to our definitions.

   512 \newcommand{\Bord}{\operatorname{Bord}}

   513 \begin{example}[The bordism $n$-category]

   514 \rm

   515 \label{ex:bordism-category}

   516 For a $k$-ball $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional

   517 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse

   518 to $\bd X$. \nn{spheres}

   519 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes (rel boundary) of such submanifolds;

   520 we identify $W$ and $W'$ if $\bd W = \bd W'$ and there is a homeomorphism

   521 $W \to W'$ which restricts to the identity on the boundary

   522 \end{example}

   523 

   524 \begin{itemize}

   525 

   526 \item \nn{Continue converting these into examples}

   527 

   528 \item 

   519 \begin{example}{Chains of maps to a space}

   541 \begin{example}[Chains of maps to a space]

   520 We can modify Example \ref{ex:maps-to-a-space} above by defining $\cC(X; c)$ for an $n$-ball $X$  to be the chain complex 

   542 \rm

   521 $C_*(\Maps_c(X\times F \to T))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,

   543 \label{ex:chains-of-maps-to-a-space}

   544 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$.

   545 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$.

   546 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex

   547 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,

   525 \begin{example}{Blob complexes of balls (with a fiber)}

   551 See ??? below, recovering $C_*(\Maps{M \to T})$ as (up to homotopy) the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$.

   552 

   553 \begin{example}[Blob complexes of balls (with a fiber)]

   554 \rm

   555 \label{ex:blob-complexes-of-balls}

   533 \begin{defn}

   563 This example will be essential for ???, which relates ...

   564 

   565 \begin{example}

   535 \end{defn}

   567 \end{example}

   544 

   576 In this section we describe how to extend an $n$-category as described above (of either the plain or $A_\infty$ variation) to a system of fields. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension, from $k$-balls and $k$-spheres to arbitrary $k$-manifolds.

   545 We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows.

   577 

   546 

   578 We will first define the `cell-decomposition' poset $\cJ(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. An $n$-category $\cC$ provides a functor from this poset to the category of sets, and we  will define $\cC(W)$ as a suitable colimit (or homotopy colimit) of this functor. We'll later give a more explicit description of this colimit. In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complex to $n$-manifolds with boundary data), then the resulting system of fields is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).

   547 Let $W$ be a $k$-manifold, $1\le k \le n$.

   579 

   548 We will define a set $\cC(W)$.

   580 \begin{defn}

   549 (If $k = n$ and our $k$-categories are enriched, then

   581 Say that a `permissible decomposition' of $W$ is a cell decomposition

   550 $\cC(W)$ will have additional structure; see below.)

   551 $\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$,

   552 which we define next.

   553 

   554 Define a permissible decomposition of $W$ to be a cell decomposition

   560 of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.

   588 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$.

   561 This defines a partial ordering $\cJ(W)$, which we will think of as a category.

   589 

   562 (The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique

   590 The category $\cJ(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

   563 morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.

   591 See Figure \ref{partofJfig} for an example.

   564 See Figure \ref{partofJfig}.)

   592 \end{defn}

   575 $\cC$ determines 

   603 

   576 a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets 

   604 

   605 An $n$-category $\cC$ determines 

   606 a functor $\psi_{\cC;W}$ from $\cJ(W)$ to the category of sets 

   578 For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset

   608 For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.

   579 \[

   609 

   580 	\psi_\cC(x) \sub \prod_a \cC(X_a)

   610 \begin{defn}

   581 \]

   611 Define the functor $\psi_{\cC;W} : \cJ(W) \to \Set$ as follows.

   582 such that the restrictions to the various pieces of shared boundaries amongst the

   612 For a decomposition $x = \bigcup_a X_a$ in $\cJ(W)$, $\psi_{\cC;W}(x)$ is the subset

   583 $X_a$ all agree.

   613 \begin{equation}

   584 (Think fibered product.)

   614 \label{eq:psi-C}

   585 If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$

   615 	\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)_{\bdy X_a}

   586 via the composition maps of $\cC$.

   616 \end{equation}

   587 (If $\dim(W) = n$ then we need to also make use of the monoidal

   617 where the restrictions to the various pieces of shared boundaries amongst the cells

   588 product in the enriching category.

   618 $X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).

   589 \nn{should probably be more explicit here})

   619 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.

   590 

   620 \end{defn}

   591 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.

   621 

   592 When $k<n$ or $k=n$ and we are in the plain (non-$A_\infty$) case, this means that

   622 When the $n$-category $\cC$ is enriched in some monoidal category $(A,\boxtimes)$, and $W$ is an $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$

   593 for each decomposition $x$ there is a map

   623 we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. (Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)

   594 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps

   624 

   625 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.

   626 

   627 \begin{defn}[System of fields functor]

   628 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cC(W)$ is the usual colimit of the functor $\psi_{\cC;W}$.

   629 That is, for each decomposition $x$ there is a map

   630 $\psi_{\cC;W}(x)\to \cC(W)$, these maps are compatible with the refinement maps

   596 When $k=n$ and we are in the $A_\infty$ case, it means

   632 \end{defn}

   597 homotopy colimit.

   633 

   598 

   634 \begin{defn}[System of fields functor, $A_\infty$ case]

   599 More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take

   635 When $\cC$ is an $A_\infty$ $n$-category, $\cC(W)$ for $W$ a $k$-manifold with $k < n$ is defined as above, as the colimit of $\psi_{\cC;W}$. When $W$ is an $n$-manifold, the chain complex $\cC(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$.

   600 \[

   636 \end{defn}

   601 	\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K

   637 

   602 \]

   638 We can specify boundary data $c \in \cC(\bdy W)$, and define functors $\psi_{\cC;W,c}$ with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$.

   603 where $K$ is generated by all things of the form $a - g(a)$, where

   639 

   604 $a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x)

   640 We can now give a more concrete description of the colimit in each case. If $\cC$ is enriched over vector spaces, and $W$ is an $n$-manifold, we can take the vector space $\cC(W,c)$ to be the direct sum over all permissible decompositions of $W$

   605 \to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$.

   641 \begin{equation*}

   606 

   642 	\cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K

   607 In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit

   643 \end{equation*}

   608 is as follows.

   644 where $K$ is the vector space spanned by elements $a - g(a)$, with

   645 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)

   646 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.

   647 

   648 In the $A_\infty$ case enriched over chain complexes, the concrete description of the homotopy colimit

   649 is slightly more involved.

   610 Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions.

   651 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.

   611 Such sequences (for all $m$) form a simplicial set.

   652 Such sequences (for all $m$) form a simplicial set in $\cJ(W)$.

   612 Let

   653 Define $V$ as a vector space via

   613 \[

   654 \[

   614 	V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,

   655 	V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,

   615 \]

   656 \]

   616 where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$.

   657 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. (Our homological conventions are obtuse: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.)

   617 We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$

   658 We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$

   622 	\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) ,

   663 	\bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) ,

   625 is the usual map.

   666 is the usual gluing map coming from the antirefinement $x_0 < x_1$.

   926 	W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) ,

   967 	W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) ,

   947 	\psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib}))

   988 	\psi_\cN(x) \sub \left(\prod_a \cC(X_a)\right) \times \left(\prod_{ib} \cN_i(M_{ib})\right)