20 It is easy to show that examples of topological origin |
20 It is easy to show that examples of topological origin |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
22 satisfy our axioms. |
22 satisfy our axioms. |
23 For examples of a more purely algebraic origin, one would typically need the combinatorial |
23 For examples of a more purely algebraic origin, one would typically need the combinatorial |
24 results that we have avoided here. |
24 results that we have avoided here. |
|
25 |
|
26 See \S\ref{n-cat-names} for a discussion of $n$-category terminology. |
25 |
27 |
26 %\nn{Say something explicit about Lurie's work here? |
28 %\nn{Say something explicit about Lurie's work here? |
27 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
29 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
28 |
30 |
29 \medskip |
31 \medskip |
376 \end{tikzpicture} |
378 \end{tikzpicture} |
377 $$ |
379 $$ |
378 \caption{Examples of pinched products}\label{pinched_prods} |
380 \caption{Examples of pinched products}\label{pinched_prods} |
379 \end{figure} |
381 \end{figure} |
380 (The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
382 (The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
381 where we construct a traditional category from a topological category.) |
383 where we construct a traditional category from a disk-like category.) |
382 Define a {\it pinched product} to be a map |
384 Define a {\it pinched product} to be a map |
383 \[ |
385 \[ |
384 \pi: E\to X |
386 \pi: E\to X |
385 \] |
387 \] |
386 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
388 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled |
666 balls and, at level $n$, quotienting out by the local relations: |
668 balls and, at level $n$, quotienting out by the local relations: |
667 \begin{align*} |
669 \begin{align*} |
668 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
670 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
669 \end{align*} |
671 \end{align*} |
670 This $n$-category can be thought of as the local part of the fields. |
672 This $n$-category can be thought of as the local part of the fields. |
671 Conversely, given a topological $n$-category we can construct a system of fields via |
673 Conversely, given a disk-like $n$-category we can construct a system of fields via |
672 a colimit construction; see \S \ref{ss:ncat_fields} below. |
674 a colimit construction; see \S \ref{ss:ncat_fields} below. |
673 |
675 |
674 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
676 In the $n$-category axioms above we have intermingled data and properties for expository reasons. |
675 Here's a summary of the definition which segregates the data from the properties. |
677 Here's a summary of the definition which segregates the data from the properties. |
676 |
678 |
853 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
855 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
854 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
856 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
855 \end{example} |
857 \end{example} |
856 |
858 |
857 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
859 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. |
858 Notice that with $F$ a point, the above example is a construction turning a topological |
860 Notice that with $F$ a point, the above example is a construction turning an ordinary |
859 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
861 $n$-category $\cC$ into an $A_\infty$ $n$-category. |
860 We think of this as providing a ``free resolution" |
862 We think of this as providing a ``free resolution" |
861 of the topological $n$-category. |
863 of the ordinary $n$-category. |
862 %\nn{say something about cofibrant replacements?} |
864 %\nn{say something about cofibrant replacements?} |
863 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
865 In fact, there is also a trivial, but mostly uninteresting, way to do this: |
864 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
866 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, |
865 and take $\CD{B}$ to act trivially. |
867 and take $\CD{B}$ to act trivially. |
866 |
868 |
867 Beware that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ |
869 Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$ |
868 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
870 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. |
869 It's easy to see that with $n=0$, the corresponding system of fields is just |
871 It's easy to see that with $n=0$, the corresponding system of fields is just |
870 linear combinations of connected components of $T$, and the local relations are trivial. |
872 linear combinations of connected components of $T$, and the local relations are trivial. |
871 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
873 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
872 |
874 |
925 The remaining data for the $A_\infty$ $n$-category |
927 The remaining data for the $A_\infty$ $n$-category |
926 --- composition and $\Diff(X\to X')$ action --- |
928 --- composition and $\Diff(X\to X')$ action --- |
927 also comes from the $\cE\cB_n$ action on $A$. |
929 also comes from the $\cE\cB_n$ action on $A$. |
928 %\nn{should we spell this out?} |
930 %\nn{should we spell this out?} |
929 |
931 |
930 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
932 Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
931 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
933 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
932 an $\cE\cB_n$-algebra. |
934 an $\cE\cB_n$-algebra. |
933 %\nn{The paper is already long; is it worth giving details here?} |
935 %\nn{The paper is already long; is it worth giving details here?} |
934 |
936 |
935 If we apply the homotopy colimit construction of the next subsection to this example, |
937 If we apply the homotopy colimit construction of the next subsection to this example, |
1193 in the context of an $m{+}1$-dimensional TQFT. |
1195 in the context of an $m{+}1$-dimensional TQFT. |
1194 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1196 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1195 This will be explained in more detail as we present the axioms. |
1197 This will be explained in more detail as we present the axioms. |
1196 |
1198 |
1197 Throughout, we fix an $n$-category $\cC$. |
1199 Throughout, we fix an $n$-category $\cC$. |
1198 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
1200 For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category. |
1199 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1201 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1200 |
1202 |
1201 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1203 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1202 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1204 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1203 We call $B$ the ball and $N$ the marking. |
1205 We call $B$ the ball and $N$ the marking. |
1507 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1509 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$), |
1508 there is no left/right module distinction. |
1510 there is no left/right module distinction. |
1509 |
1511 |
1510 \medskip |
1512 \medskip |
1511 |
1513 |
1512 We now give some examples of modules over topological and $A_\infty$ $n$-categories. |
1514 We now give some examples of modules over ordinary and $A_\infty$ $n$-categories. |
1513 |
1515 |
1514 \begin{example}[Examples from TQFTs] |
1516 \begin{example}[Examples from TQFTs] |
1515 \rm |
1517 \rm |
1516 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1518 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold, |
1517 and $\cF(W)$ the $j$-category associated to $W$. |
1519 and $\cF(W)$ the $j$-category associated to $W$. |
1550 |
1552 |
1551 |
1553 |
1552 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1554 \subsection{Modules as boundary labels (colimits for decorated manifolds)} |
1553 \label{moddecss} |
1555 \label{moddecss} |
1554 |
1556 |
1555 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1557 Fix an ordinary $n$-category or $A_\infty$ $n$-category $\cC$. |
1556 Let $W$ be a $k$-manifold ($k\le n$), |
1558 Let $W$ be a $k$-manifold ($k\le n$), |
1557 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1559 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1558 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1560 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1559 |
1561 |
1560 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
1562 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
1718 this is much less true for higher dimensional spheres, |
1720 this is much less true for higher dimensional spheres, |
1719 so we prefer the term ``sphere module" for the general case. |
1721 so we prefer the term ``sphere module" for the general case. |
1720 |
1722 |
1721 %The results of this subsection are not needed for the rest of the paper, |
1723 %The results of this subsection are not needed for the rest of the paper, |
1722 %so we will skimp on details in a couple of places. We have included this mostly |
1724 %so we will skimp on details in a couple of places. We have included this mostly |
1723 %for the sake of comparing our notion of a topological $n$-category to other definitions. |
1725 %for the sake of comparing our notion of a disk-like $n$-category to other definitions. |
1724 |
1726 |
1725 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1727 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces. |
1726 |
1728 |
1727 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1729 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe |
1728 these first. |
1730 these first. |