diff -r 9698f584e732 -r f7da004e1f14 text/ncat.tex --- a/text/ncat.tex Fri Jun 04 08:15:08 2010 -0700 +++ b/text/ncat.tex Fri Jun 04 11:42:07 2010 -0700 @@ -24,7 +24,9 @@ \medskip -There are many existing definitions of $n$-categories, with various intended uses. In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. Generally, these sets are indexed by instances of a certain typical shape. +There are many existing definitions of $n$-categories, with various intended uses. +In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. +Generally, these sets are indexed by instances of a certain typical shape. Some $n$-category definitions model $k$-morphisms on the standard bihedrons (interval, bigon, and so on). Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$, a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$, @@ -33,8 +35,10 @@ Still other definitions (see, for example, \cite{MR2094071}) model the $k$-morphisms on more complicated combinatorial polyhedra. -For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic -to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the +For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. +Thus we expect to associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic +to the standard $k$-ball. +By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the standard $k$-ball. We {\it do not} assume that it is equipped with a preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. @@ -79,7 +83,10 @@ Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. (Actually, this is only true in the oriented case, with 1-morphisms parameterized by oriented 1-balls.) -For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. +For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. +For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. +(sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. +We prefer to not make the distinction in the first place. Instead, we will combine the domain and range into a single entity which we call the boundary of a morphism. @@ -98,7 +105,9 @@ homeomorphisms to the category of sets and bijections. \end{lem} -We postpone the proof \todo{} of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in the other Axioms at lower levels. +We postpone the proof \todo{} of this result until after we've actually given all the axioms. +Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, +along with the data described in the other Axioms at lower levels. %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. @@ -284,7 +293,9 @@ The next axiom is related to identity morphisms, though that might not be immediately obvious. \begin{axiom}[Product (identity) morphisms] -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. +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. \begin{enumerate} \item If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram @@ -478,7 +489,8 @@ (and their boundaries), while for fields we consider all manifolds. Second, in category definition we directly impose isotopy invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields -but instead remember a subspace of local relations which contain differences of isotopic fields. (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) +but instead remember a subspace of local relations which contain differences of isotopic fields. +(Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to balls and, at level $n$, quotienting out by the local relations: \begin{align*} @@ -497,7 +509,8 @@ \begin{example}[Maps to a space] \rm \label{ex:maps-to-a-space}% -Fix a `target space' $T$, any topological space. We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. +Fix a `target space' $T$, any topological space. +We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows. For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of all continuous maps from $X$ to $T$. For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo @@ -506,14 +519,17 @@ For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. -Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. +Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. +Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. \end{example} \begin{example}[Maps to a space, with a fiber] \rm \label{ex:maps-to-a-space-with-a-fiber}% We can modify the example above, by fixing a -closed $m$-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. +closed $m$-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. \end{example} \begin{example}[Linearized, twisted, maps to a space] @@ -530,7 +546,8 @@ \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} \end{example} -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. +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. \begin{example}[Traditional $n$-categories] \rm \label{ex:traditional-n-categories} @@ -550,7 +567,10 @@ \nn{refer elsewhere for details?} -Recall we described a system of fields and local relations based on a `traditional $n$-category' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} +Recall we described a system of fields and local relations based on a `traditional $n$-category' +$C$ in Example \ref{ex:traditional-n-categories(fields)} above. +Constructing a system of fields from $\cC$ recovers that example. +\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.} \end{example} Finally, we describe a version of the bordism $n$-category suitable to our definitions. @@ -593,7 +613,8 @@ \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} \end{example} -See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. +See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to +homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. \begin{example}[Blob complexes of balls (with a fiber)] \rm @@ -606,9 +627,19 @@ where $\bc^\cE_*$ denotes the blob complex based on $\cE$. \end{example} -This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial, but mostly uninteresting, way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. +This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. +Notice that with $F$ a point, the above example is a construction turning a topological +$n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$. +We think of this as providing a `free resolution' of the topological $n$-category. +\todo{Say more here!} +In fact, there is also a trivial, but mostly uninteresting, way to do this: +we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, +and take $\CD{B}$ to act trivially. -Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. +Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. +It's easy to see that with $n=0$, the corresponding system of fields is just +linear combinations of connected components of $T$, and the local relations are trivial. +There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. \begin{example}[The bordism $n$-category, $A_\infty$ version] \rm @@ -639,15 +670,30 @@ %\subsection{From $n$-categories to systems of fields} \subsection{From balls to manifolds} \label{ss:ncat_fields} \label{ss:ncat-coend} -In this section we describe how to extend an $n$-category $\cC$ as described above (of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. This extension is a certain colimit, and we've chosen the notation to remind you of this. +In this section we describe how to extend an $n$-category $\cC$ as described above +(of either the plain or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. +This extension is a certain colimit, and we've chosen the notation to remind you of this. That is, we show that functors $\cC_k$ satisfying the axioms above have a canonical extension -from $k$-balls to arbitrary $k$-manifolds. Recall that we've already anticipated this construction in the previous section, inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, so that we can state the boundary axiom for $\cC$ on $k+1$-balls. -In the case of plain $n$-categories, this construction factors into a construction of a system of fields and local relations, followed by the usual TQFT definition of a vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. -For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. +from $k$-balls to arbitrary $k$-manifolds. +Recall that we've already anticipated this construction in the previous section, +inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, +so that we can state the boundary axiom for $\cC$ on $k+1$-balls. +In the case of plain $n$-categories, this construction factors into a construction of a +system of fields and local relations, followed by the usual TQFT definition of a +vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. +For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. +Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution', +an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above). +We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant +for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex for $M$ with coefficients in $\cC$. We will first define the `cell-decomposition' poset $\cell(W)$ 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 in the $A_\infty$ case) 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 complexes to $n$-manifolds with boundary data), then the resulting colimit 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). +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 in the $A_\infty$ case) 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 complexes to $n$-manifolds with boundary data), +then the resulting colimit 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). \begin{defn} Say that a `permissible decomposition' of $W$ is a cell decomposition @@ -659,7 +705,8 @@ Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$. -The category $\cell(W)$ has objects the permissible decompositions of $W$, and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. +The category $\cell(W)$ has objects the permissible decompositions of $W$, +and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. See Figure \ref{partofJfig} for an example. \end{defn} @@ -695,7 +742,8 @@ When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and -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.) +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.) Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we fix a field on $\bd W$ (i.e. fix an element of the colimit associated to $\bd W$). @@ -710,12 +758,17 @@ \end{defn} \begin{defn}[System of fields functor, $A_\infty$ case] -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 $\psi_{\cC;W}$. +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 $\psi_{\cC;W}$. \end{defn} -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 $W$. +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 $W$. -We 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 $W$ +We 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 $W$ \begin{equation*} \cC(W,c) = \left( \bigoplus_x \psi_{\cC;W,c}(x)\right) \big/ K \end{equation*} @@ -732,7 +785,9 @@ \[ V = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , \] -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 non-standard: if a complex $U$ is concentrated in degree $0$, the complex $U[m]$ is concentrated in degree $m$.) +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 non-standard: if a complex $U$ is concentrated in degree $0$, +the complex $U[m]$ is concentrated in degree $m$.) We endow $V$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ summands plus another term using the differential of the simplicial set of $m$-sequences. More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ @@ -752,7 +807,8 @@ permissible decomposition (filtration degree 0). Then we glue these together with mapping cylinders coming from gluing maps (filtration degree 1). -Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on. +Then we kill the extra homology we just introduced with mapping +cylinders between the mapping cylinders (filtration degree 2), and so on. $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. @@ -781,7 +837,9 @@ \nn{should also develop $\pi_{\le n}(T, S)$ as a module for $\pi_{\le n}(T)$, where $S\sub T$.} -Throughout, we fix an $n$-category $\cC$. For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. We state the final axiom, on actions of homeomorphisms, differently in the two cases. +Throughout, we fix an $n$-category $\cC$. +For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. +We state the final axiom, on actions of homeomorphisms, differently in the two cases. Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ @@ -819,7 +877,8 @@ the category of marked $k$-hemispheres and homeomorphisms to the category of sets and bijections.} \end{lem} -The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. We use the same type of colimit construction. +The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. +We use the same type of colimit construction. In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$. @@ -1040,13 +1099,22 @@ \end{example} \begin{example} -Suppose $S$ is a topological space, with a subspace $T$. We can define a module $\pi_{\leq n}(S,T)$ so that on each marked $k$-ball $(B,N)$ for $k