# HG changeset patch # User Scott Morrison # Date 1280344587 25200 # Node ID cb76847c439e99f6f9376858f7126cfcc1040015 # Parent 606f685e3764fa5a60b3b1032092c1c5b3d3e3bb many small fixes in ncat.tex diff -r 606f685e3764 -r cb76847c439e text/evmap.tex --- a/text/evmap.tex Wed Jul 28 12:15:58 2010 -0700 +++ b/text/evmap.tex Wed Jul 28 12:16:27 2010 -0700 @@ -434,7 +434,7 @@ Note that if $a \ge 2r$ then $\Nbd_a(S) \sup B_r(y)$. Let $z\in \Nbd_a(S) \setmin B_r(y)$. Consider the triangle -with vertices $z$, $y$ and $s$ with $s\in S$. +with vertices $z$, $y$ and $s$ with $s\in S$ such that $z \in B_a(s)$. The length of the edge $yz$ is greater than $r$ which is greater than the length of the edge $ys$. It follows that the angle at $z$ is less than $\pi/2$ (less than $\pi/3$, in fact), diff -r 606f685e3764 -r cb76847c439e text/ncat.tex --- a/text/ncat.tex Wed Jul 28 12:15:58 2010 -0700 +++ b/text/ncat.tex Wed Jul 28 12:16:27 2010 -0700 @@ -10,7 +10,7 @@ \label{ss:n-cat-def} Before proceeding, we need more appropriate definitions of $n$-categories, -$A_\infty$ $n$-categories, modules for these, and tensor products of these modules. +$A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. (As is the case throughout this paper, by ``$n$-category" we mean some notion of a ``weak" $n$-category with ``strong duality".) @@ -24,6 +24,8 @@ For examples of a more purely algebraic origin, one would typically need the combinatorial results that we have avoided here. +\nn{Say something explicit about Lurie's work here? It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} + \medskip There are many existing definitions of $n$-categories, with various intended uses. @@ -58,7 +60,7 @@ \end{axiom} -(Note: We usually omit the subscript $k$.) +(Note: We often omit the subscript $k$.) We are being deliberately vague about what flavor of $k$-balls we are considering. @@ -70,14 +72,14 @@ For each flavor of manifold there is a corresponding flavor of $n$-category. For simplicity, we will concentrate on the case of PL unoriented manifolds. -(The ambitious reader may want to keep in mind two other classes of balls. +An ambitious reader may want to keep in mind two other classes of balls. The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). -This will be used below to describe the blob complex of a fiber bundle with +This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with base space $Y$. The second is balls equipped with a section of the tangent bundle, or the frame -bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. +bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle. These can be used to define categories with less than the ``strong" duality we assume here, -though we will not develop that idea fully in this paper.) +though we will not develop that idea fully in this paper. Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries of morphisms). @@ -88,7 +90,7 @@ 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. +We prefer not to 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. @@ -118,9 +120,8 @@ These maps, for various $X$, comprise a natural transformation of functors. \end{axiom} -(Note that the first ``$\bd$" above is part of the data for the category, -while the second is the ordinary boundary of manifolds.) - +Note that the first ``$\bd$" above is part of the data for the category, +while the second is the ordinary boundary of manifolds. Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. Most of the examples of $n$-categories we are interested in are enriched in the following sense. @@ -130,14 +131,14 @@ \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} and all the structure maps of the $n$-category should be compatible with the auxiliary category structure. -Note that this auxiliary structure is only in dimension $n$; -$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. +Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then +$\cC(Y; c)$ is just a plain set. \medskip -(In order to simplify the exposition we have concentrated on the case of +In order to simplify the exposition we have concentrated on the case of unoriented PL manifolds and avoided the question of what exactly we mean by -the boundary a manifold with extra structure, such as an oriented manifold. +the boundary of a manifold with extra structure, such as an oriented manifold. In general, all manifolds of dimension less than $n$ should be equipped with the germ of a thickening to dimension $n$, and this germ should carry whatever structure we have on $n$-manifolds. @@ -147,7 +148,7 @@ For example, the boundary of an oriented $n$-ball should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent bundle and a choice of direction in this bundle indicating -which side the $n$-ball lies on.) +which side the $n$-ball lies on. \medskip @@ -188,7 +189,7 @@ \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} Note that we insist on injectivity above. -The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. +The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...} Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". @@ -199,7 +200,7 @@ We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ a {\it restriction} map and write $\res_{B_i}(a)$ (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. -More generally, we also include under the rubric ``restriction map" the +More generally, we also include under the rubric ``restriction map" the boundary maps of Axiom \ref{nca-boundary} above, another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition of restriction maps. @@ -251,15 +252,16 @@ \begin{axiom}[Strict associativity] \label{nca-assoc} The composition (gluing) maps above are strictly associative. -Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$, -any sequence of gluings of the smaller balls yields the same result. +Given any splitting of a ball $B$ into smaller balls +$$\bigsqcup B_i \to B,$$ +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. \end{axiom} \begin{figure}[!ht] $$\mathfig{.65}{ncat/strict-associativity}$$ \caption{An example of strict associativity.}\label{blah6}\end{figure} -We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. +We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. %Compositions of boundary and restriction maps will also be called restriction maps. @@ -271,22 +273,22 @@ ``splittable along $Y$'' or ``transverse to $Y$''. We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. -More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. +More generally, let $\alpha$ be a splitting of $X$ into smaller balls. Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from the smaller balls to $X$. We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". -In situations where the subdivision is notationally anonymous, we will write +In situations where the splitting is notationally anonymous, we will write $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) -the unnamed subdivision. -If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; +the unnamed splitting. +If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; this can also be denoted $\cC(X)\spl$ if the context contains an anonymous -subdivision of $\bd X$ and no competing subdivision of $X$. +decomposition of $\bd X$ and no competing splitting of $X$. The above two composition axioms are equivalent to the following one, which we state in slightly vague form. \xxpar{Multi-composition:} -{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball +{Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball into small $k$-balls, there is a map from an appropriate subset (like a fibered product) of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, @@ -384,11 +386,9 @@ (We thank Kevin Costello for suggesting this approach.) Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, -and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension +and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension $l \le m$, with $l$ depending on $x$. - It is easy to see that a composition of pinched products is again a pinched product. - A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction $\pi:E'\to \pi(E')$ is again a pinched product. A {union} of pinched products is a decomposition $E = \cup_i E_i$ @@ -523,15 +523,14 @@ \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. -Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. +Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. \end{axiom} This axiom needs to be strengthened to force product morphisms to act as the identity. Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. Let $J$ be a 1-ball (interval). We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. -(Here we use the ``pinched" version of $Y\times J$. -\nn{do we need notation for this?}) +(Here we use $Y\times J$ with boundary entirely pinched.) We define a map \begin{eqnarray*} \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ @@ -625,7 +624,7 @@ %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} \end{axiom} -We should strengthen the above axiom to apply to families of collar maps. +We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. To do this we need to explain how collar maps form a topological space. Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, and we can replace the class of all intervals $J$ with intervals contained in $\r$. @@ -652,10 +651,10 @@ invariance in dimension $n$, while in the fields definition we 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 +Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to balls and, at level $n$, quotienting out by the local relations: \begin{align*} -\cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k