# HG changeset patch # User Kevin Walker # Date 1308333703 21600 # Node ID 40729de8e06739e73c3d3509d5e60530da656a49 # Parent d30537de52c7248d47959728815e80f7d1cc016f finish fam-o-homeo axiom revisions and discussion diff -r d30537de52c7 -r 40729de8e067 blob to-do --- a/blob to-do Thu Jun 16 11:11:41 2011 -0600 +++ b/blob to-do Fri Jun 17 12:01:43 2011 -0600 @@ -4,8 +4,6 @@ * Consider moving A_\infty stuff to a subsection -* consider putting conditions for enriched n-cat all in one place - * Peter's suggestion for A_inf definition * Boundary of colimit -- not so easy to see! @@ -33,6 +31,8 @@ * leftover: we used to require that composition of A-infinity n-morphisms was injective (just like lower morphisms). Should we stick this back in? I don't think we use it anywhere. +* should we require, for A-inf n-cats, that families which preserve product morphisms act trivially? + * SCOTT will go through appendix C.2 and make it better diff -r d30537de52c7 -r 40729de8e067 blob_changes_v3 --- a/blob_changes_v3 Thu Jun 16 11:11:41 2011 -0600 +++ b/blob_changes_v3 Fri Jun 17 12:01:43 2011 -0600 @@ -26,6 +26,7 @@ - more details on axioms for enriched n-cats - added details to the construction of traditional 1-categories from disklike 1-categories (Appendix C.1) - extended the lemmas of Appendix B (about adapting families of homeomorphisms to open covers) to the topological category +- modified families-of-homeomorphisms-action axiom for A-infinity n-categories, and added discussion of alternatives diff -r d30537de52c7 -r 40729de8e067 text/kw_macros.tex --- a/text/kw_macros.tex Thu Jun 16 11:11:41 2011 -0600 +++ b/text/kw_macros.tex Fri Jun 17 12:01:43 2011 -0600 @@ -63,7 +63,7 @@ % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}; +\applytolist{declaremathop}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{Homeo}{sign}{supp}{Nbd}{res}{rad}{Compat}{Coll}; \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator*{\hocolim}{hocolim} diff -r d30537de52c7 -r 40729de8e067 text/ncat.tex --- a/text/ncat.tex Thu Jun 16 11:11:41 2011 -0600 +++ b/text/ncat.tex Fri Jun 17 12:01:43 2011 -0600 @@ -590,7 +590,7 @@ We define a map \begin{eqnarray*} \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ - a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . + a & \mapsto & s_{Y,J}(a \bullet ((a|_Y)\times J)) . \end{eqnarray*} (See Figure \ref{glue-collar}.) \begin{figure}[t] @@ -654,7 +654,7 @@ Let $X$ be an $n$-ball, $b \in \cC(X)$, and $f: X\to X$ be a homeomorphism which acts trivially on the restriction $\bd b$ of $b$ to $\bd X$. Suppose furthermore that $f$ is isotopic to the identity through homeomorphisms which -trivially on $\bd b$. +act trivially on $\bd b$. Then $f(b) = b$. In addition, collar maps act trivially on $\cC(X)$. \end{axiom} @@ -696,7 +696,9 @@ Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the category $\bbc$ of {\it $n$-balls with boundary conditions}. Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". -Its morphisms are homeomorphisms $f:X\to X$ such that $f|_{\bd X}(c) = c$. +The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X,c; X', c')$, are +homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. +%Let $\pi_0(\bbc)$ denote \begin{axiom}[Enriched $n$-categories] \label{axiom:enriched} @@ -705,6 +707,8 @@ and modifies the axioms for $k=n$ as follows: \begin{itemize} \item Morphisms. We have a functor $\cC_n$ from $\bbc$ ($n$-balls with boundary conditions) to $\cS$. +%[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. +%In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. Let $Y_i = \bd B_i \setmin Y$. Note that $\bd B = Y_1\cup Y_2$. @@ -727,46 +731,74 @@ to require that families of homeomorphisms act and obtain an $A_\infty$ $n$-category. +\noop{ We believe that abstract definitions should be guided by diverse collections of concrete examples, and a lack of diversity in our present collection of examples of $A_\infty$ $n$-categories makes us reluctant to commit to an all-encompassing general definition. Instead, we will give a relatively narrow definition which covers the examples we consider in this paper. After stating it, we will briefly discuss ways in which it can be made more general. +} -Assume that our $n$-morphisms are enriched over chain complexes. -Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and -$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. - -\nn{need to loosen for bbc reasons} +Recall the category $\bbc$ of balls with boundary conditions. +Note that the morphisms $\Homeo(X,c; X', c')$ from $(X, c)$ to $(X', c')$ form a topological space. +Let $\cS$ be an appropriate $\infty$-category (e.g.\ chain complexes) +and let $\cJ$ be an $\infty$-functor from topological spaces to $\cS$ +(e.g.\ the singular chain functor $C_*$). \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] \label{axiom:families} -For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes +For each pair of $n$-balls $X$ and $X'$ and each pair $c\in \cl{\cC}(\bd X)$ and $c'\in \cl{\cC}(\bd X')$ we have an $\cS$-morphism \[ - C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . + \cJ(\Homeo(X,c; X', c')) \ot \cC(X; c) \to \cC(X'; c') . \] +Similarly, we have an $\cS$-morphism +\[ + \cJ(\Coll(X,c)) \ot \cC(X; c) \to \cC(X; c), +\] +where $\Coll(X,c)$ denotes the space of collar maps. +(See below for further discussion.) These action maps are required to be associative up to coherent homotopy, and also compatible with composition (gluing) in the sense that a diagram like the one in Theorem \ref{thm:CH} commutes. -%\nn{repeat diagram here?} -%\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} -On $C_0(\Homeo_\bd(X))\ot \cC(X; c)$ the action should coincide -with the one coming from Axiom \ref{axiom:morphisms}. +% say something about compatibility with product morphisms? \end{axiom} -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 $\r$. -Having chains on the space of collar maps act gives rise to coherence maps involving -weak identities. -We will not pursue this in detail here. +We now describe the topology on $\Coll(X; c)$. +We retain notation from the above definition of collar map. +Each collaring homeomorphism $X \cup (Y\times J) \to X$ determines a map from points $p$ of $\bd X$ to +(possibly zero-width) embedded intervals in $X$ terminating at $p$. +If $p \in Y$ this interval is the image of $\{p\}\times J$. +If $p \notin Y$ then $p$ is assigned the zero-width interval $\{p\}$. +Such collections of intervals have a natural topology, and $\Coll(X; c)$ inherits its topology from this. +Note in particular that parts of the collar are allowed to shrink continuously to zero width. +(This is the real content; if nothing shrinks to zero width then the action of families of collar +maps follows from the action of families of homeomorphisms and compatibility with gluing.) -One potential variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. -(In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} +The $k=n$ case of Axiom \ref{axiom:morphisms} posits a {\it strictly} associative action of {\it sets} +$\Homeo(X,c; X', c') \times \cC(X; c) \to \cC(X'; c')$, and at first it might seem that this would force the above +action of $\cJ(\Homeo(X,c; X', c'))$ to be strictly associative as well (assuming the two actions are compatible). +In fact, compatibility implies less than this. +For simplicity, assume that $\cJ$ is $C_*$, the singular chains functor. +(This is the example most relevant to this paper.) +Then compatibility implies that the action of $C_*(\Homeo(X,c; X', c'))$ agrees with the action +of $C_0(\Homeo(X,c; X', c'))$ coming from Axiom \ref{axiom:morphisms}, so we only require associativity in degree zero. +And indeed, this is true for our main example of an $A_\infty$ $n$-category based on the blob construction. +Stating this sort of compatibility for general $\cS$ and $\cJ$ requires further assumptions, +such as the forgetful functor from $\cS$ to sets having a left adjoint, and $\cS$ having an internal Hom. +An alternative (due to Peter Teichner) is to say that Axiom \ref{axiom:families} +supersedes the $k=n$ case of Axiom \ref{axiom:morphisms}; in dimension $n$ we just have a +functor $\bbc \to \cS$ of $A_\infty$ 1-categories. +(This assumes some prior notion of $A_\infty$ 1-category.) +We are not currently aware of any examples which require this sort of greater generality, so we think it best +to refrain from settling on a preferred version of the axiom until +we have a greater variety of examples to guide the choice. + +Another variant of the above axiom would be to drop the ``up to homotopy" and require a strictly associative action. +In fact, the alternative construction of the blob complex described in \S \ref{ss:alt-def} gives $n$-categories as in Example \ref{ex:blob-complexes-of-balls} which satisfy this stronger axiom; -since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across.) +since that construction is only homotopy equivalent to the usual one, only the weaker axiom carries across. +\noop{ Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category into a ordinary $n$-category (enriched over graded groups). In a different direction, if we enrich over topological spaces instead of chain complexes, @@ -774,15 +806,7 @@ instead of $C_*(\Homeo_\bd(X))$. Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex type $A_\infty$ $n$-category. - -One possibility for generalizing the above axiom to encompass a wider variety of examples goes as follows. -(Credit for these ideas goes to Peter Teichner, but of course the blame for any flaws goes to us.) -Let $\cS$ be an $A_\infty$ 1-category. -(We assume some prior notion of $A_\infty$ 1-category.) -Note that the category $\cB\cB\cC$ of balls with boundary conditions, defined above, is enriched in the category -of topological spaces, and hence can also be regarded as an $A_\infty$ 1-category. -\nn{...} - +} \medskip