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%!TEX root = ../blob1.tex
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\def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip}
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\section{$n$-categories (maybe)}
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\label{sec:ncats}
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\nn{experimental section. maybe this should be rolled into other sections.
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maybe it should be split off into a separate paper.}
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108
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\subsection{Definition of $n$-categories}
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Before proceeding, we need more appropriate definitions of $n$-categories,
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$A_\infty$ $n$-categories, modules for these, and tensor products of these modules.
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(As is the case throughout this paper, by ``$n$-category" we mean
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a weak $n$-category with strong duality.)
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Consider first ordinary $n$-categories.
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We need a set (or sets) of $k$-morphisms for each $0\le k \le n$.
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We must decide on the ``shape" of the $k$-morphisms.
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Some $n$-category definitions model $k$-morphisms on the standard bihedron (interval, bigon, ...).
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Other definitions have a separate set of 1-morphisms for each interval $[0,l] \sub \r$,
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a separate set of 2-morphisms for each rectangle $[0,l_1]\times [0,l_2] \sub \r^2$,
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and so on.
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(This allows for strict associativity.)
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Still other definitions \nn{need refs for all these; maybe the Leinster book}
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model the $k$-morphisms on more complicated combinatorial polyhedra.
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We will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to
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the standard $k$-ball.
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In other words,
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\xxpar{Morphisms (preliminary version):}
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{For any $k$-manifold $X$ homeomorphic
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to the standard $k$-ball, we have a set of $k$-morphisms
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$\cC_k(X)$.}
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Terminology: By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the
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standard $k$-ball.
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We {\it do not} assume that it is equipped with a
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preferred homeomorphism to the standard $k$-ball.
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The same goes for ``a $k$-sphere" below.
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Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on
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the boundary), we want a corresponding
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bijection of sets $f:\cC(X)\to \cC(Y)$.
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(This will imply ``strong duality", among other things.)
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So we replace the above with
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\xxpar{Morphisms:}
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{For each $0 \le k \le n$, we have a functor $\cC_k$ from
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the category of $k$-balls and
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homeomorphisms to the category of sets and bijections.}
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(Note: We usually omit the subscript $k$.)
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We are being deliberately vague about what flavor of manifolds we are considering.
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They could be unoriented or oriented or Spin or $\mbox{Pin}_\pm$.
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They could be topological or PL or smooth.
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\nn{need to check whether this makes much difference --- see pseudo-isotopy below}
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(If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need
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to be fussier about corners.)
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For each flavor of manifold there is a corresponding flavor of $n$-category.
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We will concentrate of the case of PL unoriented manifolds.
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Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries
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of morphisms).
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The 0-sphere is unusual among spheres in that it is disconnected.
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Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range.
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(Actually, this is only true in the oriented case, with 1-morphsims parameterized
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by oriented 1-balls.)
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For $k>1$ and in the presence of strong duality the domain/range division makes less sense.
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\nn{maybe say more here; rotate disk, Frobenius reciprocity blah blah}
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We prefer to combine the domain and range into a single entity which we call the
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boundary of a morphism.
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Morphisms are modeled on balls, so their boundaries are modeled on spheres:
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\xxpar{Boundaries (domain and range), part 1:}
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{For each $0 \le k \le n-1$, we have a functor $\cC_k$ from
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the category of $k$-spheres and
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homeomorphisms to the category of sets and bijections.}
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(In order to conserve symbols, we use the same symbol $\cC_k$ for both morphisms and boundaries.)
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\xxpar{Boundaries, part 2:}
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{For each $k$-ball $X$, we have a map of sets $\bd: \cC(X)\to \cC(\bd X)$.
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These maps, for various $X$, comprise a natural transformation of functors.}
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(Note that the first ``$\bd$" above is part of the data for the category,
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while the second is the ordinary boundary of manifolds.)
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Given $c\in\cC(\bd(X))$, let $\cC(X; c) \deq \bd^{-1}(c)$.
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Most of the examples of $n$-categories we are interested in are enriched in the following sense.
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The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
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all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category
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(e.g.\ vector spaces, or modules over some ring, or chain complexes),
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and all the structure maps of the $n$-category should be compatible with the auxiliary
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category structure.
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Note that this auxiliary structure is only in dimension $n$;
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$\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
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\medskip
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\nn{
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%At the moment I'm a little confused about orientations, and more specifically
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%about the role of orientation-reversing maps of boundaries when gluing oriented manifolds.
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Maybe need a discussion about what the boundary of a manifold with a
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structure (e.g. orientation) means.
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Tentatively, I think we need to redefine the oriented boundary of an oriented $n$-manifold.
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Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal
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first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold
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equipped with an orientation of its once-stabilized tangent bundle.
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Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of
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their $k$ times stabilized tangent bundles.
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Probably should also have a framing of the stabilized dimensions in order to indicate which
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side the bounded manifold is on.
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For the moment just stick with unoriented manifolds.}
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\medskip
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We have just argued that the boundary of a morphism has no preferred splitting into
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domain and range, but the converse meets with our approval.
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That is, given compatible domain and range, we should be able to combine them into
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the full boundary of a morphism:
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\xxpar{Domain $+$ range $\to$ boundary:}
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{Let $S = B_1 \cup_E B_2$, where $S$ is a $k$-sphere ($0\le k\le n-1$),
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$B_i$ is a $k$-ball, and $E = B_1\cap B_2$ is a $k{-}1$-sphere.
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Let $\cC(B_1) \times_{\cC(E)} \cC(B_2)$ denote the fibered product of the
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two maps $\bd: \cC(B_i)\to \cC(E)$.
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Then (axiom) we have an injective map
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\[
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\gl_E : \cC(B_1) \times_{\cC(E)} \cC(B_2) \to \cC(S)
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\]
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which is natural with respect to the actions of homeomorphisms.}
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Note that we insist on injectivity above.
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Let $\cC(S)_E$ denote the image of $\gl_E$.
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We will refer to elements of $\cC(S)_E$ as ``splittable along $E$" or ``transverse to $E$".
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We will call the projection $\cC(S)_E \to \cC(B_i)$
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a {\it restriction} map and write $\res_{B_i}(a)$
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(or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$.
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These restriction maps can be thought of as
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domain and range maps, relative to the choice of splitting $S = B_1 \cup_E B_2$.
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If $B$ is a $k$-ball and $E \sub \bd B$ splits $\bd B$ into two $k{-}1$-balls
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as above, then we define $\cC(B)_E = \bd^{-1}(\cC(\bd B)_E)$.
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150 |
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Next we consider composition of morphisms.
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For $n$-categories which lack strong duality, one usually considers
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$k$ different types of composition of $k$-morphisms, each associated to a different direction.
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(For example, vertical and horizontal composition of 2-morphisms.)
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In the presence of strong duality, these $k$ distinct compositions are subsumed into
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one general type of composition which can be in any ``direction".
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\xxpar{Composition:}
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|
159 |
{Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$)
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160 |
and $Y = B_1\cap B_2$ is a $k{-}1$-ball.
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161 |
Let $E = \bd Y$, which is a $k{-}2$-sphere.
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Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$.
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We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$.
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Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps.
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Then (axiom) we have a map
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\[
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\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E
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\]
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which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
|
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to the intersection of the boundaries of $B$ and $B_i$.
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If $k < n$ we require that $\gl_Y$ is injective.
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(For $k=n$, see below.)}
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|
173 |
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|
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\xxpar{Strict associativity:}
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{The composition (gluing) maps above are strictly associative.}
|
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109
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Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$.
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In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$
|
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a {\it restriction} map and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$.
|
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Compositions of boundary and restriction maps will also be called restriction maps.
|
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For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a
|
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restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$.
|
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|
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%More notation and terminology:
|
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%We will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ a {\it restriction}
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%map
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187 |
|
102
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The above two axioms are equivalent to the following axiom,
|
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which we state in slightly vague form.
|
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|
|
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\xxpar{Multi-composition:}
|
|
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{Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball
|
|
193 |
into small $k$-balls, there is a
|
|
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map from an appropriate subset (like a fibered product)
|
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of $\cC(B_1)\times\cdots\times\cC(B_m)$ to $\cC(B)$,
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196 |
and these various $m$-fold composition maps satisfy an
|
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operad-type strict associativity condition.}
|
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|
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The next axiom is related to identity morphisms, though that might not be immediately obvious.
|
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|
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\xxpar{Product (identity) morphisms:}
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{Let $X$ be a $k$-ball and $D$ be an $m$-ball, with $k+m \le n$.
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Then we have a map $\cC(X)\to \cC(X\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cC(X)$.
|
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If $f:X\to X'$ and $\tilde{f}:X\times D \to X'\times D'$ are maps such that the diagram
|
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\[ \xymatrix{
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X\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & X'\times D' \ar[d]^{\pi} \\
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X \ar[r]^{f} & X'
|
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} \]
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commutes, then we have
|
|
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\[
|
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\tilde{f}(a\times D) = f(a)\times D' .
|
|
212 |
\]
|
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Product morphisms are compatible with gluing (composition) in both factors:
|
|
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\[
|
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(a'\times D)\bullet(a''\times D) = (a'\bullet a'')\times D
|
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216 |
\]
|
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and
|
|
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\[
|
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(a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') .
|
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\]
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\nn{problem: if pinched boundary, then only one factor}
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Product morphisms are associative:
|
|
223 |
\[
|
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224 |
(a\times D)\times D' = a\times (D\times D') .
|
|
225 |
\]
|
|
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(Here we are implicitly using functoriality and the obvious homeomorphism
|
|
227 |
$(X\times D)\times D' \to X\times(D\times D')$.)
|
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|
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Product morphisms are compatible with restriction:
|
|
229 |
\[
|
|
230 |
\res_{X\times E}(a\times D) = a\times E
|
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231 |
\]
|
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232 |
for $E\sub \bd D$ and $a\in \cC(X)$.
|
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}
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|
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|
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\nn{need even more subaxioms for product morphisms?}
|
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|
|
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All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories.
|
|
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The last axiom (below), concerning actions of
|
|
239 |
homeomorphisms in the top dimension $n$, distinguishes the two cases.
|
|
240 |
|
|
241 |
We start with the plain $n$-category case.
|
|
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|
|
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\xxpar{Isotopy invariance in dimension $n$ (preliminary version):}
|
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|
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{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
|
95
|
245 |
to the identity on $\bd X$ and is isotopic (rel boundary) to the identity.
|
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|
246 |
Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$.}
|
|
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|
|
248 |
We will strengthen the above axiom in two ways.
|
|
249 |
(Amusingly, these two ways are related to each of the two senses of the term
|
|
250 |
``pseudo-isotopy".)
|
|
251 |
|
|
252 |
First, we require that $f$ act trivially on $\cC(X)$ if it is pseudo-isotopic to the identity
|
|
253 |
in the sense of homeomorphisms of mapping cylinders.
|
|
254 |
This is motivated by TQFT considerations:
|
|
255 |
If the mapping cylinder of $f$ is homeomorphic to the mapping cylinder of the identity,
|
|
256 |
then these two $n{+}1$-manifolds should induce the same map from $\cC(X)$ to itself.
|
|
257 |
\nn{is there a non-TQFT reason to require this?}
|
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|
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|
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|
259 |
Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity.
|
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|
260 |
Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball.
|
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|
261 |
Let $J$ be a 1-ball (interval).
|
|
262 |
We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$.
|
|
263 |
We define a map
|
|
264 |
\begin{eqnarray*}
|
|
265 |
\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
|
|
266 |
a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
|
|
267 |
\end{eqnarray*}
|
|
268 |
\nn{need to say something somewhere about pinched boundary convention for products}
|
|
269 |
We will call $\psi_{Y,J}$ an extended isotopy.
|
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|
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\nn{or extended homeomorphism? see below.}
|
|
271 |
\nn{maybe remark that in some examples (e.g.\ ones based on sub cell complexes)
|
|
272 |
extended isotopies are also plain isotopies, so
|
|
273 |
no extension necessary}
|
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|
274 |
It can be thought of as the action of the inverse of
|
|
275 |
a map which projects a collar neighborhood of $Y$ onto $Y$.
|
|
276 |
(This sort of collapse map is the other sense of ``pseudo-isotopy".)
|
|
277 |
\nn{need to check this}
|
|
278 |
|
|
279 |
The revised axiom is
|
|
280 |
|
|
281 |
\xxpar{Pseudo and extended isotopy invariance in dimension $n$:}
|
103
|
282 |
{Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts
|
96
|
283 |
to the identity on $\bd X$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity.
|
|
284 |
Then $f$ acts trivially on $\cC(X)$.}
|
|
285 |
|
|
286 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
|
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|
287 |
|
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|
288 |
\smallskip
|
|
289 |
|
|
290 |
For $A_\infty$ $n$-categories, we replace
|
|
291 |
isotopy invariance with the requirement that families of homeomorphisms act.
|
|
292 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes.
|
|
293 |
|
|
294 |
\xxpar{Families of homeomorphisms act.}
|
|
295 |
{For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes
|
|
296 |
\[
|
|
297 |
C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) .
|
|
298 |
\]
|
|
299 |
Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$
|
|
300 |
which fix $\bd X$.
|
|
301 |
These action maps are required to be associative up to homotopy
|
|
302 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
|
|
303 |
a diagram like the one in Proposition \ref{CDprop} commutes.
|
|
304 |
\nn{repeat diagram here?}
|
|
305 |
\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?}}
|
|
306 |
|
|
307 |
We should strengthen the above axiom to apply to families of extended homeomorphisms.
|
109
|
308 |
To do this we need to explain how extended homeomorphisms form a topological space.
|
97
|
309 |
Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology,
|
|
310 |
and we can replace the class of all intervals $J$ with intervals contained in $\r$.
|
|
311 |
\nn{need to also say something about collaring homeomorphisms.}
|
|
312 |
\nn{this paragraph needs work.}
|
|
313 |
|
103
|
314 |
Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category
|
|
315 |
into a plain $n$-category (enriched over graded groups).
|
97
|
316 |
\nn{say more here?}
|
|
317 |
In the other direction, if we enrich over topological spaces instead of chain complexes,
|
|
318 |
we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting
|
|
319 |
instead of $C_*(\Homeo_\bd(X))$.
|
|
320 |
Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex
|
|
321 |
type $A_\infty$ $n$-category.
|
|
322 |
|
99
|
323 |
\medskip
|
97
|
324 |
|
99
|
325 |
The alert reader will have already noticed that our definition of (plain) $n$-category
|
|
326 |
is extremely similar to our definition of topological fields.
|
|
327 |
The only difference is that for the $n$-category definition we restrict our attention to balls
|
|
328 |
(and their boundaries), while for fields we consider all manifolds.
|
|
329 |
\nn{also: difference at the top dimension; fix this}
|
|
330 |
Thus a system of fields determines an $n$-category simply by restricting our attention to
|
|
331 |
balls.
|
|
332 |
The $n$-category can be thought of as the local part of the fields.
|
|
333 |
Conversely, given an $n$-category we can construct a system of fields via
|
|
334 |
\nn{gluing, or a universal construction}
|
109
|
335 |
\nn{see subsection below}
|
99
|
336 |
|
|
337 |
\nn{Next, say something about $A_\infty$ $n$-categories and ``homological" systems
|
|
338 |
of fields.
|
|
339 |
The universal (colimit) construction becomes our generalized definition of blob homology.
|
|
340 |
Need to explain how it relates to the old definition.}
|
97
|
341 |
|
95
|
342 |
\medskip
|
|
343 |
|
101
|
344 |
\nn{these examples need to be fleshed out a bit more}
|
|
345 |
|
|
346 |
Examples of plain $n$-categories:
|
|
347 |
\begin{itemize}
|
|
348 |
|
|
349 |
\item Let $F$ be a closed $m$-manifold (e.g.\ a point).
|
|
350 |
Let $T$ be a topological space.
|
|
351 |
For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of
|
|
352 |
all maps from $X\times F$ to $T$.
|
|
353 |
For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo
|
103
|
354 |
homotopies fixed on $\bd X \times F$.
|
101
|
355 |
(Note that homotopy invariance implies isotopy invariance.)
|
|
356 |
For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to
|
|
357 |
be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection.
|
|
358 |
|
|
359 |
\item We can linearize the above example as follows.
|
|
360 |
Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$
|
|
361 |
(e.g.\ the trivial cocycle).
|
|
362 |
For $X$ of dimension less than $n$ define $\cC(X)$ as before.
|
|
363 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be
|
|
364 |
the $R$-module of finite linear combinations of maps from $X\times F$ to $T$,
|
|
365 |
modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy
|
|
366 |
$h: X\times F\times I \to T$, then $a \sim \alpha(h)b$.
|
|
367 |
\nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
|
|
368 |
|
|
369 |
\item Given a traditional $n$-category $C$ (with strong duality etc.),
|
|
370 |
define $\cC(X)$ (with $\dim(X) < n$)
|
|
371 |
to be the set of all $C$-labeled sub cell complexes of $X$.
|
|
372 |
For $X$ an $n$-ball and $c\in \cC(\bd X)$, define $\cC(X)$ to finite linear
|
|
373 |
combinations of $C$-labeled sub cell complexes of $X$
|
|
374 |
modulo the kernel of the evaluation map.
|
|
375 |
Define a product morphism $a\times D$ to be the product of the cell complex of $a$ with $D$,
|
|
376 |
and with the same labeling as $a$.
|
102
|
377 |
More generally, start with an $n{+}m$-category $C$ and a closed $m$-manifold $F$.
|
|
378 |
Define $\cC(X)$, for $\dim(X) < n$,
|
|
379 |
to be the set of all $C$-labeled sub cell complexes of $X\times F$.
|
|
380 |
Define $\cC(X; c)$, for $X$ an $n$-ball,
|
|
381 |
to be the dual Hilbert space $A(X\times F; c)$.
|
101
|
382 |
\nn{refer elsewhere for details?}
|
|
383 |
|
|
384 |
\item Variation on the above examples:
|
103
|
385 |
We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$,
|
101
|
386 |
for example product boundary conditions or take the union over all boundary conditions.
|
102
|
387 |
\nn{maybe should not emphasize this case, since it's ``better" in some sense
|
|
388 |
to think of these guys as affording a representation
|
|
389 |
of the $n{+}1$-category associated to $\bd F$.}
|
101
|
390 |
|
|
391 |
\end{itemize}
|
|
392 |
|
|
393 |
|
|
394 |
Examples of $A_\infty$ $n$-categories:
|
|
395 |
\begin{itemize}
|
|
396 |
|
|
397 |
\item Same as in example \nn{xxxx} above (fiber $F$, target space $T$),
|
|
398 |
but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex
|
|
399 |
$C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary,
|
|
400 |
and $C_*$ denotes singular chains.
|
|
401 |
|
|
402 |
\item
|
|
403 |
Given a plain $n$-category $C$,
|
|
404 |
define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball
|
|
405 |
and $\bc^C_*$ denotes the blob complex based on $C$.
|
|
406 |
|
|
407 |
\end{itemize}
|
95
|
408 |
|
108
|
409 |
|
|
410 |
|
|
411 |
|
|
412 |
|
|
413 |
|
|
414 |
\subsection{From $n$-categories to systems of fields}
|
113
|
415 |
\label{ss:ncat_fields}
|
108
|
416 |
|
|
417 |
We can extend the functors $\cC$ above from $k$-balls to arbitrary $k$-manifolds as follows.
|
|
418 |
|
|
419 |
Let $W$ be a $k$-manifold, $1\le k \le n$.
|
|
420 |
We will define a set $\cC(W)$.
|
|
421 |
(If $k = n$ and our $k$-categories are enriched, then
|
|
422 |
$\cC(W)$ will have additional structure; see below.)
|
|
423 |
$\cC(W)$ will be the colimit of a functor defined on a category $\cJ(W)$,
|
|
424 |
which we define next.
|
|
425 |
|
|
426 |
Define a permissible decomposition of $W$ to be a decomposition
|
|
427 |
\[
|
|
428 |
W = \bigcup_a X_a ,
|
|
429 |
\]
|
|
430 |
where each $X_a$ is a $k$-ball.
|
|
431 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
|
|
432 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
|
|
433 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category.
|
|
434 |
(The objects of $\cJ(W)$ are permissible decompositions of $W$, and there is a unique
|
|
435 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
|
|
436 |
\nn{need figures}
|
|
437 |
|
|
438 |
$\cC$ determines
|
|
439 |
a functor $\psi_\cC$ from $\cJ(W)$ to the category of sets
|
|
440 |
(possibly with additional structure if $k=n$).
|
|
441 |
For a decomposition $x = (X_a)$ in $\cJ(W)$, define $\psi_\cC(x)$ to be the subset
|
|
442 |
\[
|
|
443 |
\psi_\cC(x) \sub \prod_a \cC(X_a)
|
|
444 |
\]
|
|
445 |
such that the restrictions to the various pieces of shared boundaries amongst the
|
|
446 |
$X_a$ all agree.
|
|
447 |
(Think fibered product.)
|
|
448 |
If $x$ is a refinement of $y$, define a map $\psi_\cC(x)\to\psi_\cC(y)$
|
|
449 |
via the composition maps of $\cC$.
|
112
|
450 |
(If $\dim(W) = n$ then we need to also make use of the monoidal
|
|
451 |
product in the enriching category.
|
|
452 |
\nn{should probably be more explicit here})
|
108
|
453 |
|
|
454 |
Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$.
|
112
|
455 |
In the plain (non-$A_\infty$) case, this means that
|
|
456 |
for each decomposition $x$ there is a map
|
108
|
457 |
$\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps
|
|
458 |
above, and $\cC(W)$ is universal with respect to these properties.
|
112
|
459 |
In the $A_\infty$ case, it means
|
|
460 |
\nn{.... need to check if there is a def in the literature before writing this down}
|
|
461 |
|
|
462 |
More concretely, in the plain case enriched over vector spaces, and with $\dim(W) = n$, we can take
|
|
463 |
\[
|
|
464 |
\cC(W) = \left( \oplus_x \psi_\cC(x)\right) \big/ K
|
|
465 |
\]
|
|
466 |
where $K$ is generated by all things of the form $a - g(a)$, where
|
|
467 |
$a\in \psi_\cC(x)$ for some decomposition $x$, $x\le y$, and $g: \psi_\cC(x)
|
|
468 |
\to \psi_\cC(y)$ is value of $\psi_\cC$ on the antirefinement $x\to y$.
|
111
|
469 |
|
112
|
470 |
In the $A_\infty$ case enriched over chain complexes, the concrete description of the colimit
|
|
471 |
is as follows.
|
113
|
472 |
Define an $m$-sequence to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions.
|
112
|
473 |
Such sequences (for all $m$) form a simplicial set.
|
|
474 |
Let
|
|
475 |
\[
|
|
476 |
V = \bigoplus_{(x_i)} \psi_\cC(x_0) ,
|
|
477 |
\]
|
113
|
478 |
where the sum is over all $m$-sequences and all $m$, and each summand is degree shifted by $m$.
|
112
|
479 |
We endow $V$ with a differential which is the sum of the differential of the $\psi_\cC(x_0)$
|
|
480 |
summands plus another term using the differential of the simplicial set of $m$-sequences.
|
|
481 |
More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$
|
|
482 |
summand of $V$ (with $\bar{x} = (x_0,\dots,x_k)$), define
|
|
483 |
\[
|
|
484 |
\bd (a, \bar{x}) = (\bd a, \bar{x}) \pm (g(a), d_0(\bar{x})) + \sum_{j=1}^k \pm (a, d_j(\bar{x})) ,
|
|
485 |
\]
|
|
486 |
where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$
|
|
487 |
is the usual map.
|
|
488 |
\nn{need to say this better}
|
|
489 |
\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which
|
|
490 |
combine only two balls at a time; for $n=1$ this version will lead to usual definition
|
|
491 |
of $A_\infty$ category}
|
108
|
492 |
|
113
|
493 |
We will call $m$ the filtration degree of the complex.
|
|
494 |
We can think of this construction as starting with a disjoint copy of a complex for each
|
|
495 |
permissible decomposition (filtration degree 0).
|
|
496 |
Then we glue these together with mapping cylinders coming from gluing maps
|
|
497 |
(filtration degree 1).
|
|
498 |
Then we kill the extra homology we just introduced with mapping cylinder between the mapping cylinders (filtration degree 2).
|
|
499 |
And so on.
|
|
500 |
|
108
|
501 |
$\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds.
|
|
502 |
|
|
503 |
It is easy to see that
|
|
504 |
there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps
|
|
505 |
comprise a natural transformation of functors.
|
|
506 |
|
|
507 |
\nn{need to finish explaining why we have a system of fields;
|
|
508 |
need to say more about ``homological" fields?
|
|
509 |
(actions of homeomorphisms);
|
|
510 |
define $k$-cat $\cC(\cdot\times W)$}
|
|
511 |
|
|
512 |
|
|
513 |
|
|
514 |
\subsection{Modules}
|
95
|
515 |
|
101
|
516 |
Next we define [$A_\infty$] $n$-category modules (a.k.a.\ representations,
|
|
517 |
a.k.a.\ actions).
|
102
|
518 |
The definition will be very similar to that of $n$-categories.
|
109
|
519 |
\nn{** need to make sure all revisions of $n$-cat def are also made to module def.}
|
110
|
520 |
\nn{should they be called $n$-modules instead of just modules? probably not, but worth considering.}
|
102
|
521 |
|
104
|
522 |
Our motivating example comes from an $(m{-}n{+}1)$-dimensional manifold $W$ with boundary
|
102
|
523 |
in the context of an $m{+}1$-dimensional TQFT.
|
|
524 |
Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
|
|
525 |
This will be explained in more detail as we present the axioms.
|
|
526 |
|
|
527 |
Fix an $n$-category $\cC$.
|
|
528 |
|
|
529 |
Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
|
|
530 |
(standard $k$-ball, northern hemisphere in boundary of standard $k$-ball).
|
|
531 |
We call $B$ the ball and $N$ the marking.
|
|
532 |
A homeomorphism between marked $k$-balls is a homeomorphism of balls which
|
|
533 |
restricts to a homeomorphism of markings.
|
|
534 |
|
|
535 |
\xxpar{Module morphisms}
|
|
536 |
{For each $0 \le k \le n$, we have a functor $\cM_k$ from
|
|
537 |
the category of marked $k$-balls and
|
|
538 |
homeomorphisms to the category of sets and bijections.}
|
|
539 |
|
|
540 |
(As with $n$-categories, we will usually omit the subscript $k$.)
|
|
541 |
|
104
|
542 |
For example, let $\cD$ be the $m{+}1$-dimensional TQFT which assigns to a $k$-manifold $N$ the set
|
|
543 |
of maps from $N$ to $T$, modulo homotopy (and possibly linearized) if $k=m$.
|
|
544 |
Let $W$ be an $(m{-}n{+}1)$-dimensional manifold with boundary.
|
|
545 |
Let $\cC$ be the $n$-category with $\cC(X) \deq \cD(X\times \bd W)$.
|
|
546 |
Let $\cM(B, N) \deq \cD((B\times \bd W)\cup (N\times W))$.
|
|
547 |
(The union is along $N\times \bd W$.)
|
110
|
548 |
(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be
|
|
549 |
the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
|
102
|
550 |
|
103
|
551 |
Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
|
|
552 |
Call such a thing a {marked $k{-}1$-hemisphere}.
|
102
|
553 |
|
|
554 |
\xxpar{Module boundaries, part 1:}
|
|
555 |
{For each $0 \le k \le n-1$, we have a functor $\cM_k$ from
|
104
|
556 |
the category of marked $k$-hemispheres and
|
102
|
557 |
homeomorphisms to the category of sets and bijections.}
|
|
558 |
|
104
|
559 |
In our example, let $\cM(H) \deq \cD(H\times\bd W \cup \bd H\times W)$.
|
|
560 |
|
102
|
561 |
\xxpar{Module boundaries, part 2:}
|
|
562 |
{For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cM(\bd M)$.
|
|
563 |
These maps, for various $M$, comprise a natural transformation of functors.}
|
|
564 |
|
110
|
565 |
Given $c\in\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
|
102
|
566 |
|
|
567 |
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
|
|
568 |
then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
|
|
569 |
and $c\in \cC(\bd M)$.
|
|
570 |
|
|
571 |
\xxpar{Module domain $+$ range $\to$ boundary:}
|
|
572 |
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k$-hemisphere ($0\le k\le n-1$),
|
104
|
573 |
$M_i$ is a marked $k$-ball, and $E = M_1\cap M_2$ is a marked $k{-}1$-hemisphere.
|
|
574 |
Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the
|
|
575 |
two maps $\bd: \cM(M_i)\to \cM(E)$.
|
102
|
576 |
Then (axiom) we have an injective map
|
|
577 |
\[
|
|
578 |
\gl_E : \cM(M_1) \times_{\cM(E)} \cM(M_2) \to \cM(H)
|
|
579 |
\]
|
|
580 |
which is natural with respect to the actions of homeomorphisms.}
|
|
581 |
|
110
|
582 |
Let $\cM(H)_E$ denote the image of $\gl_E$.
|
|
583 |
We will refer to elements of $\cM(H)_E$ as ``splittable along $E$" or ``transverse to $E$".
|
|
584 |
|
|
585 |
|
103
|
586 |
\xxpar{Axiom yet to be named:}
|
|
587 |
{For each marked $k$-hemisphere $H$ there is a restriction map
|
|
588 |
$\cM(H)\to \cC(H)$.
|
|
589 |
($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.)
|
|
590 |
These maps comprise a natural transformation of functors.}
|
102
|
591 |
|
103
|
592 |
Note that combining the various boundary and restriction maps above
|
110
|
593 |
(for both modules and $n$-categories)
|
103
|
594 |
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$
|
|
595 |
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$.
|
110
|
596 |
The subset is the subset of morphisms which are appropriately splittable (transverse to the
|
|
597 |
cutting submanifolds).
|
103
|
598 |
This fact will be used below.
|
102
|
599 |
|
104
|
600 |
In our example, the various restriction and gluing maps above come from
|
|
601 |
restricting and gluing maps into $T$.
|
|
602 |
|
|
603 |
We require two sorts of composition (gluing) for modules, corresponding to two ways
|
103
|
604 |
of splitting a marked $k$-ball into two (marked or plain) $k$-balls.
|
|
605 |
First, we can compose two module morphisms to get another module morphism.
|
|
606 |
|
|
607 |
\nn{need figures for next two axioms}
|
|
608 |
|
|
609 |
\xxpar{Module composition:}
|
|
610 |
{Let $M = M_1 \cup_Y M_2$, where $M$, $M_1$ and $M_2$ are marked $k$-balls ($0\le k\le n$)
|
|
611 |
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball.
|
|
612 |
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere.
|
|
613 |
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$.
|
|
614 |
We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$.
|
|
615 |
Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps.
|
|
616 |
Then (axiom) we have a map
|
|
617 |
\[
|
|
618 |
\gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E
|
|
619 |
\]
|
|
620 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
|
|
621 |
to the intersection of the boundaries of $M$ and $M_i$.
|
|
622 |
If $k < n$ we require that $\gl_Y$ is injective.
|
|
623 |
(For $k=n$, see below.)}
|
|
624 |
|
|
625 |
Second, we can compose an $n$-category morphism with a module morphism to get another
|
|
626 |
module morphism.
|
|
627 |
We'll call this the action map to distinguish it from the other kind of composition.
|
|
628 |
|
|
629 |
\xxpar{$n$-category action:}
|
|
630 |
{Let $M = X \cup_Y M'$, where $M$ and $M'$ are marked $k$-balls ($0\le k\le n$),
|
|
631 |
$X$ is a plain $k$-ball,
|
|
632 |
and $Y = X\cap M'$ is a $k{-}1$-ball.
|
|
633 |
Let $E = \bd Y$, which is a $k{-}2$-sphere.
|
|
634 |
We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$.
|
|
635 |
Let $\cC(X)_E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps.
|
|
636 |
Then (axiom) we have a map
|
|
637 |
\[
|
|
638 |
\gl_Y :\cC(X)_E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E
|
|
639 |
\]
|
|
640 |
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
|
|
641 |
to the intersection of the boundaries of $X$ and $M'$.
|
|
642 |
If $k < n$ we require that $\gl_Y$ is injective.
|
|
643 |
(For $k=n$, see below.)}
|
|
644 |
|
|
645 |
\xxpar{Module strict associativity:}
|
|
646 |
{The composition and action maps above are strictly associative.}
|
|
647 |
|
110
|
648 |
Note that the above associativity axiom applies to mixtures of module composition,
|
|
649 |
action maps and $n$-category composition.
|
|
650 |
See Figure xxxx.
|
|
651 |
|
|
652 |
The above three axioms are equivalent to the following axiom,
|
103
|
653 |
which we state in slightly vague form.
|
|
654 |
\nn{need figure for this}
|
|
655 |
|
|
656 |
\xxpar{Module multi-composition:}
|
|
657 |
{Given any decomposition
|
|
658 |
\[
|
|
659 |
M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q
|
|
660 |
\]
|
|
661 |
of a marked $k$-ball $M$
|
|
662 |
into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a
|
|
663 |
map from an appropriate subset (like a fibered product)
|
|
664 |
of
|
|
665 |
\[
|
|
666 |
\cC(X_1)\times\cdots\times\cC(X_p) \times \cM(M_1)\times\cdots\times\cM(M_q)
|
|
667 |
\]
|
|
668 |
to $\cM(M)$,
|
|
669 |
and these various multifold composition maps satisfy an
|
|
670 |
operad-type strict associativity condition.}
|
|
671 |
|
|
672 |
(The above operad-like structure is analogous to the swiss cheese operad
|
|
673 |
\nn{need citation}.)
|
|
674 |
\nn{need to double-check that this is true.}
|
|
675 |
|
|
676 |
\xxpar{Module product (identity) morphisms:}
|
|
677 |
{Let $M$ be a marked $k$-ball and $D$ be a plain $m$-ball, with $k+m \le n$.
|
|
678 |
Then we have a map $\cM(M)\to \cM(M\times D)$, usually denoted $a\mapsto a\times D$ for $a\in \cM(M)$.
|
|
679 |
If $f:M\to M'$ and $\tilde{f}:M\times D \to M'\times D'$ are maps such that the diagram
|
|
680 |
\[ \xymatrix{
|
|
681 |
M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\
|
|
682 |
M \ar[r]^{f} & M'
|
|
683 |
} \]
|
|
684 |
commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.}
|
|
685 |
|
111
|
686 |
\nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.}
|
103
|
687 |
|
110
|
688 |
\nn{** marker --- resume revising here **}
|
|
689 |
|
103
|
690 |
There are two alternatives for the next axiom, according whether we are defining
|
|
691 |
modules for plain $n$-categories or $A_\infty$ $n$-categories.
|
|
692 |
In the plain case we require
|
|
693 |
|
|
694 |
\xxpar{Pseudo and extended isotopy invariance in dimension $n$:}
|
|
695 |
{Let $M$ be a marked $n$-ball and $f: M\to M$ be a homeomorphism which restricts
|
|
696 |
to the identity on $\bd M$ and is pseudo-isotopic or extended isotopic (rel boundary) to the identity.
|
|
697 |
Then $f$ acts trivially on $\cM(M)$.}
|
|
698 |
|
|
699 |
\nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.}
|
|
700 |
|
|
701 |
We emphasize that the $\bd M$ above means boundary in the marked $k$-ball sense.
|
|
702 |
In other words, if $M = (B, N)$ then we require only that isotopies are fixed
|
|
703 |
on $\bd B \setmin N$.
|
|
704 |
|
|
705 |
For $A_\infty$ modules we require
|
|
706 |
|
|
707 |
\xxpar{Families of homeomorphisms act.}
|
|
708 |
{For each marked $n$-ball $M$ and each $c\in \cM(\bd M)$ we have a map of chain complexes
|
|
709 |
\[
|
|
710 |
C_*(\Homeo_\bd(M))\ot \cM(M; c) \to \cM(M; c) .
|
|
711 |
\]
|
|
712 |
Here $C_*$ means singular chains and $\Homeo_\bd(M)$ is the space of homeomorphisms of $M$
|
|
713 |
which fix $\bd M$.
|
|
714 |
These action maps are required to be associative up to homotopy
|
|
715 |
\nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that
|
|
716 |
a diagram like the one in Proposition \ref{CDprop} commutes.
|
|
717 |
\nn{repeat diagram here?}
|
|
718 |
\nn{restate this with $\Homeo(M\to M')$? what about boundary fixing property?}}
|
|
719 |
|
|
720 |
\medskip
|
102
|
721 |
|
104
|
722 |
Note that the above axioms imply that an $n$-category module has the structure
|
|
723 |
of an $n{-}1$-category.
|
|
724 |
More specifically, let $J$ be a marked 1-ball, and define $\cE(X)\deq \cM(X\times J)$,
|
|
725 |
where $X$ is a $k$-ball or $k{-}1$-sphere and in the product $X\times J$ we pinch
|
|
726 |
above the non-marked boundary component of $J$.
|
|
727 |
\nn{give figure for this, or say more?}
|
|
728 |
Then $\cE$ has the structure of an $n{-}1$-category.
|
102
|
729 |
|
105
|
730 |
All marked $k$-balls are homeomorphic, unless $k = 1$ and our manifolds
|
|
731 |
are oriented or Spin (but not unoriented or $\text{Pin}_\pm$).
|
|
732 |
In this case ($k=1$ and oriented or Spin), there are two types
|
|
733 |
of marked 1-balls, call them left-marked and right-marked,
|
|
734 |
and hence there are two types of modules, call them right modules and left modules.
|
|
735 |
In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$),
|
|
736 |
there is no left/right module distinction.
|
|
737 |
|
108
|
738 |
|
|
739 |
\subsection{Modules as boundary labels}
|
112
|
740 |
\label{moddecss}
|
108
|
741 |
|
|
742 |
Let $\cC$ be an [$A_\infty$] $n$-category, let $W$ be a $k$-manifold ($k\le n$),
|
|
743 |
and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to each boundary
|
|
744 |
component $\bd_i W$ of $W$.
|
|
745 |
|
|
746 |
We will define a set $\cC(W, \cN)$ using a colimit construction similar to above.
|
|
747 |
\nn{give ref}
|
|
748 |
(If $k = n$ and our $k$-categories are enriched, then
|
|
749 |
$\cC(W, \cN)$ will have additional structure; see below.)
|
|
750 |
|
|
751 |
Define a permissible decomposition of $W$ to be a decomposition
|
|
752 |
\[
|
|
753 |
W = (\bigcup_a X_a) \cup (\bigcup_{i,b} M_{ib}) ,
|
|
754 |
\]
|
|
755 |
where each $X_a$ is a plain $k$-ball (disjoint from $\bd W$) and
|
|
756 |
each $M_{ib}$ is a marked $k$-ball intersecting $\bd_i W$,
|
|
757 |
with $M_{ib}\cap\bd_i W$ being the marking.
|
|
758 |
Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
|
|
759 |
of $y$, or write $x \le y$, if each ball of $y$ is a union of balls of $x$.
|
|
760 |
This defines a partial ordering $\cJ(W)$, which we will think of as a category.
|
|
761 |
(The objects of $\cJ(D)$ are permissible decompositions of $W$, and there is a unique
|
|
762 |
morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.)
|
|
763 |
\nn{need figures}
|
|
764 |
|
|
765 |
$\cN$ determines
|
|
766 |
a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets
|
|
767 |
(possibly with additional structure if $k=n$).
|
|
768 |
For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset
|
|
769 |
\[
|
111
|
770 |
\psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib}))
|
108
|
771 |
\]
|
|
772 |
such that the restrictions to the various pieces of shared boundaries amongst the
|
|
773 |
$X_a$ and $M_{ib}$ all agree.
|
|
774 |
(Think fibered product.)
|
|
775 |
If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$
|
|
776 |
via the gluing (composition or action) maps from $\cC$ and the $\cN_i$.
|
|
777 |
|
|
778 |
Finally, define $\cC(W, \cN)$ to be the colimit of $\psi_\cN$.
|
|
779 |
In other words, for each decomposition $x$ there is a map
|
|
780 |
$\psi(x)\to \cC(W, \cN)$, these maps are compatible with the refinement maps
|
|
781 |
above, and $\cC(W, \cN)$ is universal with respect to these properties.
|
|
782 |
|
112
|
783 |
More generally, each $\cN_i$ could label some codimension zero submanifold of $\bd W$.
|
|
784 |
\nn{need to say more?}
|
|
785 |
|
108
|
786 |
\nn{boundary restrictions, $k$-cat $\cC(\cdot\times W; N)$ etc.}
|
|
787 |
|
|
788 |
\subsection{Tensor products}
|
105
|
789 |
|
112
|
790 |
Next we consider tensor products.
|
|
791 |
|
|
792 |
\nn{what about self tensor products /coends ?}
|
105
|
793 |
|
108
|
794 |
\nn{maybe ``tensor product" is not the best name?}
|
|
795 |
|
106
|
796 |
\nn{start with (less general) tensor products; maybe change this later}
|
105
|
797 |
|
108
|
798 |
|
107
|
799 |
Let $\cM$ and $\cM'$ be modules for an $n$-category $\cC$.
|
|
800 |
(If $k=1$ and manifolds are oriented, then one should be
|
|
801 |
a left module and the other a right module.)
|
109
|
802 |
We will define an $n{-}1$-category $\cM\ot_\cC\cM'$, which depends (functorially)
|
107
|
803 |
on a choice of 1-ball (interval) $J$.
|
|
804 |
|
112
|
805 |
Let $p$ and $p'$ be the boundary points of $J$.
|
|
806 |
Given a $k$-ball $X$, let $(X\times J, \cM, \cM')$ denote $X\times J$ with
|
|
807 |
$X\times\{p\}$ labeled by $\cM$ and $X\times\{p'\}$ labeled by $\cM'$, as in Subsection \ref{moddecss}.
|
|
808 |
Let
|
106
|
809 |
\[
|
112
|
810 |
\cT(X) \deq \cC(X\times J, \cM, \cM') ,
|
106
|
811 |
\]
|
112
|
812 |
where the right hand side is the colimit construction defined in Subsection \ref{moddecss}.
|
|
813 |
It is not hard to see that $\cT$ becomes an $n{-}1$-category.
|
|
814 |
\nn{maybe follows from stuff (not yet written) in previous subsection?}
|
106
|
815 |
|
107
|
816 |
|
|
817 |
|
|
818 |
|
|
819 |
|
101
|
820 |
|
|
821 |
\medskip
|
|
822 |
\hrule
|
|
823 |
\medskip
|
|
824 |
|
95
|
825 |
\nn{to be continued...}
|
101
|
826 |
\medskip
|
98
|
827 |
|
|
828 |
|
|
829 |
Stuff that remains to be done (either below or in an appendix or in a separate section or in
|
|
830 |
a separate paper):
|
|
831 |
\begin{itemize}
|
|
832 |
\item traditional $n$-cat defs (e.g. *-1-cat, pivotal 2-cat) imply our def of plain $n$-cat
|
|
833 |
\item conversely, our def implies other defs
|
105
|
834 |
\item do same for modules; maybe an appendix on relating topological
|
|
835 |
vs traditional defs, $n = 1,2$, $A_\infty$ or not, cats, modules, tensor products
|
98
|
836 |
\item traditional $A_\infty$ 1-cat def implies our def
|
99
|
837 |
\item ... and vice-versa (already done in appendix)
|
98
|
838 |
\item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?)
|
|
839 |
\item spell out what difference (if any) Top vs PL vs Smooth makes
|
99
|
840 |
\item explain relation between old-fashioned blob homology and new-fangled blob homology
|
105
|
841 |
\item define $n{+}1$-cat of $n$-cats; discuss Morita equivalence
|
98
|
842 |
\end{itemize}
|
|
843 |
|
|
844 |
|