32 Still other definitions (see, for example, \cite{MR2094071}) |
32 Still other definitions (see, for example, \cite{MR2094071}) |
33 model the $k$-morphisms on more complicated combinatorial polyhedra. |
33 model the $k$-morphisms on more complicated combinatorial polyhedra. |
34 |
34 |
35 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball: |
35 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball: |
36 |
36 |
37 \begin{preliminary-axiom}{\ref{axiom:morphisms}}{Morphisms} |
37 \begin{axiom}[Morphisms]{\textup{\textbf{[preliminary]}}} |
38 For any $k$-manifold $X$ homeomorphic |
38 For any $k$-manifold $X$ homeomorphic |
39 to the standard $k$-ball, we have a set of $k$-morphisms |
39 to the standard $k$-ball, we have a set of $k$-morphisms |
40 $\cC_k(X)$. |
40 $\cC_k(X)$. |
41 \end{preliminary-axiom} |
41 \end{axiom} |
42 |
42 |
43 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
43 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
44 standard $k$-ball. |
44 standard $k$-ball. |
45 We {\it do not} assume that it is equipped with a |
45 We {\it do not} assume that it is equipped with a |
46 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
46 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
50 the boundary), we want a corresponding |
50 the boundary), we want a corresponding |
51 bijection of sets $f:\cC(X)\to \cC(Y)$. |
51 bijection of sets $f:\cC(X)\to \cC(Y)$. |
52 (This will imply ``strong duality", among other things.) |
52 (This will imply ``strong duality", among other things.) |
53 So we replace the above with |
53 So we replace the above with |
54 |
54 |
|
55 \addtocounter{axiom}{-1} |
55 \begin{axiom}[Morphisms] |
56 \begin{axiom}[Morphisms] |
56 \label{axiom:morphisms} |
57 \label{axiom:morphisms} |
57 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
58 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
58 the category of $k$-balls and |
59 the category of $k$-balls and |
59 homeomorphisms to the category of sets and bijections. |
60 homeomorphisms to the category of sets and bijections. |
332 The last axiom (below), concerning actions of |
333 The last axiom (below), concerning actions of |
333 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
334 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
334 |
335 |
335 We start with the plain $n$-category case. |
336 We start with the plain $n$-category case. |
336 |
337 |
337 \begin{preliminary-axiom}{\ref{axiom:extended-isotopies}}{Isotopy invariance in dimension $n$} |
338 \begin{axiom}[Isotopy invariance in dimension $n$]{\textup{\textbf{[preliminary]}}} |
338 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
339 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
339 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
340 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
340 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
341 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
341 \end{preliminary-axiom} |
342 \end{axiom} |
342 |
343 |
343 This axiom needs to be strengthened to force product morphisms to act as the identity. |
344 This axiom needs to be strengthened to force product morphisms to act as the identity. |
344 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
345 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
345 Let $J$ be a 1-ball (interval). |
346 Let $J$ be a 1-ball (interval). |
346 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
347 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
407 It can be thought of as the action of the inverse of |
408 It can be thought of as the action of the inverse of |
408 a map which projects a collar neighborhood of $Y$ onto $Y$. |
409 a map which projects a collar neighborhood of $Y$ onto $Y$. |
409 |
410 |
410 The revised axiom is |
411 The revised axiom is |
411 |
412 |
|
413 \addtocounter{axiom}{-1} |
412 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
414 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
413 \label{axiom:extended-isotopies} |
415 \label{axiom:extended-isotopies} |
414 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
416 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
415 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
417 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
416 Then $f$ acts trivially on $\cC(X)$. |
418 Then $f$ acts trivially on $\cC(X)$. |
719 We will call $m$ the filtration degree of the complex. |
721 We will call $m$ the filtration degree of the complex. |
720 We can think of this construction as starting with a disjoint copy of a complex for each |
722 We can think of this construction as starting with a disjoint copy of a complex for each |
721 permissible decomposition (filtration degree 0). |
723 permissible decomposition (filtration degree 0). |
722 Then we glue these together with mapping cylinders coming from gluing maps |
724 Then we glue these together with mapping cylinders coming from gluing maps |
723 (filtration degree 1). |
725 (filtration degree 1). |
724 Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2). |
726 Then we kill the extra homology we just introduced with mapping cylinders between the mapping cylinders (filtration degree 2), and so on. |
725 And so on. |
|
726 |
727 |
727 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
728 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
728 |
729 |
729 It is easy to see that |
730 It is easy to see that |
730 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
731 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
733 \nn{need to finish explaining why we have a system of fields; |
734 \nn{need to finish explaining why we have a system of fields; |
734 need to say more about ``homological" fields? |
735 need to say more about ``homological" fields? |
735 (actions of homeomorphisms); |
736 (actions of homeomorphisms); |
736 define $k$-cat $\cC(\cdot\times W)$} |
737 define $k$-cat $\cC(\cdot\times W)$} |
737 |
738 |
|
739 Recall that Axiom \ref{} for an $n$-category provided functors $\cC$ from $k$-spheres to sets for $0 \leq k < n$. We claim now that these functors automatically agree with the colimits we have associated to spheres in this section. \todo{} \todo{In fact, we probably should do this for balls as well!} For the remainder of this section we will write $\underrightarrow{\cC}(W)$ for the colimit associated to an arbitary manifold $W$, to distinguish it, in the case that $W$ is a ball or a sphere, from $\cC(W)$, which is part of the definition of the $n$-category. After the next three lemmas, there will be no further need for this notational distinction. |
|
740 |
|
741 \begin{lem} |
|
742 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ |
|
743 \end{lem} |
|
744 |
|
745 \begin{lem} |
|
746 For a topological $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) = \cC(B).$$ |
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747 \end{lem} |
|
748 |
|
749 \begin{lem} |
|
750 For an $A_\infty$ $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) \quism \cC(B).$$ |
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751 \end{lem} |
738 |
752 |
739 |
753 |
740 \subsection{Modules} |
754 \subsection{Modules} |
741 |
755 |
742 Next we define plain and $A_\infty$ $n$-category modules. |
756 Next we define plain and $A_\infty$ $n$-category modules. |