603 \smallskip |
603 \smallskip |
604 |
604 |
605 For $A_\infty$ $n$-categories, we replace |
605 For $A_\infty$ $n$-categories, we replace |
606 isotopy invariance with the requirement that families of homeomorphisms act. |
606 isotopy invariance with the requirement that families of homeomorphisms act. |
607 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
607 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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608 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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609 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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610 |
608 |
611 |
609 \addtocounter{axiom}{-1} |
612 \addtocounter{axiom}{-1} |
610 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
613 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.} |
611 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
614 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
612 \[ |
615 \[ |
613 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
616 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
614 \] |
617 \] |
615 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
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616 which fix $\bd X$. |
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617 These action maps are required to be associative up to homotopy |
618 These action maps are required to be associative up to homotopy |
618 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
619 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
619 a diagram like the one in Proposition \ref{CHprop} commutes. |
620 a diagram like the one in Proposition \ref{CHprop} commutes. |
620 \nn{repeat diagram here?} |
621 \nn{repeat diagram here?} |
621 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
622 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
622 \end{axiom} |
623 \end{axiom} |
623 |
624 |
624 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
625 We should strengthen the above axiom to apply to families of collar maps. |
625 To do this we need to explain how extended homeomorphisms form a topological space. |
626 To do this we need to explain how collar maps form a topological space. |
626 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
627 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
627 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
628 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
628 \nn{need to also say something about collaring homeomorphisms.} |
629 Having chains on the space of collar maps act gives rise to coherence maps involving |
629 \nn{this paragraph needs work.} |
630 weak identities. |
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631 We will not pursue this in this draft of the paper. |
630 |
632 |
631 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
633 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
632 into a plain $n$-category (enriched over graded groups). |
634 into a plain $n$-category (enriched over graded groups). |
633 \nn{say more here?} |
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634 In a different direction, if we enrich over topological spaces instead of chain complexes, |
635 In a different direction, if we enrich over topological spaces instead of chain complexes, |
635 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
636 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
636 instead of $C_*(\Homeo_\bd(X))$. |
637 instead of $C_*(\Homeo_\bd(X))$. |
637 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
638 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
638 type $A_\infty$ $n$-category. |
639 type $A_\infty$ $n$-category. |
639 |
640 |
640 \medskip |
641 \medskip |
641 |
642 |
642 The alert reader will have already noticed that our definition of a (plain) $n$-category |
643 The alert reader will have already noticed that our definition of a (plain) $n$-category |
643 is extremely similar to our definition of a topological system of fields. |
644 is extremely similar to our definition of a system of fields. |
644 There are two essential differences. |
645 There are two differences. |
645 First, for the $n$-category definition we restrict our attention to balls |
646 First, for the $n$-category definition we restrict our attention to balls |
646 (and their boundaries), while for fields we consider all manifolds. |
647 (and their boundaries), while for fields we consider all manifolds. |
647 Second, in category definition we directly impose isotopy |
648 Second, in category definition we directly impose isotopy |
648 invariance in dimension $n$, while in the fields definition we have do not expect isotopy invariance on fields |
649 invariance in dimension $n$, while in the fields definition we |
649 but instead remember a subspace of local relations which contain differences of isotopic fields. |
650 instead remember a subspace of local relations which contain differences of isotopic fields. |
650 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
651 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
651 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
652 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
652 balls and, at level $n$, quotienting out by the local relations: |
653 balls and, at level $n$, quotienting out by the local relations: |
653 \begin{align*} |
654 \begin{align*} |
654 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
655 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |