72 to be fussier about corners.) |
72 to be fussier about corners.) |
73 For each flavor of manifold there is a corresponding flavor of $n$-category. |
73 For each flavor of manifold there is a corresponding flavor of $n$-category. |
74 We will concentrate on the case of PL unoriented manifolds. |
74 We will concentrate on the case of PL unoriented manifolds. |
75 |
75 |
76 (The ambitious reader may want to keep in mind two other classes of balls. |
76 (The ambitious reader may want to keep in mind two other classes of balls. |
77 The first is balls equipped with a map to some other space $Y$. \todo{cite something of Teichner's?} |
77 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
78 This will be used below to describe the blob complex of a fiber bundle with |
78 This will be used below to describe the blob complex of a fiber bundle with |
79 base space $Y$. |
79 base space $Y$. |
80 The second is balls equipped with a section of the the tangent bundle, or the frame |
80 The second is balls equipped with a section of the the tangent bundle, or the frame |
81 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
81 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
82 These can be used to define categories with less than the ``strong" duality we assume here, |
82 These can be used to define categories with less than the ``strong" duality we assume here, |
84 |
84 |
85 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
85 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
86 of morphisms). |
86 of morphisms). |
87 The 0-sphere is unusual among spheres in that it is disconnected. |
87 The 0-sphere is unusual among spheres in that it is disconnected. |
88 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
88 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
89 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
89 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
90 by oriented 1-balls.) |
90 by oriented 1-balls.) |
91 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. |
91 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. |
92 |
92 |
93 Instead, we will combine the domain and range into a single entity which we call the |
93 Instead, we will combine the domain and range into a single entity which we call the |
94 boundary of a morphism. |
94 boundary of a morphism. |
121 |
121 |
122 Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
122 Given $c\in\cC(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
123 |
123 |
124 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
124 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
125 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
125 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
126 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary category |
126 all $c\in \cC(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
127 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
127 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
128 and all the structure maps of the $n$-category should be compatible with the auxiliary |
128 and all the structure maps of the $n$-category should be compatible with the auxiliary |
129 category structure. |
129 category structure. |
130 Note that this auxiliary structure is only in dimension $n$; |
130 Note that this auxiliary structure is only in dimension $n$; |
131 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
131 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
140 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
140 Instead of an ordinary oriented $(n-1)$-manifold via the inward (or outward) normal |
141 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
141 first (or last) convention, perhaps it is better to define the boundary to be an $(n-1)$-manifold |
142 equipped with an orientation of its once-stabilized tangent bundle. |
142 equipped with an orientation of its once-stabilized tangent bundle. |
143 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
143 Similarly, in dimension $n-k$ we would have manifolds equipped with an orientation of |
144 their $k$ times stabilized tangent bundles. |
144 their $k$ times stabilized tangent bundles. |
145 (cf. [Stolz and Teichner].) |
145 (cf. \cite{MR2079378}.) |
146 Probably should also have a framing of the stabilized dimensions in order to indicate which |
146 Probably should also have a framing of the stabilized dimensions in order to indicate which |
147 side the bounded manifold is on. |
147 side the bounded manifold is on. |
148 For the moment just stick with unoriented manifolds.} |
148 For the moment just stick with unoriented manifolds.} |
149 \medskip |
149 \medskip |
150 |
150 |
778 \nn{need to finish explaining why we have a system of fields; |
778 \nn{need to finish explaining why we have a system of fields; |
779 need to say more about ``homological" fields? |
779 need to say more about ``homological" fields? |
780 (actions of homeomorphisms); |
780 (actions of homeomorphisms); |
781 define $k$-cat $\cC(\cdot\times W)$} |
781 define $k$-cat $\cC(\cdot\times W)$} |
782 |
782 |
783 \nn{need to revise stuff below, since we no longer have the sphere axiom} |
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784 |
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785 Recall that Axiom \ref{axiom:spheres} 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. |
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786 |
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787 \begin{lem} |
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788 For a $k$-ball or $k$-sphere $W$, with $0\leq k < n$, $$\underrightarrow{\cC}(W) = \cC(W).$$ |
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789 \end{lem} |
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790 |
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791 \begin{lem} |
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792 For a topological $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) = \cC(B).$$ |
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793 \end{lem} |
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794 |
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795 \begin{lem} |
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796 For an $A_\infty$ $n$-category $\cC$, and an $n$-ball $B$, $$\underrightarrow{\cC}(B) \quism \cC(B).$$ |
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797 \end{lem} |
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798 |
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799 |
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800 \subsection{Modules} |
783 \subsection{Modules} |
801 |
784 |
802 Next we define plain and $A_\infty$ $n$-category modules. |
785 Next we define plain and $A_\infty$ $n$-category modules. |
803 The definition will be very similar to that of $n$-categories, |
786 The definition will be very similar to that of $n$-categories, |
804 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |
787 but with $k$-balls replaced by {\it marked $k$-balls,} defined below. |