15 a ``weak" $n$-category with ``strong duality".) |
15 a ``weak" $n$-category with ``strong duality".) |
16 |
16 |
17 The definitions presented below tie the categories more closely to the topology |
17 The definitions presented below tie the categories more closely to the topology |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
19 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
19 collections of generalized associativity axioms; we prefer maximal sets of axioms to minimal sets. |
20 For examples of topological origin |
20 It is easy to show that examples of topological origin |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
22 it is easy to show that they |
|
23 satisfy our axioms. |
22 satisfy our axioms. |
24 For examples of a more purely algebraic origin, one would typically need the combinatorial |
23 For examples of a more purely algebraic origin, one would typically need the combinatorial |
25 results that we have avoided here. |
24 results that we have avoided here. |
26 |
25 |
27 \nn{Say something explicit about Lurie's work here? It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
26 %\nn{Say something explicit about Lurie's work here? |
|
27 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
28 |
28 |
29 \medskip |
29 \medskip |
30 |
30 |
31 There are many existing definitions of $n$-categories, with various intended uses. |
31 There are many existing definitions of $n$-categories, with various intended uses. |
32 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
32 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
188 \end{tikzpicture} |
188 \end{tikzpicture} |
189 $$ |
189 $$ |
190 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
190 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
191 |
191 |
192 Note that we insist on injectivity above. |
192 Note that we insist on injectivity above. |
193 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...} |
193 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
|
194 %\nn{we might want a more official looking proof...} |
194 |
195 |
195 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
196 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
196 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
197 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
197 |
198 |
198 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
199 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
888 Alternatively and more simply, we could define $\cC^A(X)$ to be |
889 Alternatively and more simply, we could define $\cC^A(X)$ to be |
889 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
890 $\Diff(B^n\to X)\times A$ modulo the diagonal action of $\Diff(B^n)$. |
890 The remaining data for the $A_\infty$ $n$-category |
891 The remaining data for the $A_\infty$ $n$-category |
891 --- composition and $\Diff(X\to X')$ action --- |
892 --- composition and $\Diff(X\to X')$ action --- |
892 also comes from the $\cE\cB_n$ action on $A$. |
893 also comes from the $\cE\cB_n$ action on $A$. |
893 \nn{should we spell this out?} |
894 %\nn{should we spell this out?} |
894 |
895 |
895 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
896 Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms |
896 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
897 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to |
897 an $\cE\cB_n$-algebra. |
898 an $\cE\cB_n$-algebra. |
898 \nn{The paper is already long; is it worth giving details here?} |
899 %\nn{The paper is already long; is it worth giving details here?} |
899 |
900 |
900 If we apply the homotopy colimit construction of the next subsection to this example, |
901 If we apply the homotopy colimit construction of the next subsection to this example, |
901 we get an instance of Lurie's topological chiral homology construction. |
902 we get an instance of Lurie's topological chiral homology construction. |
902 \end{example} |
903 \end{example} |
903 |
904 |