equal
deleted
inserted
replaced
5 |
5 |
6 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob |
6 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob |
7 complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}. |
7 complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}. |
8 \nn{say something about this being anticlimatically tautological?} |
8 \nn{say something about this being anticlimatically tautological?} |
9 We will show below |
9 We will show below |
10 \nn{give ref} |
10 in Corollary \ref{cor:new-old} |
11 that this agrees (up to homotopy) with our original definition of the blob complex |
11 that this agrees (up to homotopy) with our original definition of the blob complex |
12 in the case of plain $n$-categories. |
12 in the case of plain $n$-categories. |
13 When we need to distinguish between the new and old definitions, we will refer to the |
13 When we need to distinguish between the new and old definitions, we will refer to the |
14 new-fangled and old-fashioned blob complex. |
14 new-fangled and old-fashioned blob complex. |
15 |
15 |
202 \nn{need to say something about dim $< n$ above} |
202 \nn{need to say something about dim $< n$ above} |
203 |
203 |
204 \medskip |
204 \medskip |
205 |
205 |
206 \begin{cor} |
206 \begin{cor} |
|
207 \label{cor:new-old} |
207 The new-fangled and old-fashioned blob complexes are homotopic. |
208 The new-fangled and old-fashioned blob complexes are homotopic. |
208 \end{cor} |
209 \end{cor} |
209 \begin{proof} |
210 \begin{proof} |
210 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
211 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point. |
211 \end{proof} |
212 \end{proof} |