[J.Lurie, Higher topos theory in topology]

]]>In other words, should the j's be switched?

Also, is it traditional to write i before j, for example,
0-->i_*i^! F --> -->j_* j^* F-->0.

In other words, should the j's be switched?

]]>In the second paragraph the inequality $\text{dim}S_{\mathfrak{m}}\geq \text{dim}S'*{\mathfrak{n}}$ does not imply $n-\text{dim}S*{\mathfrak{m}}\geq n - \text{dim}S'_{\mathfrak{n}}$. But as in reality the dimensions are equal it still works. The equality follows from Lemma 10.111.7.