280 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) 

   280 In the second approach we use a decorated colimit (as in \S \ref{ssec:spherecat}) 

   281 and various sphere modules based on $F \to E \to Y$

   281 and various sphere modules based on $F \to E \to Y$

   282 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$.

   282 or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$.

   283 Information about the specific map to $Y$ has been taken out of the categories

   283 Information about the specific map to $Y$ has been taken out of the categories

   284 and put into sphere modules and decorations.

   284 and put into sphere modules and decorations.

   286 

   286 

   287 %Let $F \to E \to Y$ be a fiber bundle as above.

   287 %Let $F \to E \to Y$ be a fiber bundle as above.

   288 %Choose a decomposition $Y = \cup X_i$

   288 %Choose a decomposition $Y = \cup X_i$

   289 %such that the restriction of $E$ to $X_i$ is a product $F\times X_i$,

   289 %such that the restriction of $E$ to $X_i$ is a product $F\times X_i$,

   290 %and choose trivializations of these products as well.

   290 %and choose trivializations of these products as well.

   441 

   441 

   442 It is now easy to see that $\psi\circ\phi$ is the identity on the nose.

   442 It is now easy to see that $\psi\circ\phi$ is the identity on the nose.

   443 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.

   443 Another acyclic models argument shows that $\phi\circ\psi$ is homotopic to the identity.

   444 (See the proof of Theorem \ref{thm:product} for more details.)

   444 (See the proof of Theorem \ref{thm:product} for more details.)

   445 \end{proof}

   445 \end{proof}