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Floris van Doorn 2017-03-24 09:27:16 -04:00
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Notes/smash.tex
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@ -215,6 +215,8 @@ f'\smash 0\arrow[dl,equals] \\
If $x$ varies over $\gluel_a$ the proof is very similar. Only in the end we need to fill the If $x$ varies over $\gluel_a$ the proof is very similar. Only in the end we need to fill the
following cube instead (TODO). following cube instead (TODO).
To show that this homotopy is pointed, (TODO)
\end{proof} \end{proof}
\begin{thm}\label{thm:smash-functor-right} \begin{thm}\label{thm:smash-functor-right}
@ -357,6 +359,13 @@ $$(A\pmap B\pmap C)\lpmap{({-})\smash B}(A\smash B\pmap (B\pmap C)\smash B)\lpma
$$\inv{e}_{A,B,C}:(A\smash B\pmap C)\lpmap{B\pmap({-})}((B\pmap A\smash B)\pmap (B\pmap $$\inv{e}_{A,B,C}:(A\smash B\pmap C)\lpmap{B\pmap({-})}((B\pmap A\smash B)\pmap (B\pmap
C))\lpmap{\eta\pmap(B\pmap C)}(A\pmap B\pmap C).$$ It is easy to show that $e$ and $\inv{e}$ are C))\lpmap{\eta\pmap(B\pmap C)}(A\pmap B\pmap C).$$ It is easy to show that $e$ and $\inv{e}$ are
inverses as unpointed maps from the unit-counit laws and naturality of $\eta$ and $\epsilon$. inverses as unpointed maps from the unit-counit laws and naturality of $\eta$ and $\epsilon$.
% For $f : A\pmap B\pmap C$ we have
% \begin{align*}
% \inv{e}(e(f))&\equiv(\eta\pmap(B\pmap C))\o (B\pmap((A\smash B\pmap\epsilon)\of\smash B))\\
% &= (\eta\pmap(B\pmap C))\o (B\pmap(A\smash B\pmap\epsilon))\o(B\pmapf\smash B)\\
% % &= (\eta\pmap(B\pmap C))\o (B\pmap(A\smash B\pmap\epsilon))\o(B\pmapf\smash B)\\
% \end{align*}
\end{proof} \end{proof}
\begin{lem}\label{e-natural} \begin{lem}\label{e-natural}
$e$ is natural in $A$, $B$ and $C$. $e$ is natural in $A$, $B$ and $C$.