Proof (0MUN) · ScalingStacks

0MUN

Proof: Let $M$ be an object of $\mathcal{T}$ and write $M=M_{0}$ . Let $X\in S^{\perp_{\infty}}$ and suppose we have a morphism $\alpha_{0}:M_{0}\rightarrow X$ . By the argument of Proposition 3.2 we can construct the following commutative diagram:

with $M_{n}\in S^{\perp_{n}}$ for each $n\geqslant 1$ . We now construct the homotopy colimit, $\textnormal{hocolim}(M_{i})$ . By construction, the composite

\coprod_{i=0}^{\infty}M_{i}\stackrel{{\scriptstyle 1-\textnormal{shift}}}{{% \longrightarrow}}\coprod_{i=0}^{\infty}M_{i}\stackrel{{\scriptstyle\langle% \alpha_{i}\rangle}}{{\longrightarrow}}X

is zero, so that we have the following commutative diagram:

\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 17.6529pt\hbox{\ignorespaces% \ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{% \entry@#!@&&\entry@@#!@\cr&&&\\&&&\crcr}}}\ignorespaces{\hbox{\kern-17.6529pt% \raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\coprod_{i=0}% ^{\infty}M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}\ignorespaces% \ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces% \ignorespaces\ignorespaces{\hbox{\kern 17.00844pt\raise 5.43056pt\hbox{{}\hbox% {\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2% .43056pt\hbox{$\scriptstyle{1-\textnormal{shift}}$}}}\kern 3.0pt}}}}}}% \ignorespaces{\hbox{\kern 41.6529pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0% .0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{% \lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces% \ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{% \hbox{\kern 21.74998pt\raise-24.568pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt% \hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.25555pt\hbox{$% \scriptstyle{0}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 51.77107pt\raise% -33.77823pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1% }}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{% \lx@xy@drawline@}}{\hbox{\kern 41.6529pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt% \raise 0.0pt\hbox{$\textstyle{\coprod_{i=0}^{\infty}M_{i}\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces}$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{% \hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 100.95868pt\raise 0.0pt\hbox{% \hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{% \lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces% \ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces% \ignorespaces{\hbox{\kern 59.30579pt\raise-16.84972pt\hbox{{}\hbox{\kern 0.0pt% \raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$% \scriptstyle{\langle\alpha_{i}\rangle}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox% {\kern 59.30579pt\raise-28.85277pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{% \lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule% }}{\hbox{\kern 100.95868pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox% {$\textstyle{\textnormal{hocolim}(M_{i})\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% }$}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{% \lx@xy@droprule}}\ignorespaces{\hbox{\kern 185.62418pt\raise 0.0pt\hbox{\hbox{% \kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{% \lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces% \ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}% \ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 91.66127pt\raise-25.5001pt% \hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0% .0pt\raise-2.43056pt\hbox{$\scriptstyle{\exists}$}}}\kern 3.0pt}}}}}}% \ignorespaces{\hbox{\kern 66.8405pt\raise-34.63913pt\hbox{\hbox{\kern 0.0pt% \raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{% \hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 185% .62418pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{% \Sigma\coprod M_{i}}$}}}}}{\hbox{\kern-3.0pt\raise-38.68608pt\hbox{\hbox{\kern 3% .0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}{\hbox{\kern 51.77107pt\raise-38.6860% 8pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X}$}}}}}{\hbox{\kern 1% 28.29143pt\raise-38.68608pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$% \textstyle{}$}}}}}{\hbox{\kern 201.01239pt\raise-38.68608pt\hbox{\hbox{\kern 3% .0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}\ignorespaces}}}}\ignorespaces.

That is, every morphism $M\rightarrow X$ factors through $\textnormal{hocolim}(M_{i})\rightarrow X$ .

Now we have:

$\displaystyle\textnormal{Hom}(S,\Sigma^{j}\textnormal{hocolim}(M_{i}))$	$\displaystyle\cong$	$\displaystyle\textnormal{Hom}(S,\textnormal{hocolim}(\Sigma^{j}M_{i}))$
	$\displaystyle\cong$	$\displaystyle\textnormal{colim}\ \textnormal{Hom}(S,\Sigma^{j}M_{i})$
	$\displaystyle=$	$\displaystyle 0$

for $j\geqslant 1$ . We obtain the first isomorphism because the homotopy colimit commutes with the suspension functor and the second isomorphism by Lemma 3.5. The final equality is a consequence of the fact that $\textnormal{Hom}(S,\Sigma^{j}M_{i})=0$ for $i$ sufficiently large and $j\geqslant 1$ . Hence we have $\textnormal{hocolim}(M_{i})\in S^{\perp_{\infty}}$ . Therefore, setting $\bar{M}=\textnormal{hocolim}(M_{i})$ , we obtain an $S^{\perp_{\infty}}$ -preenvelope $\mu:M\rightarrow\bar{M}$ . $\Box$