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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 2nd 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI am having trouble understanding the proof of Propositions 6.1.6.7 in [[Higher Topos Theory]]. After the displayed cartesian rectangle, he says "Since colimits are universal, we conclude that $Z'$ is a coproduct of objects $Z_\alpha' = X_\alpha \times_{Y_\alpha} Z$, where $\alpha$ ranges over $A_0$." But I don't even see a morphism $Z\to Y_\alpha$ that we could use to form this pullback; all we have is a morphism $Z\to Y_{A_0}$. Can anyone help? <a href="https://ncatlab.org/nlab/revision/diff/relatively+k-compact+morphism+in+an+%28infinity%2C1%29-category/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/relatively+k-compact+morphism+in+an+%28infinity%2C1%29-category/4">v4</a>, <a href="https://ncatlab.org/nlab/show/relatively+k-compact+morphism+in+an+%28infinity%2C1%29-category">current</a>

   I am having trouble understanding the proof of Propositions 6.1.6.7 in Higher Topos Theory. After the displayed cartesian rectangle, he says “Since colimits are universal, we conclude that $Z'$ is a coproduct of objects $Z_\alpha' = X_\alpha \times_{Y_\alpha} Z$, where $\alpha$ ranges over $A_0$.” But I don’t even see a morphism $Z\to Y_\alpha$ that we could use to form this pullback; all we have is a morphism $Z\to Y_{A_0}$. Can anyone help?

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 2nd 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI also don't understand the treatment of cardinalties. The statement of the proposition says "if $\kappa$ is sufficently large", which generally means "there exists a $\lambda$ such that for all $\kappa\gt\lambda$". However, the proof proceeds by letting $\kappa'$ be such that $\mathcal{X}$ is locally $\kappa'$-presentable, letting $\kappa''\gt\kappa'$ be such that pullbacks preserve $\kappa''$-filtered colimits, and letting $\kappa\ge\kappa''$ be such that pullbacks of $\kappa''$-compact objects are $\kappa$-compact, and concluding that $\kappa$-compact objects are stable under pullbacks. Firstly I don't see why this construction as stated guarantees that any sufficiently large $\kappa$ will work. And secondly I don't see why this construction is sufficient to ensure that $\kappa$-compact objects are stable under pullbacks; for instance, Proposition 5.4.7.4 shows that $\tau$-compact objects are stable under $\kappa$-small limits not for all sufficiently $\tau$ but only when $\tau \gg \kappa$.

   I also don’t understand the treatment of cardinalties. The statement of the proposition says “if $\kappa$ is sufficently large”, which generally means “there exists a $\lambda$ such that for all $\kappa\gt\lambda$”. However, the proof proceeds by letting $\kappa'$ be such that $\mathcal{X}$ is locally $\kappa'$-presentable, letting $\kappa''\gt\kappa'$ be such that pullbacks preserve $\kappa''$-filtered colimits, and letting $\kappa\ge\kappa''$ be such that pullbacks of $\kappa''$-compact objects are $\kappa$-compact, and concluding that $\kappa$-compact objects are stable under pullbacks. Firstly I don’t see why this construction as stated guarantees that any sufficiently large $\kappa$ will work. And secondly I don’t see why this construction is sufficient to ensure that $\kappa$-compact objects are stable under pullbacks; for instance, Proposition 5.4.7.4 shows that $\tau$-compact objects are stable under $\kappa$-small limits not for all sufficiently $\tau$ but only when $\tau \gg \kappa$.