K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST)) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:59:26Z http://mathoverflow.net/feeds/question/28871 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28871/k-good-trees-and-k-compactness-of-colimits-over-k-small-downwards-closed-subposet K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST)) Harry Gindi 2010-06-20T18:21:12Z 2010-06-22T20:07:44Z <h2>Question:</h2> <hr> <p>Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to the proof of Lemma A.1.5.8 of <em>Higher Topos Theory</em> by Lurie, for any $\kappa$-small downward-closed $B\subseteq A$, the colimit of the restricted diagram, $\varinjlim D|_B$ is $\kappa$-compact in $(X\downarrow C)$. </p> <p>Why is this true? (It is stated without proof.)</p> <h2>Definitions:</h2> <hr> <p>For your convenience, here are the definitions:</p> <p>Recall that an object $X$ in $C$ is called $\kappa$-compact if $h^X(\cdot):=\hom(X,\cdot)$ preserves all $\kappa$-filtered colimits (where $\kappa$-filtered means "$&lt;\kappa$"-filtered, since the terminology is different depending on the source).</p> <p>Recall that an $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$ consists of the following data:</p> <ul> <li>An object $X$ in C (the root)</li> <li>A partially ordered set $A$ whose order structure is well-founded (the index)</li> <li>A diagram $D:A\to (X\downarrow C)$ such that given any element $\alpha\in A$, the canonical map <code>$$\varinjlim D|_{\{\beta:\beta&lt;\alpha\}}\to D(\alpha)$$</code> is the pushout of some map $U_\alpha\to V_\alpha\in S$.</li> </ul> <p>We say that an $S$-tree is $\kappa$-good if for all of the morphisms $U_\alpha\to V_\alpha$ above, $U_\alpha$ and $V_\alpha$ are $\kappa$-compact, and such that for any $\alpha\in A$, the subset <code>$\{\beta: \beta &lt; \alpha \}\subseteq A$</code> is $\kappa$-small.</p> <p><strong>Edit</strong>: It's easy to reduce the proof to showing that $D(\alpha)$ is $\kappa$-compact, since projective limits of diagrams $B\to Set$ are $|Arr(B)|$-accessible (and therefore $\kappa$-accessible since $B$ is $\kappa$-small), we perform the computation for $I$ a $\kappa$-filtered poset, and $F:I\to C$, assuming that $D(\alpha)$ is $\kappa$-compact for all $\alpha\in B$:</p> <p><code>$$\begin{matrix}\ \varinjlim_I Hom_C(\varinjlim_B D, F)&amp;\cong&amp;\varinjlim_I\varprojlim_{B^{op}} Hom_C(D,F)\\ &amp;\cong&amp; \varprojlim_{B^{op}}\varinjlim_I Hom_C(D,F)\\ &amp;\cong&amp; \varprojlim_{B^{op}} Hom_C(D,\varinjlim_IF)\\ &amp;\cong&amp; Hom_C(\varinjlim_B D,\varinjlim_IF) \end{matrix}$$</code></p> <p><strong>Edit 2</strong>: I think the above reduction actually won't work, since it doesn't use the hypothesis that B is downward-closed.</p> http://mathoverflow.net/questions/28871/k-good-trees-and-k-compactness-of-colimits-over-k-small-downwards-closed-subposet/28922#28922 Answer by Todd Trimble for K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST)) Todd Trimble 2010-06-21T06:45:47Z 2010-06-21T09:32:20Z <p>Does this work? </p> <p>To prove $D(\alpha)$ is $\kappa$-compact for all $\alpha$ in $A$, assume otherwise, that there exists some counterexample. Then, by the fact $A$ is well-ordered, there is a minimal counterexample (i.e., there is a minimal element $\alpha$ in the set of $\gamma \in A$ such that $D(\gamma)$ is not $\kappa$-compact). This means $D_\beta$ is $\kappa$-compact for all $\beta \lt \alpha$. Since <code>$\{\beta: \beta \lt \alpha\}$</code> has cardinality less than $\kappa$, we have that </p> <p>$$colim_{\beta: \beta \lt \alpha} D(\beta)$$ </p> <p>is $\kappa$-compact. Now, given a diagram of the form</p> <p><code>$$V_\alpha \leftarrow U_\alpha \to colim_{\{\beta: \beta \lt \alpha\}} D(\beta)$$</code></p> <p>in the category of $\kappa$-compact objects, its pushout is also $\kappa$-compact. But the hypothesis is that $D(\alpha)$ is the pushout for some such diagram, so $D(\alpha)$ is $\kappa$-compact, and we have reached a contradiction. </p> <p>So $D(\alpha)$ is $\kappa$-small for all $\alpha \in A$. It follows that $colim_{\beta \in B} D(\beta)$ is $\kappa$-compact for any subposet $B \subseteq A$ whenever this is a $\kappa$-small colimit. (The restriction to downward-closed $B$ is not much loss of generality, because if $B \subseteq A$ is full, then the colimit over such a $B$ is isomorphic to the colimit over its downward closure, since $B$ is cofinal in its downward closure.) </p>