Compact space in site -> compact object in topos - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:43:11Z http://mathoverflow.net/feeds/question/95563 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95563/compact-space-in-site-compact-object-in-topos Compact space in site -> compact object in topos Urs Schreiber 2012-04-30T08:32:15Z 2012-04-30T08:32:15Z <p>Given a site $C$, there are various standard notions for an object $X \in C$ being <em>compact</em>. For instance:</p> <ol> <li><p>Every covering family $\lbrace U_i \to X \rbrace$ has a finite subfamily that is still covering.</p></li> <li><p>The functor $C(X,-)$ commutes with filtered colimits.</p></li> <li><p>After Yoneda-embedding, the functor $Sh_C(X, -)$ commutes with filtered colimits.</p></li> <li><p>After $\infty$-Yoneda-embedding, the functor $\infty Sh_C(X, -)$ commutes with filtered $\infty$-colimits.</p></li> </ol> <p>These notions are closely related but subtly different. For instance for $C = Top$ it is well known that the first two are not equivalent without further fine-tuning.</p> <p>What can one say about the relation of 1. to 3. and 4. ?</p> <p>It seems to me that one can say for instance: compactness in the first sense implies that $Sh_C(X,-)$ commutes with <em>mono-filtered</em> colimits, and this should generalize to the $\infty$-case in the suitable sense.</p> <p>What else can one say?</p>