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More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F2680%2Fquasicompact%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: quasicompact</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 3 of 3</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_22689"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>Apr 30th 2011</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2680/quasicompact/?Focus=22689#Comment_22689">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_22689"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I changed [[quasicompact]] to [[quasicompact morphism]] though it is also about quasicompact schemes etc. as before and moved the query box here: >[[Mike Shulman|Mike]]: To accord with terminological conventions, this page should probably be either "quasicompact space" or "quasicompact object." >[[Zoran Skoda]]: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice. >_Toby_: By the «Each definition gets its own page.» convention, I\'m not even sure that this shouldn\'t just redirect to [[compact space]] or [[compact object]]. My impression is that assuming that 'compact' implies Hausdorff is either (like assuming that 'ring' implies commutative) restricted to fields where it\'s a common assumption or to languages (I\'m thinking mostly of Bourbaki in French here) other than English. On the other hand, if it\'s used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces). >[[Zoran Skoda]]: Convention that 'compact' includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for "paracompact" the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form "compactum". Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won't work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact. There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the "same" example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to [[compact object]] and saying this is for algebraic geometry won't do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism). >_Toby_: Ah, so when you\'ve got both Zariski and complex topologies around, you can easily distinguish the former by the prefix 'quasi'; that\'s cute. Anyway, perhaps we\'ll move this to [[quasicompact morphism]] if you write mostly about that, but I won\'t try to move anything for now.</code></div><div> <p>I changed <a href="https://ncatlab.org/nlab/show/quasicompact">quasicompact</a> to <a href="https://ncatlab.org/nlab/show/quasicompact morphism">quasicompact morphism</a> though it is also about quasicompact schemes etc. as before and moved the query box here:</p> <blockquote> <p><a href="https://ncatlab.org/nlab/show/Mike Shulman">Mike</a>: To accord with terminological conventions, this page should probably be either “quasicompact space” or “quasicompact object.”</p> <p><a href="https://ncatlab.org/nlab/show/Zoran Skoda">Zoran Skoda</a>: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice.</p> <p><em>Toby</em>: By the «Each definition gets its own page.» convention, I'm not even sure that this shouldn't just redirect to <a href="https://ncatlab.org/nlab/show/compact space">compact space</a> or <a href="https://ncatlab.org/nlab/show/compact object">compact object</a>. My impression is that assuming that ’compact’ implies Hausdorff is either (like assuming that ’ring’ implies commutative) restricted to fields where it's a common assumption or to languages (I'm thinking mostly of Bourbaki in French here) other than English. On the other hand, if it's used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces).</p> <p><a href="https://ncatlab.org/nlab/show/Zoran Skoda">Zoran Skoda</a>: Convention that ’compact’ includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for “paracompact” the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form “compactum”. Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won’t work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact.<br/> There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the “same” example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to <a href="https://ncatlab.org/nlab/show/compact object">compact object</a> and saying this is for algebraic geometry won’t do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism).</p> <p><em>Toby</em>: Ah, so when you've got both Zariski and complex topologies around, you can easily distinguish the former by the prefix ’quasi’; that's cute. Anyway, perhaps we'll move this to <a href="https://ncatlab.org/nlab/show/quasicompact morphism">quasicompact morphism</a> if you write mostly about that, but I won't try to move anything for now.</p> </blockquote> </div> </div> </li><li id="Comment_31452" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/10/">zskoda</a></li> <li><span>CommentTime</span>May 17th 2012</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2680/quasicompact/?Focus=31452#Comment_31452">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_31452"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/10/">zskoda</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>An interesting variant of notion of quasi-compactness in model theory, in the context of "topological structures" sec.2.1 is in sec. 4.1 of * [[Boris Zilber]], _On model theory, non-commutative geometry and physics_, [pdf](https://http>//people.maths.ox.ac.uk/zilber/bul-survey.pdf)</code></div><div> <p>An interesting variant of notion of quasi-compactness in model theory, in the context of “topological structures” sec.2.1 is in sec. 4.1 of</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Boris Zilber">Boris Zilber</a>, <em>On model theory, non-commutative geometry and physics</em>, <a href="https://http>//people.maths.ox.ac.uk/zilber/bul-survey.pdf">pdf</a></li> </ul> </div> </div> </li><li id="Comment_31466"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/7/">TobyBartels</a></li> <li><span>CommentTime</span>May 17th 2012</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/2680/quasicompact/?Focus=31466#Comment_31466">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_31466"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/7/">TobyBartels</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I just realised that this did not get moved properly; it was duplicated instead. I've moved [[quasicompact]] to [[quasicompact > history]]</code></div><div> <p>I just realised that this did not get moved properly; it was duplicated instead. I’ve moved <a href="https://ncatlab.org/nlab/show/quasicompact">quasicompact</a> to <a href="https://ncatlab.org/nlab/show/quasicompact > history">quasicompact > history</a></p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 3 of 3</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>