Triangulating Baez and Rovelli

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data-date-string="Apr 14, 2004" data-time-string="1:55 PM" title="Apr 14, 2004 at 1:55 PM">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187341/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #1 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" > <div class="message-userDetails"> <span class="message-name"><a href="/members/marcus.66/" class="username " dir="auto" data-user-id="66" data-xf-init="member-tooltip">marcus</a></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>  <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> 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js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187341" data-lb-caption-desc="marcus · Apr 14, 2004 at 1:55 PM"> <article class="message-body js-selectToQuote"> <div > <div class="bbWrapper">Baez and Rovelli have each tossed out two freshly minted mathematical ideas.<br /> These appear to be new in the sense of not having been studied yet (new theorems available to prove, one assumes)<br /> both ideas seem to have possibilities<br /> we ought to be able to treat the two ideas as "surveyer stakes" and<br /> triangulate from them where some future development might be<br /> <br /> Baez idea is *-category<br /> which is a category with paired morphisms<br /> each morphism in the category has a mate<br /> (the morphisms are like socks in the sock drawer ought to be but arent)<br /> <br /> if f goes from X to Y, then the mate of f goes from Y to X<br /> it is a symmetric relation----f** = f<br /> it is reflexive on identities----if 1 is the identity morphism on an object X then<br /> it is its own mate----1* = 1<br /> the pairing respects composition---if f goes from X to Y and g goes from Y to Z so that gf is defined going from X to Z then the mate of gf is f*g*----you go back by g* from Z to Y and then back by f* from Y to X<br /> <br /> the reason your eyes may be glazed is that it is so obvious---we used to call these things Columbus eggs for some reason, for some reason nobody thought of it till recently but now it seems like a really obvious thing to require of a category<br /> <br /> the category of Hilbertspaces with linear operators between them which is the Home Town of quantum mechanics is a star-category<br /> so is the home town of gen rel, Baez notes in his paper<br /> <br /> but the category of sets is not a star-category (although it is like a paradigm for category theory, oh well)<br /> <br /> ---------------<br /> Rovelli's idea, which plays a key role in his new book QG<br /> and also is the subject of the recent Fairbairn Rovelli paper<br /> is extending a certain gauge group<br /> to be the extended diffeomorphisms which are<br /> smooth except on a finite set<br /> <br /> and this boils the quantum states of space (the pure states of the geometry of the universe or whatever) down to a countable list of colored knots<br /> <br /> if we are going to triangulate from these two ideas we should<br /> think if there is a likely category in which the objects are colored knots<br /> <br /> and how do you get from one knot to another?<br /> <br /> by cobordisms?<br /> by "moves"?<br /> by 2-knots?<br /> by spinfoams...well?<br /> <br /> so what are the morphisms that get you from one knot to another<br /> and doubtless they are paired so you get a star-category a la Baez</div> </div> <div class="js-selectToQuoteEnd"> </div> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="3049969336" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2004-04-14T13:58:41-0500" data-time="1081969121" data-date-string="Apr 14, 2004" data-time-string="1:58 PM" title="Apr 14, 2004 at 1:58 PM">Apr 14, 2004</time> </div> </div> <footer class="message-footer"> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <div class="message-cell message-cell--main" id="post-physorg"> <a href="https://phys.org/physics-news/" target="_blank">Physics news</a> on Phys.org <ul class="phys_items"> <li><a href="https://phys.org/news/2025-02-unexpected-nucleus-prompts-reevaluation-atomic.html" target="_blank">Unexpected shape of lead-208 nucleus prompts reevaluation of atomic nuclei models</a></li> <li><a href="https://phys.org/news/2025-02-scientists-elusive-liquid-transition-deep.html" target="_blank">Scientists map elusive liquid-liquid transition point using deep neural network</a></li> <li><a href="https://phys.org/news/2025-02-links-disparate-pion-reactions-nuclear.html" target="_blank">New calculation links disparate pion reactions in nuclear physics</a></li> </ul> </div> <article class="message message--post js-post js-inlineModContainer " data-author="matt grime" data-content="post-187370" id="js-post-187370" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187370/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187370"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T14:50:20-0500" data-time="1081972220" data-date-string="Apr 14, 2004" data-time-string="2:50 PM" title="Apr 14, 2004 at 2:50 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187370/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #2 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/matt-grime.6037/"> <meta itemprop="url" content="https://www.physicsforums.com/members/matt-grime.6037/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/matt-grime.6037/" class="username " dir="auto" data-user-id="6037" data-xf-init="member-tooltip"><span itemprop="name">matt grime</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>9,361</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>6</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187370" data-lb-caption-desc="matt grime · Apr 14, 2004 at 2:50 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">try looking up the definition of a contravariant functor (that is fully faithful)</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="marcus" data-content="post-187416" id="js-post-187416" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187416/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187416"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T15:59:56-0500" data-time="1081976396" data-date-string="Apr 14, 2004" data-time-string="3:59 PM" title="Apr 14, 2004 at 3:59 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187416/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #3 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/marcus.66/"> <meta itemprop="url" content="https://www.physicsforums.com/members/marcus.66/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/marcus.66/" class="username " dir="auto" data-user-id="66" data-xf-init="member-tooltip"><span itemprop="name">marcus</span></a></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>  <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-heart" data-xf-init="tooltip" title="Memorial: Member has Passed"> <i class="fa--xf fal fa-heart" aria-hidden="true"></i><span> Dearly Missed</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>24,753</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>792</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187416" data-lb-caption-desc="marcus · Apr 14, 2004 at 3:59 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="" data-quote="matt grime" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> matt grime said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> try looking up the definition of a contravariant functor (that is fully faithful) </div> </div> </blockquote><br /> Hi Matt, I know the defn of a contravariant functor.<br /> <br /> Probably also a bunch of other people who visit here at PF also know the term<br /> (arivero had a category theory thread here a while ago)<br /> <br /> You may be making an interesting and valid point but I do not see clearly what it is. Maybe you could elaborate.<br /> <br /> Baez has defined a term "star-category" which as far as I know is a type of category (it is not a functor---a functor is a function between categories) and as far as I know it is a type of object which as such has not been studied a lot.<br /> <br /> A *-category is clearly not the same thing as a "contravariant functor!<br /> But perhaps you would like to rephrase Baez definition of *-category in<br /> functorial language?<br /> <br /> Why not spell it out, if you want. It would bring along more people with you.</div> </div> <div class="js-selectToQuoteEnd"> </div> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="5961342082" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="matt grime" data-content="post-187438" id="js-post-187438" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187438/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187438"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T16:30:41-0500" data-time="1081978241" data-date-string="Apr 14, 2004" data-time-string="4:30 PM" title="Apr 14, 2004 at 4:30 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187438/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #4 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/matt-grime.6037/"> <meta itemprop="url" content="https://www.physicsforums.com/members/matt-grime.6037/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/matt-grime.6037/" class="username " dir="auto" data-user-id="6037" data-xf-init="member-tooltip"><span itemprop="name">matt grime</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>9,361</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>6</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187438" data-lb-caption-desc="matt grime · Apr 14, 2004 at 4:30 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">The star operation is a contravariant functor.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="marcus" data-content="post-187446" id="js-post-187446" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187446/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187446"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T16:37:43-0500" data-time="1081978663" data-date-string="Apr 14, 2004" data-time-string="4:37 PM" title="Apr 14, 2004 at 4:37 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187446/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #5 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/marcus.66/"> <meta itemprop="url" content="https://www.physicsforums.com/members/marcus.66/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/marcus.66/" class="username " dir="auto" data-user-id="66" data-xf-init="member-tooltip"><span itemprop="name">marcus</span></a></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>  <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-heart" data-xf-init="tooltip" title="Memorial: Member has Passed"> <i class="fa--xf fal fa-heart" aria-hidden="true"></i><span> Dearly Missed</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>24,753</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>792</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187446" data-lb-caption-desc="marcus · Apr 14, 2004 at 4:37 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">Yes!<br /> and of a very special type.<br /> <br /> since this is possibly interesting I have put some basic definitions<br /> over in the math forum<br /> in a thread which meteor started<br /> <br /> <a href="https://www.physicsforums.com/showthread.php?p=187436#post187436" class="link link--internal">https://www.physicsforums.com/showthread.php?p=187436#post187436</a><br /> <br /> the question now is, what is so special about the * operator<br /> it is not just any old contravariant functor.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="matt grime" data-content="post-187453" id="js-post-187453" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187453/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187453"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T16:41:52-0500" data-time="1081978912" data-date-string="Apr 14, 2004" data-time-string="4:41 PM" title="Apr 14, 2004 at 4:41 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187453/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #6 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/matt-grime.6037/"> <meta itemprop="url" content="https://www.physicsforums.com/members/matt-grime.6037/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/matt-grime.6037/" class="username " dir="auto" data-user-id="6037" data-xf-init="member-tooltip"><span itemprop="name">matt grime</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>9,361</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>6</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187453" data-lb-caption-desc="matt grime · Apr 14, 2004 at 4:41 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">because it takes Hilb to Hilb and not Hilb to Hilb^{op}?</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="marcus" data-content="post-187463" id="js-post-187463" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187463/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187463"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T16:53:42-0500" data-time="1081979622" data-date-string="Apr 14, 2004" data-time-string="4:53 PM" title="Apr 14, 2004 at 4:53 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187463/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #7 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/marcus.66/"> <meta itemprop="url" content="https://www.physicsforums.com/members/marcus.66/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/marcus.66/" class="username " dir="auto" data-user-id="66" data-xf-init="member-tooltip"><span itemprop="name">marcus</span></a></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>  <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-heart" data-xf-init="tooltip" title="Memorial: Member has Passed"> <i class="fa--xf fal fa-heart" aria-hidden="true"></i><span> Dearly Missed</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>24,753</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>792</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187463" data-lb-caption-desc="marcus · Apr 14, 2004 at 4:53 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="" data-quote="matt grime" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> matt grime said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> because it takes Hilb to Hilb and not Hilb to Hilb^{op}? </div> </div> </blockquote><br /> You are free here---doing constructive mathematics. Baez gives us the definition of a certain kind of category, he calls it "star-category" and in defining it he uses a pairing relation on the morphisms <b>in that category</b>.<br /> this pairing is not, on the face of it, a functor<br /> (functors take categories to categories)<br /> but perhaps it is a functor of a certain restricted kind<br /> (one with no effect on objects?)<br /> he gives certain axioms that the pairing relation should satisfy that make the category be a star-category<br /> <br /> but you may be able to use functorial language to paraphrase Baez<br /> and give another, perhaps more general or suggestive, definition of<br /> star-category.<br /> <br /> at least somebody should try to do this, to see if it is a good idea to do it.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2004-04-14T16:56:22-0500" data-time="1081979782" data-date-string="Apr 14, 2004" data-time-string="4:56 PM" title="Apr 14, 2004 at 4:56 PM" itemprop="dateModified">Apr 14, 2004</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="matt grime" data-content="post-187469" id="js-post-187469" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187469/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187469"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T16:57:22-0500" data-time="1081979842" data-date-string="Apr 14, 2004" data-time-string="4:57 PM" title="Apr 14, 2004 at 4:57 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187469/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #8 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/matt-grime.6037/"> <meta itemprop="url" content="https://www.physicsforums.com/members/matt-grime.6037/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/matt-grime.6037/" class="username " dir="auto" data-user-id="6037" data-xf-init="member-tooltip"><span itemprop="name">matt grime</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>9,361</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>6</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187469" data-lb-caption-desc="matt grime · Apr 14, 2004 at 4:57 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">do you want to do this here or in the hijacked thread in maths?<br /> <br /> the star is a contravariant idempotent functor from Hilb to Hilb that is the identity on objects.</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="marcus" data-content="post-187476" id="js-post-187476" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187476/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187476"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T17:04:27-0500" data-time="1081980267" data-date-string="Apr 14, 2004" data-time-string="5:04 PM" title="Apr 14, 2004 at 5:04 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187476/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #9 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/marcus.66/"> <meta itemprop="url" content="https://www.physicsforums.com/members/marcus.66/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/marcus.66/" class="username " dir="auto" data-user-id="66" data-xf-init="member-tooltip"><span itemprop="name">marcus</span></a></span>   <span class="threadstarter" data-xf-init="tooltip" title="Thread Starter"><i class="fal fa-user-edit"></i></span>  <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-star" data-xf-init="tooltip" title="Click to Support PF"> <i class="fa--xf fal fa-star" aria-hidden="true"></i><span> Gold Member</span> </div> <div class="post-badge badge-heart" data-xf-init="tooltip" title="Memorial: Member has Passed"> <i class="fa--xf fal fa-heart" aria-hidden="true"></i><span> Dearly Missed</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>24,753</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>792</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187476" data-lb-caption-desc="marcus · Apr 14, 2004 at 5:04 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper"><blockquote data-attributes="" data-quote="matt grime" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-title"> matt grime said: </div> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> do you want to do this here or in the hijacked thread in maths?<br /> <br /> the star is a contravariant idempotent functor from Hilb to Hilb that is the identity on objects. </div> </div> </blockquote><br /> I think the thread in maths would be the right place. Meteor called it "Category Theory" so it would be a good place for discussion of <br /> things like this. Unless Meteor, who started it, objects in which case I shall start a "Category Theory 2" thread. Let's go there and check your definition.<br /> <br /> <a href="https://www.physicsforums.com/showthread.php?p=187506#post187506" class="link link--internal">https://www.physicsforums.com/showthread.php?p=187506#post187506</a></div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2004-04-14T17:55:32-0500" data-time="1081983332" data-date-string="Apr 14, 2004" data-time-string="5:55 PM" title="Apr 14, 2004 at 5:55 PM" itemprop="dateModified">Apr 14, 2004</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> </div> </div> </div> </article>  <article class="message message--post js-post js-inlineModContainer " data-author="matt grime" data-content="post-187486" id="js-post-187486" itemscope itemtype="https://schema.org/Comment" itemid="https://www.physicsforums.com/posts/187486/"> <meta itemprop="parentItem" itemscope itemid="https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/" /> <span class="u-anchorTarget" id="post-187486"></span> <div class="message-header"> <header class="message-attribution message-attribution--split"> <ul class="message-attribution-main listInline "> <li class="u-concealed"> <time class="u-dt" dir="auto" datetime="2004-04-14T17:11:06-0500" data-time="1081980666" data-date-string="Apr 14, 2004" data-time-string="5:11 PM" title="Apr 14, 2004 at 5:11 PM" itemprop="datePublished">Apr 14, 2004</time> </li> </ul> <ul class="message-attribution-opposite message-attribution-opposite--list "> <li> <span style="cursor:pointer;" data-xf-init="share-tooltip" data-href="/posts/187486/share" aria-label="Share" role="button" rel="nofollow"> <i class="fa--xf fal fa-share-alt" aria-hidden="true"></i> </span> </li> <li> #10 </li> </ul> </header> </div> <div class="message-inner"> <div class="message-cell message-cell--user"> <section class="message-user" itemprop="author" itemscope itemtype="https://schema.org/Person" itemid="https://www.physicsforums.com/members/matt-grime.6037/"> <meta itemprop="url" content="https://www.physicsforums.com/members/matt-grime.6037/" /> <div class="message-userDetails"> <span class="message-name"><a href="/members/matt-grime.6037/" class="username " dir="auto" data-user-id="6037" data-xf-init="member-tooltip"><span itemprop="name">matt grime</span></a></span>    <div class="post-badges"> <div class="post-badge badge-university" data-xf-init="tooltip" title="Verified Expert in Science and Math"> <i class="fa--xf fal fa-university" aria-hidden="true"></i><span> Science Advisor</span> </div> <div class="post-badge badge-cap" data-xf-init="tooltip" title="Verified Expert with Homework Help"> <i class="fa--xf fal fa-graduation-cap" aria-hidden="true"></i><span> Homework Helper</span> </div> </div>  </div> <div class="message-userExtras"> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-comments fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Messages"></i></dt> <dd>9,361</dd> </dl> <dl class="pairs pairs--justified"> <dt><i class="fa--xf fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>6</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-187486" data-lb-caption-desc="matt grime · Apr 14, 2004 at 5:11 PM"> <article class="message-body js-selectToQuote"> <div itemprop="text"> <div class="bbWrapper">I don't believe it is the right thread as your post does not address any of the issues raised in the previous post by others. I think that implies you ought to start a new thread if you've got new topics to raise, but my answer above contains all you need to know at the moment. Moreover the title does not indicate the type of subject inside the thread</div> </div> <div class="js-selectToQuoteEnd"> </div> </article> </div> <div class="message-lastEdit"> Last edited: <time class="u-dt" dir="auto" datetime="2004-04-14T17:13:53-0500" data-time="1081980833" data-date-string="Apr 14, 2004" data-time-string="5:13 PM" title="Apr 14, 2004 at 5:13 PM" itemprop="dateModified">Apr 14, 2004</time> </div> </div> <footer class="message-footer"> <div class="message-microdata" itemprop="interactionStatistic" itemtype="https://schema.org/InteractionCounter" itemscope> <meta itemprop="userInteractionCount" content="0" /> <meta itemprop="interactionType" content="https://schema.org/LikeAction" /> </div> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> <script> if (typeof parseNewPost == 'function') { $(document).ajaxStop(parseNewPost(187486)); } </script> </div> </div> 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Are you sure you want to submit this form?", attach: "Attach files", rich_text_box: "Rich text box", close: "Close", link_copied_to_clipboard: "Link copied to clipboard.", text_copied_to_clipboard: "Text copied to clipboard.", loading: "Loading…", you_have_exceeded_maximum_number_of_selectable_items: "You have exceeded the maximum number of selectable items.", processing: "Processing", 'processing...': "Processing…", showing_x_of_y_items: "Showing {count} of {total} items", showing_all_items: "Showing all items", no_items_to_display: "No items to display", number_button_up: "Increase", number_button_down: "Decrease", push_enable_notification_title: "Push notifications enabled successfully at Physics Forums: Science Discussion, Homework Help, Articles", push_enable_notification_body: "Thank you for enabling push notifications!", pull_down_to_refresh: "Pull down to refresh", release_to_refresh: "Release to refresh", refreshing: "Refreshing…" }); </script> <form style="display:none" hidden="hidden"> <input type="text" name="_xfClientLoadTime" value="" id="_xfClientLoadTime" title="_xfClientLoadTime" tabindex="-1" /> </form>  <script type="application/ld+json"> { "@context": "https://schema.org", "@type": "WebPage", "url": "https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/", "mainEntity": { "@type": "DiscussionForumPosting", "@id": "https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/", "headline": "Triangulating Baez and Rovelli", "datePublished": "2004-04-14T18:55:24+00:00", "url": "https://www.physicsforums.com/threads/triangulating-baez-and-rovelli.20655/", "articleSection": "Beyond the Standard Models", "author": { "@type": "Person", "@id": "https://www.physicsforums.com/members/marcus.66/", "name": "marcus", "url": "https://www.physicsforums.com/members/marcus.66/" }, "interactionStatistic": [ { "@type": "InteractionCounter", "interactionType": "https://schema.org/ViewAction", "userInteractionCount": 3782 }, { "@type": "InteractionCounter", "interactionType": "https://schema.org/CommentAction", "userInteractionCount": 9 }, { "@type": "InteractionCounter", "interactionType": "https://schema.org/LikeAction", "userInteractionCount": 0 } ], "dateModified": "2004-04-14T18:58:41+00:00", "articleBody": "Baez and Rovelli have each tossed out two freshly minted mathematical ideas.\r\nThese appear to be new in the sense of not having been studied yet (new theorems available to prove, one assumes)\r\nboth ideas seem to have possibilities\r\nwe ought to be able to treat the two ideas as \"surveyer stakes\" and\r\ntriangulate from them where some future development might be\r\n\r\nBaez idea is *-category\r\nwhich is a category with paired morphisms\r\neach morphism in the category has a mate\r\n(the morphisms are like socks in the sock drawer ought to be but arent)\r\n\r\nif f goes from X to Y, then the mate of f goes from Y to X\r\nit is a symmetric relation----f** = f\r\nit is reflexive on identities----if 1 is the identity morphism on an object X then\r\nit is its own mate----1* = 1\r\nthe pairing respects composition---if f goes from X to Y and g goes from Y to Z so that gf is defined going from X to Z then the mate of gf is f*g*----you go back by g* from Z to Y and then back by f* from Y to X\r\n\r\nthe reason your eyes may be glazed is that it is so obvious---we used to call these things Columbus eggs for some reason, for some reason nobody thought of it till recently but now it seems like a really obvious thing to require of a category\r\n\r\nthe category of Hilbertspaces with linear operators between them which is the Home Town of quantum mechanics is a star-category\r\nso is the home town of gen rel, Baez notes in his paper\r\n\r\nbut the category of sets is not a star-category (although it is like a paradigm for category theory, oh well)\r\n\r\n---------------\r\nRovelli's idea, which plays a key role in his new book QG\r\nand also is the subject of the recent Fairbairn Rovelli paper\r\nis extending a certain gauge group\r\nto be the extended diffeomorphisms which are\r\nsmooth except on a finite set\r\n\r\nand this boils the quantum states of space (the pure states of the geometry of the universe or whatever) down to a countable list of colored knots\r\n\r\nif we are going to triangulate from these two ideas we should\r\nthink if there is a likely category in which the objects are colored knots\r\n\r\nand how do you get from one knot to another?\r\n\r\nby cobordisms?\r\nby \"moves\"?\r\nby 2-knots?\r\nby spinfoams...well?\r\n\r\nso what are the morphisms that get you from one knot to another\r\nand doubtless they are paired so you get a star-category a la Baez" }, "publisher": { "@type": "Organization", "name": "Physics Forums: Science Discussion, Homework Help, Articles", "alternateName": "Physics Forums", "description": "Join Physics Forums, where students, scientists, and enthusiasts come together to explore and discuss the current 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Triangulating Baez and Rovelli