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class="available-ntent"><div dir="to" class="body markup"><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="306" height="33.5" data-attrs="{"src":"" class="sizg-normal" alt srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" fetchprry="high" /></picture><div></div></div></a></figure></div><p>Math is queer, and not jt bee of puns volvg homos.</p><p>But before we talk about that, let’s get a broar picture of mathematics self.</p><p>Image mathematics to be a nse fort. Each tree is a problem or object that ptur a rpective field (topologil spac topology, sets set theory, groups and rgs algebra, and so forth). To study the objects is to look at the var ponents that make up the tree, often wh a magnifyg glass.</p><p>There’s an exprsn that turns up monly among math nversatnalists:</p><blockquote><p><em>Can’t see the fort for the tre.</em></p></blockquote><p>When you’re so zoomed , even if you are borrowg techniqu om lookg at other tre—you miss the bigger picture.</p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="352" height="198" data-attrs="{"src":" Dora the Explorer Season 8 Episo 16: Dora's Night Light Adventure - Full show on Paramount Pl","tle":null,"type":null,"href":null,"belowTheFold":false,"topImage":false,"ternalRedirect":null}" class="sizg-normal" alt="Watch Dora the Explorer Season 8 Episo 16: Dora's Night Light Adventure - Full show on Paramount Pl" tle="Watch Dora the Explorer Season 8 Episo 16: Dora's Night Light Adventure - Full show on Paramount Pl" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /></picture><div></div></div></a><figptn class="image-ptn">Dora’s Night Light Adventure</figptn></figure></div><p>Category theory is seeg the fort of mathematics self— looks at ways which the tre are related to one another: the roots that nnect and tertwe and the on that don’t, the patterns or lack of among the stcture, shap, and twists between tre.</p><p>Equipped wh a picture, let’s now troduce some more formal statements to enpsulate tegory theory a basic way:</p><p>Rell that a functn tak an put to an output. A functn is a directnal relatnship— tells how thgs the put relate to thgs the output (but not necsarily the other way around unls the functn satisfi certa properti).</p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="199" height="95.46125461254613" data-attrs="{"src":" transformed functns wh arrows - mathrecreatn","tle":null,"type":null,"href":null,"belowTheFold":te,"topImage":false,"ternalRedirect":null}" class="sizg-normal" alt="unrstandg transformed functns wh arrows - mathrecreatn" tle="unrstandg transformed functns wh arrows - mathrecreatn" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div></div></div></a><figptn class="image-ptn">the functn f relat x to y</figptn></figure></div><p><span>A </span><strong>morphism</strong><span> is a generalized ia of a functn which clus the relatn of ‘functn’ unr s umbrella—more formally, ’s a stcture-prervg mappg (relatn) om one mathematil stcture to another of the same kd.</span></p><p>Some exampl of morphisms and the objects they relate:</p><ul><li><p>Functns and sets</p></li><li><p>Lear transformatns and vector spac</p></li><li><p>Contuo functns and topologil spac</p></li></ul><p><span>A </span><strong>tegory</strong><span> pris of two thgs: the objects of the tegory and the morphisms (relatns/mappgs) which relate two objects (stead of </span><em>put</em><span> and </span><em>output</em><span>, objects are lled </span><em>source</em><span> and </span><em>target</em><strong>).</strong></p><p>In fact, those pots above each make up their own tegory: the tegory of sets, the tegory of vector spac, the tegory of tops (ugh).</p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="176" height="176" data-attrs="{"src":" theory - Wikipedia","tle":"Category theory - Wikipedia","type":null,"href":null,"belowTheFold":te,"topImage":false,"ternalRedirect":null}" class="sizg-normal" alt="Category theory - Wikipedia" tle="Category theory - Wikipedia" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div></div></div></a><figptn class="image-ptn">A tegory wh objects X, Y, Z and morphisms f and g</figptn></figure></div><p>Now we ask: what if we looked at morphisms between different tegori themselv?</p><p><span>A </span><strong>functor</strong><span> is a morphism between two tegori, mappg objects and morphisms one tegory to those another—this is how we see the fort. </span></p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="238" height="153.2125" data-attrs="{"src":" wh Functors. In our last post we provid some… | by Mar Perone | Statebox","tle":null,"type":null,"href":null,"belowTheFold":te,"topImage":false,"ternalRedirect":null}" class="sizg-normal" alt="Fun wh Functors. In our last post we provid some… | by Mar Perone | Statebox" tle="Fun wh Functors. In our last post we provid some… | by Mar Perone | Statebox" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div></div></div></a><figptn class="image-ptn"><span>Given tegori </span><em>C</em><span> and </span><em>D</em><span>, </span><strong>F</strong><span> is a functor such that </span><strong>F</strong><span>(</span><em>C</em><span>) = </span><em>D.</em></figptn></figure></div><p>We saw morphisms reprented as arrows—now we draw more arrows (dotted) between those arrows as well as their objects.</p><p>But let’s get even more meta:</p><p><span>A </span><strong>natural transformatn</strong><span> is a morphism between two functors. Practilly, provis a way to transform a functor to another while prervg posnal stctur of the morphisms. In other words, we n make a relatnship look like another and e to fd that they are really not that different (sound faiar?)</span></p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="296" height="193.74175824175825" data-attrs="{"src":"" class="sizg-normal" alt srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div></div></div></a><figptn class="image-ptn">Y homo</figptn></figure></div><p>Category theory origally veloped as a tool to e problems, but as grew as a field, enabled mathematicians to bridge realms prevly thought to be unnnected.</p><p>Set theory abstracts llectns of objects (lled elements), topology abstracts space, rgs abstract arhmetic, and so on. If mathematics abstracts objects and properti about those objects, tegory theory abstracts abstractn self—such that is lled “the math of math.”</p><p>So strong is s power of abstractn that any mathematil statement n be exprsed tegory theory—while may end up beg overly technil and nvoluted, do show how powerful and exprsive tegory theory is.</p><div class="ptned-image-ntaer"><figure><a class="image-lk image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="414" height="227.47252747252747" data-attrs="{"src":"" class="sizg-normal" alt srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div></div></div></a><figptn class="image-ptn">A func-y diagram</figptn></figure></div><p>Powerful, exprsive, reimagative—’s why tegory theory, the math of math, is both radil and queer.</p><p><span>In </span><a href=" rel="nofollow ugc noopener">LGBTQ+Math Day</a><span>, mathematician Ey Riehl talked about her work tegory theory on “Contractibily as Uniquens” which she am as “queerg uniquens.”</span></p><p>Informally and very loosely, Riehl showed a way to rive uniquens a problem posed to lose .</p><p>Formally, Riehl looked at a fundamental group a topologil space, which is a group ma of all the loops startg and endg at a base pot—if we lose the rtrictn that the paths have to be loops, we also lose the abily to pose any arbrary pair of elements back to self.</p><p>She showed that unr certa ndns, such a posn n still fact be unique and associative (a property we want sce mak nice thgs happen). </p><p><span>One, two, skip a few and g ntractibily (a property where a space n be ntuoly ntracted to a pot) and tegory theory that generaliz higher dimensnal tegori, Riehl go on to rive uniquens om the fact, drawg om s robtns and power </span><em>exprsive form</em><span> even wh an inty-losg transformatn.</span></p><p>Inty and exprsn are paramount queerns—allegorilly, I see this as a triumph of queer inty mathematics.</p><div class="ptned-image-ntaer"><figure><a class="image-lk is-viewable-img image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="256" height="307.05413105413106" data-attrs="{"src":"" class="sizg-normal" alt srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" loadg="lazy" /></picture><div class="image-lk-expand"><svg xmlns=" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="#FFFFFF" stroke-width="2" stroke-lep="round" stroke-lejo="round" class="luci luci-maximize2"><polyle pots="15 3 21 3 21 9"></polyle><polyle pots="9 21 3 21 3 15"></polyle><le x1="21" x2="14" y1="3" y2="10"></le><le x1="3" x2="10" y1="21" y2="14"></le></svg></div></div></a><figptn class="image-ptn"><span>Chen Chen poem Dora Malech scrib as “</span><a href=" rel="nofollow ugc noopener">inty form</a><span>”</span></figptn></figure></div><p><strong>The Queer Categoril Agenda</strong></p><p>At the heart of tegory theory isn’t to tegorize thgs, but to tach om the ia of a fixed label on somethg and stead look at the ways relat, meangful or not, to everythg else.</p><p>Natural transformatns pture and abstract the trsic notns of transformative inty and morphic relatnships. Extensnally, tegory theory reimag the foundatns of what ed to be thought of as necsary mathematics— ways analogo to the work of reimagg the msag once thought of as necsary and fundamental for genr and sexualy.</p><p>It’s queer, mathematil work: g the power of relatn and exprsn to not only unravel, but honor difference amed through samens.</p><div class="ptned-image-ntaer"><figure><a class="image-lk is-viewable-img image2" target="_blank" href=" data-ponent-name="Image2ToDOM" rel="nofollow ugc noopener"><div class="image2-set"><picture><source type="image/webp" srcset=" 424w, 848w, 1272w, 1456w" siz="100vw" /><img src=" width="223" height="296.7680608365019" data-attrs="{"src":" MATH DISPLAY by Se wh Miss Gil | Teachers Pay Teachers","tle":null,"type":null,"href":null,"belowTheFold":te,"topImage":false,"ternalRedirect":null}" class="sizg-normal" alt="RAINBOW MATH DISPLAY by Se wh Miss Gil | Teachers Pay Teachers" tle="RAINBOW MATH DISPLAY by Se wh Miss Gil | Teachers Pay Teachers" 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