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Woodward"/><meta property="og:url" content="https://ems.press/journals/qt/articles/2736"/><meta property="og:type" content="website"/><meta property="og:title" content="Functoriality for Lagrangian correspondences in Floer theory | EMS Press"/><meta property="og:description" content="Katrin Wehrheim, Chris T. Woodward"/><meta property="og:image" content="https://content.ems.press/assets/public/images/serial-issues/cover-qt-volume-1-issue-2.png"/><link rel="canonical" href="https://ems.press/journals/qt/articles/2736"/><meta name="citation_title" content="Functoriality for Lagrangian correspondences in Floer theory"/><meta name="citation_author" content="Katrin Wehrheim"/><meta name="citation_author" content="Chris T. 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type="application/ld+json">{"@context":"https://schema.org","@type":"Article","headline":"Functoriality for Lagrangian correspondences in Floer theory","image":"https://content.ems.press/assets/public/images/serial-issues/cover-qt-volume-1-issue-2.png","datePublished":"2010-06-22T00:00:00.000Z","author":[{"@type":"Person","name":"Katrin Wehrheim","affiliation":"MIT, Cambridge, MA"},{"@type":"Person","name":"Chris T. Woodward","affiliation":"Rutgers University"}],"name":""}</script><article class="jsx-1331919424 jsx-3100645510 detail-page"><div class="jsx-1331919424 jsx-3100645510 container with-description with-actions"><div class="jsx-1331919424 jsx-3100645510 title"><header class="jsx-2448132927 title undefined"><span class="jsx-2448132927 roofline"><div class="jsx-1179927685 breadcrumbs"><span class="jsx-1179927685 step"><a class="link jsx-2462825770" href="/journals">Journals</a></span><span class="jsx-1179927685 step"><a class="link jsx-2462825770" href="/journals/qt">qt</a></span><span class="jsx-1179927685 step"><a class="link jsx-2462825770" href="/journals/qt/issues/1334">Vol. 1, No. 2</a></span><span class="jsx-1179927685 label">pp. 129–170</span></div></span><h1 class="jsx-2448132927 primary-title">Functoriality for Lagrangian correspondences in Floer theory</h1></header></div><div style="grid-area:desc" class="jsx-1331919424 jsx-3100645510"><div class="jsx-95f2c4b78c3236a8 journal-article-top-description"><div role="group" class="jsx-2097570476 person-group undefined"><ul class="jsx-2097570476 list"><li class="jsx-2097570476 list-item"><section class="jsx-2177874377 person"><header class="jsx-2177874377 title-wrapper"><h3 class="jsx-2177874377 title">Katrin Wehrheim</h3><span class="jsx-2177874377 organisation">MIT, Cambridge, MA</span></header></section></li><li class="jsx-2097570476 list-item"><section class="jsx-2177874377 person"><header class="jsx-2177874377 title-wrapper"><h3 class="jsx-2177874377 title">Chris T. Woodward</h3><span class="jsx-2177874377 organisation">Rutgers University</span></header></section></li></ul></div></div></div><div class="jsx-1331919424 jsx-3100645510 cover"><img alt="Functoriality for Lagrangian correspondences in Floer theory cover" fetchPriority="high" width="4016" height="5669" decoding="async" data-nimg="1" style="color:transparent;width:100%;height:auto" sizes="33vw" srcSet="/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=256&q=90 256w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=384&q=90 384w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=640&q=90 640w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=750&q=90 750w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=828&q=90 828w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=1080&q=90 1080w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=1200&q=90 1200w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=1920&q=90 1920w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=2048&q=90 2048w, /_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=3840&q=90 3840w" src="/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-qt-volume-1-issue-2.png&w=3840&q=90"/><div class="jsx-1331919424 jsx-3100645510 actions"><div class="article-download-button"><a class="button jsx-1059779991 download-button jsx-1368592022 jsx-2462825770" href="/content/serial-article-files/36743?nt=1">Download PDF</a></div></div></div><main style="grid-area:main;padding-top:1.8em"><div class="jsx-2264073825 formatted-text"><h2 class="jsx-a7383d06ba4e7e2 abstract-title">Abstract</h2><p>We associate to every monotone Lagrangian correspondence a functor between Donaldson–Fukaya categories. The composition of such functors agrees with the functor associated to the geometric composition of the correspondences, if the latter is embedded. That is “categorification commutes with composition” for Lagrangian correspondences. This construction fits into a symplectic 2-category with a categorification 2-functor, in which all correspondences are composable, and embedded geometric composition is isomorphic to the actual composition. As a consequence, any functor from a bordism category to the symplectic category gives rise to a category valued topological field theory.</p></div><div class="jsx-2357363445 cite-box"><h2 class="jsx-a7383d06ba4e7e2 cite-box-title">Cite this article</h2><p class="jsx-2357363445 cite-text">Katrin Wehrheim, Chris T. Woodward, Functoriality for Lagrangian correspondences in Floer theory. Quantum Topol. 1 (2010), no. 2, pp. 129–170</p><a href="https://doi.org/10.4171/qt/4" target="_blank" rel="noreferrer" class="jsx-2357363445 doi">DOI 10.4171/QT/4</a></div></main><aside class="jsx-1279956873 jsx-1604905398 metadata-box"><h2 class="jsx-1279956873 jsx-1604905398">Dates</h2><dl class="jsx-1279956873 jsx-1604905398"><dt class="jsx-1279956873 jsx-1604905398">Submitted</dt><dd class="jsx-1279956873 jsx-1604905398">20 August 2009</dd><dt class="jsx-1279956873 jsx-1604905398">Published</dt><dd class="jsx-1279956873 jsx-1604905398">22 June 2010</dd></dl><h2 class="jsx-1279956873 jsx-1604905398">Identifiers</h2><dl class="jsx-1279956873 jsx-1604905398"><dt class="jsx-1279956873 jsx-1604905398">DOI</dt><dd class="jsx-1279956873 jsx-1604905398"><a href="https://doi.org/10.4171/qt/4" class="jsx-1279956873 jsx-1604905398">10.4171/QT/4</a></dd></dl><h2 class="jsx-1279956873 jsx-1604905398">Keywords</h2><ul class="jsx-1279956873 jsx-1604905398 keywords"><li class="jsx-1279956873 jsx-1604905398">Floer homology</li><li class="jsx-1279956873 jsx-1604905398">Lagrangian correspondences</li><li class="jsx-1279956873 jsx-1604905398">topological field theory</li></ul><h2 class="jsx-1279956873 jsx-1604905398">Mathematics Subject Classification</h2><ul class="jsx-1279956873 jsx-1604905398 keywords"><li class="jsx-1279956873 jsx-1604905398">53-XX</li><li class="jsx-1279956873 jsx-1604905398">57-XX</li><li class="jsx-1279956873 jsx-1604905398">00-XX</li></ul></aside></div></article></main><footer class="jsx-812847572 footer"><div class="jsx-812847572 inner layout-container"><nav title="Footer Navigation" class="jsx-812847572"><ul class="jsx-812847572"><li class="jsx-812847572"><a class="jsx-2462825770 unstyled" href="/contact">Contact</a></li><li class="jsx-812847572"><a class="jsx-2462825770 unstyled" href="/legal">Legal Notice</a></li><li class="jsx-812847572"><a class="jsx-2462825770 unstyled" href="/privacy">Privacy Policy</a></li><li class="jsx-812847572"><a 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