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<!DOCTYPE html><html><head><meta charset="utf-8"><meta http-equiv="x-ua-compatible" content="ie=edge"><meta property="fb:app_id" content="118554188236439"><meta name="viewport" content="width=device-width, initial-scale=1"><meta name="author" content="Maxim Sokhatsky"><meta name="twitter:site" content="@5HT"><meta name="twitter:creator" content="@5HT"><meta property="og:type" content="website"><meta property="og:image" content="https://avatars.githubusercontent.com/u/17128096?s=400&u=66a63d4cdd9625b2b4b37d724cc00fe6401e5bd8&v=4"><meta name="msapplication-TileColor" content="#ffffff"><meta name="msapplication-TileImage" content="https://anders.groupoid.space/images/ms-icon-144x144.png"><meta name="theme-color" content="#ffffff"><link rel="stylesheet" href="https://anders.groupoid.space/main.css?v=1"><link rel="apple-touch-icon" sizes="57x57" href="https://anders.groupoid.space/images/apple-icon-57x57.png"><link rel="apple-touch-icon" sizes="60x60" href="https://anders.groupoid.space/images/apple-icon-60x60.png"><link rel="apple-touch-icon" sizes="72x72" href="https://anders.groupoid.space/images/apple-icon-72x72.png"><link rel="apple-touch-icon" sizes="76x76" href="https://anders.groupoid.space/images/apple-icon-76x76.png"><link rel="apple-touch-icon" sizes="114x114" href="https://anders.groupoid.space/images/apple-icon-114x114.png"><link rel="apple-touch-icon" sizes="120x120" href="https://anders.groupoid.space/images/apple-icon-120x120.png"><link rel="apple-touch-icon" sizes="144x144" href="https://anders.groupoid.space/images/apple-icon-144x144.png"><link rel="apple-touch-icon" sizes="152x152" href="https://anders.groupoid.space/images/apple-icon-152x152.png"><link rel="apple-touch-icon" sizes="180x180" href="https://anders.groupoid.space/images//apple-icon-180x180.png"><link rel="icon" type="image/png" sizes="192x192" href="https://anders.groupoid.space/images/android-icon-192x192.png"><link rel="icon" type="image/png" sizes="32x32" href="https://anders.groupoid.space/images/favicon-32x32.png"><link rel="icon" type="image/png" sizes="96x96" href="https://anders.groupoid.space/images/favicon-96x96.png"><link rel="icon" type="image/png" sizes="16x16" href="https://anders.groupoid.space/images/favicon-16x16.png"><link rel="manifest" href="https://anders.groupoid.space/images/manifest.json"><style>svg a{fill:blue;stroke:blue}
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</style></head><body class="content"></body></html><html><head><meta property="og:title" content="GROUPOЇD"><meta property="og:description" content="L'Infini des Groupoïdes"><meta property="og:url" content="https://groupoid.space/"></head></html><title>GROUPOЇD</title><article class="main"><div class="exe"><section></section><p><a href="https://groupoid.space/institute/">L'Infini des Groupoïdes</a>
achieves a landmark synthesis, unifying synthetic and classical mathematics
in a mechanically verifiable framework <a href="https://axio.groupoid.space">AXIO/1</a>
showcasing its ability to span algebraic, analytic, geometric, categorical,
topological, and foundational domains in the set of languages:
<a href="https://anders.groupoid.space/">Anders</a> (Cubical HoTT),
<a href="https://dan.groupoid.space/">Dan</a> (Simplicial HoTT),
<a href="https://jack.groupoid.space/">Jack</a> (K-Theory, Hopf Fibrations),
<a href="https://urs.groupoid.space/">Urs</a> (Supergeometry),
<a href="https://fabien.groupoid.space/">Fabien</a> (A¹ HoTT).
Its type formers—spanning simplicial ∞-categories, stable spectra,
cohesive modalities, reals, ZFC, large cardinals, and forcing.
</p></div><br><hr size=1><br><div class="exe"><section></section><h1>MATHEMATICS</h1><p>Mathematical theories, models and languages.</p></div><div class="types"><br><br><div class="type"><ol class="type__col"><h3>Algebraic Systems</h3><li><a href="https://anders.groupoid.space/mathematics/algebra/algebra/">Algebraic structure</a></li><li><b>Group, Subgroup</b></li><li><b>Normal Group</b></li><li><b>Factorgroup</b></li><li><b>Abelian Group</b></li><li><b>Ring, Module</b></li><li><b>Field Theory</b></li><li><b>Linear Algebra</b></li><li><b>Universal Algebra</b></li><li><b>Lie, Leibniz Algebra</b></li></ol><ol class="type__col"><h3>(Co)Homotopy Theory</h3><li><a href="https://anders.groupoid.space/mathematics/homotopy/pullback/">Pullback</a></li><li><a href="https://anders.groupoid.space/mathematics/homotopy/pushout/">Pushout</a></li><li><a href="https://anders.groupoid.space/mathematics/geometry/bundle/">Limit, Fiber</a></li><li><b>Suspension, Loop</b></li><li><b>Smash, Wedge, Join</b></li><li><b>H-(co)spaces</b></li><li><b>Eilenberg-MacLane Spaces</b></li><li><b>Cell Complexes</b></li><li><b>∞-Groupoids</b></li><li><b>(Co)Homotopy</b></li><li><b>Hopf Invariant</b></li></ol><ol class="type__col"><h3>(Co)Homology Theory</h3><li><b>Chain Complex</b></li><li><a href="https://anders.groupoid.space/mathematics/algebra/homology/">(Co)Homology</a></li><li><b>Stinrod Axioms</b></li><li><b>Hom and Tensor</b></li><li><b>Resolutions</b></li><li><b>Derived Cat, Fun</b></li><li><b>Tor, Ext and Local Cohomology</b></li><li><b>Homological Algebra</b></li><li><b>Spectral Sequences</b></li><li><b>Cohomology Operations</b></li></ol></div><br><br><div class="type"><ol class="type__col"><h3>Category Theory</h3><li><a href="https://anders.groupoid.space/mathematics/categories/category/">Categories</a></li><li><a href="https://anders.groupoid.space/mathematics/categories/functor/">Functors, Adjunctions</a></li><li><b>Natural Transformations</b></li><li><b>Kan Extensions</b></li><li><b>(Co)limits</b></li><li><b>Universal Properties</b></li><li><b>Monoidal Categories</b></li><li><b>Enriched Categories</b></li><li><b>Structure Identity Principle</b></li></ol><ol class="type__col"><h3>Topos Theory</h3><li><b>Topology</b></li><li><b>Coverings</b></li><li><a href="https://anders.groupoid.space/mathematics/categories/topos/">Grothendieck Topology</a></li><li><b>Grothendieck Topos</b></li><li><b>Geometric Morphisms</b></li><li><b>Higher Topos Theory</b></li><li><b>Elementary Topos</b></li></ol><ol class="type__col"><h3>Geometry</h3><li><b>Nisnevich Site</b></li><li><b>Zariski Site</b></li><li><b>Theory of Schemes</b></li><li><b>Noetherian Scheme</b></li><li><b>A¹-Homotopy Theory</b></li><li><b>Cohesive Topos</b></li><li><b>Etale Topos</b></li><li><b>Differential Geometry</b></li><li><b>Synthetic Geometry</b></li><li><b>Local Homotopy Theory</b></li></ol></div><br><br><div class="type"><ol class="type__col"><h3>Analysis</h3><li><b>Real Analysis</b></li><li><b>Funtional analysis</b></li><li><b>Measure theory</b></li></ol><ol class="type__col"><h3>Foundations</h3><li><b>Proof Theory</b></li><li><b>Set Theory</b></li><li><b>Schönfinkel</b></li><li><b>Łukasiewicz</b></li><li><b>Frege</b></li><li><b>Hilbert</b></li><li><b>Church</b></li><li><b>Tarski</b></li></ol><ol class="type__col"><h3>Type Theory</h3><li><a href="https://laurent.groupoid.space/">Laurent Schwartz</a></li><li><b>Ernst Zermelo</b></li><li><b>Paul Cohen</b></li><li><a href="https://henk.groupoid.space/">Henk Barendregt</a></li><li><a href="https://per.groupoid.space/">Per Martin-Löf</a></li><li><a href="https://christine.groupoid.space/">Christine Paulin-Mohring</a></li><li><a href="https://anders.groupoid.space/">Anders Mörtberg</a></li><li><a href="https://dan.groupoid.space/">Dan Kan</a></li><li><a href="https://jack.groupoid.space/">Jack Morava</a></li><li><a href="https://urs.groupoid.space/">Urs Schreiber</a></li><li><a href="https://fabien.groupoid.space/">Fabien Morel</a></li></ol></div></div><div class="exe"><section></section><h1>LIBRARY</h1><p>Homotopy Library for <a href="https://anders.groupoid.space/">Anders</a>
theorem prover consists of two parts Foundations and Mathematics just
as <a href="https://homotopytypetheory.org/book/">HoTT Book</a>.
</p><h2>FOUNDATIONS</h2><p>MLTT, Modal and Univalent Foundations represent a basic
language primitives of <b>Anders</b> and its base library.
</p><section><div class="macro"><div class="macro__col"><h3 id="mltt"><b>MLTT</b></h3><ol><li><a href='https://anders.groupoid.space/foundations/mltt/pi/'>Π</a>, <a href='https://anders.groupoid.space/foundations/mltt/sigma/'>Σ</a>, <a href='https://anders.groupoid.space/foundations/mltt/id/'>=</a></li><li><a href='https://anders.groupoid.space/foundations/mltt/inductive/'>0, 1, 2, W</a></li><li><a href='https://anders.groupoid.space/foundations/mltt/either/'>+</a>, <a href='https://anders.groupoid.space/foundations/mltt/maybe/'>1+</a></li><li><a href='https://anders.groupoid.space/foundations/mltt/nat/'>N</a>, <a href='https://anders.groupoid.space/foundations/mltt/list/'>LIST</a></li><li><a href='https://anders.groupoid.space/foundations/mltt/fin/'>FIN</a>, <a href='https://anders.groupoid.space/foundations/mltt/vec/'>VEC</a></li></ol></div><div class="macro__col"><h3 id="modal"><b>MODAL</b></h3><ol><li><a href="https://anders.groupoid.space/foundations/modal/process/">PROCESS</a></li><li><a href="https://anders.groupoid.space/foundations/modal/infinitesimal/">INFINITESIMAL</a></li><li><a href="https://anders.groupoid.space/foundations/modal/localization/">LOCALIZATION</a></li></ol></div><div class="macro__col"><h3 id="univalent"><b>UNIVALENT</b></h3><ol><li><a href="https://anders.groupoid.space/foundations/univalent/path/">PATH</a></li><li><a href="https://anders.groupoid.space/foundations/univalent/glue/">GLUE</a></li><li><a href="https://anders.groupoid.space/foundations/univalent/equiv/">EQUIV</a></li><li><a href="https://anders.groupoid.space/foundations/univalent/funext/">FUNEXT</a></li><li><a href="https://anders.groupoid.space/foundations/univalent/iso/">ISO</a></li></ol></div></div></section><h2>MATHEMATICS</h2><p>The second part is dedicated to mathematical models and theories internalized in this language.
</p><section><div class="macro"><div class="macro__col"><h3 id="categories"><b>ANALYSIS</b></h3><ol><li><a href='https://anders.groupoid.space/mathematics/analysis/topology/'>TOPOLOGY</a></li><li><a href='https://anders.groupoid.space/mathematics/analysis/set/'>SET</a></li><li><a href='https://anders.groupoid.space/mathematics/analysis/rational/'>ℚ</a>,
<a href='https://anders.groupoid.space/mathematics/analysis/real/'>ℝ</a></li><li><a href='https://anders.groupoid.space/mathematics/analysis/complex/'>ℂ</a>,
<a href='https://anders.groupoid.space/mathematics/analysis/quatro/'>ℍ</a>,
<a href='https://anders.groupoid.space/mathematics/analysis/octo/'>𝕆</a></li></ol></div><div class="macro__col"><h3 id="algebra"><b>ALGEBRA</b></h3><ol><li><a href="https://anders.groupoid.space/mathematics/algebra/group/">GROUP</a></li><li><a href="https://anders.groupoid.space/mathematics/algebra/algebra/">ALGEBRA</a></li><li><a href="https://anders.groupoid.space/mathematics/algebra/homology/">HOMOLOGY</a></li></ol></div><div class="macro__col"><h3 id="geometry"><b>GEOMETRY</b></h3><ol><li><a href="https://anders.groupoid.space/mathematics/geometry/etale/">ETALE</a></li><li><a href="https://anders.groupoid.space/mathematics/geometry/bundle/">BUNDLE</a></li><li><a href="https://anders.groupoid.space/mathematics/geometry/manifold/">MANIFOLD</a></li><li><a href="https://anders.groupoid.space/mathematics/geometry/derham/">DE RHAM</a></li></ol></div></div></section><section><div class="macro"><div class="macro__col"><h3 id="homotopy"><b>HOMOTOPY</b></h3><ol><li><a href="https://anders.groupoid.space/mathematics/homotopy/coequalizer/">COEQUALIZER</a></li><li><a href="https://anders.groupoid.space/mathematics/homotopy/pushout/">PUSHOUT</a></li><li><a href="https://anders.groupoid.space/mathematics/homotopy/pullback/">PULLBACK</a></li><li><a href="https://anders.groupoid.space/mathematics/homotopy/hopf/">HOPF</a></li><li><a href="https://anders.groupoid.space/mathematics/homotopy/cw/">CW</a></li></ol></div><div class="macro__col"><h3 id="categories"><b>CATEGORIES</b></h3><ol><li><a href="https://anders.groupoid.space/mathematics/categories/category/">CATEGORY</a></li><li><a href="https://anders.groupoid.space/mathematics/categories/functor/">FUNCTOR</a></li><li><a href="https://anders.groupoid.space/mathematics/categories/groupoid/">GROUPOID</a></li><li><a href="https://anders.groupoid.space/mathematics/categories/topos/">TOPOS</a></li><li><a href="https://anders.groupoid.space/mathematics/categories/presheaf/">PRESHEAF</a></li></ol></div></div></section><br><p>The base library for <b>cubicaltt</b> is given on separate page:
<a href='https://groupoid.space/misc/library/'>Formal Mathematics: The Cubical Base Library</a><br>
</p><h1>LECTURES</h1><p>The series of articles on foundation and mathematics of Homotopy Type Theory.</p><p>— <a href='articles/mltt/mltt.pdf'>Issue I: Internalizing MLTT</a><br>
— <a href='articles/hott/hott.pdf'>Issue III: Homotopy Type Theory</a><br>
— <a href='articles/topos/topos.pdf'>Issue VIII: Formal Set Topos</a><br>
— <a href='articles/pts/pts.pdf'>Addendum I: Pure Type System</a><br>
— <a href='articles/equ/equ.pdf'>Addendum II: Many Faces of Equality</a><br></p><center><br>🧊 <br><br><br>
</center></div></article><hr><link rel="stylesheet" href="https://groupoid.space/main.css"><footer class="footer"><a href="https://5HT.co/license/"><img class="footer__logo" src="https://longchenpa.guru/seal.png" width="50"></a><span class="footer__copy">2016—2025 © <a rel="me" href="https://mathstodon.xyz/@5ht">Namdak Tönpa</a></span><script src="https://groupoid.space/highlight.js"></script><script src="https://groupoid.space/bundle.js"></script></footer>