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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="URS"><meta property="og:description" content="Super Type Theory"><meta property="og:url" content="https://urs.groupoid.space/"></head></html><title>URS</title><nav><a href='#'>URS</a></nav><article class="main"><div class="exe"><section><h1>HOMOTOPY SUPER SMOOTH <br> DIFFERENTIAL QUANTUM <br> TYPE SYSTEM</h1></section><aside><time>Published: 28 FEB 2025</time><p>We present Homotopy Supersmooth Differential Quantum Type System.
This system builds a progressive structure for formalizing mathematical and physical concepts,
from homotopy and higher categorical structures,
through geometric cohesion and differential properties,
to the rich graded and equivariant world of supergeometry.
Inspired by Urs Schreiber’s work on equivariant super homotopy theory,
this layered approach offers a modular,
type-theoretic foundation for synthetic supergeometry and beyond.
</p><p>Type theory has emerged as a powerful language for mathematics
and physics, unifying computation, logic, and structure. This
article introduces a layered type theory that extends
Martin-Löf’s intensional type theory into a framework capable
of capturing homotopy, cohesion, and supergeometry:
</p><p>1) Homotopy Type System: foundation for homotopical properties of spaces;
2) Super Smooth Type Theory: the graded universes/tensors, group actions, and super-modality operators for geometric cohesion and differential structure such as bosonic/fermionic, flat/sharp modalities;
3) Differential K-Theoretical Type Theory: homotopical foundations via spectra and groupoids, supporting differential K-theory with forms, connections, and refined cohomology structures;
4) Quantum Linear Type Theory: configuration spaces and braids for quantum statistics, supporting quantum systems and linear types for resource-sensitive quantum programming.
</p><p>Each layer builds on the previous, culminating in a system tailored
to formalize superpoints (<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="4.63ex" height="2.02ex" role="img" focusable="false" viewBox="0 -893 2046.7 893" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1109-TEX-D-211D" d="M17 665Q17 672 28 683H221Q415 681 439 677Q461 673 481 667T516 654T544 639T566 623T584 607T597 592T607 578T614 565T618 554L621 548Q626 530 626 497Q626 447 613 419Q578 348 473 326L455 321Q462 310 473 292T517 226T578 141T637 72T686 35Q705 30 705 16Q705 7 693 -1H510Q503 6 404 159L306 310H268V183Q270 67 271 59Q274 42 291 38Q295 37 319 35Q344 35 353 28Q362 17 353 3L346 -1H28Q16 5 16 16Q16 35 55 35Q96 38 101 52Q106 60 106 341T101 632Q95 645 55 648Q17 648 17 665ZM241 35Q238 42 237 45T235 78T233 163T233 337V621L237 635L244 648H133Q136 641 137 638T139 603T141 517T141 341Q141 131 140 89T134 37Q133 36 133 35H241ZM457 496Q457 540 449 570T425 615T400 634T377 643Q374 643 339 648Q300 648 281 635Q271 628 270 610T268 481V346H284Q327 346 375 352Q421 364 439 392T457 496ZM492 537T492 496T488 427T478 389T469 371T464 361Q464 360 465 360Q469 360 497 370Q593 400 593 495Q593 592 477 630L457 637L461 626Q474 611 488 561Q492 537 492 496ZM464 243Q411 317 410 317Q404 317 401 315Q384 315 370 312H346L526 35H619L606 50Q553 109 464 243Z"></path><path id="MJX-1109-TEX-I-1D45A" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1109-TEX-N-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path id="MJX-1109-TEX-I-1D45B" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="211D" xlink:href="#MJX-1109-TEX-D-211D"></use></g></g><g data-mml-node="TeXAtom" transform="translate(755,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D45A" xlink:href="#MJX-1109-TEX-I-1D45A"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(878,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-1109-TEX-N-7C"></use></g></g><g data-mml-node="mi" transform="translate(1156,0)"><use data-c="1D45B" xlink:href="#MJX-1109-TEX-I-1D45B"></use></g></g></g></g></g></svg></mjx-container>), supersymmetry, and equivariant structures,
as exemplified in Schreiber’s "Equivariant Super Homotopy Theory" (2019).
</p></aside><section><div class="macro"><div class="macro__col"><h3 id="hts"><b>HOMOTOPY</b></h3><ol><li><a href="https://urs.groupoid.space/foundations/universe/">UNIVERSE</sub></a></li><li><a href="https://urs.groupoid.space/foundations/forall/">FORALL</a></li><li><a href="https://urs.groupoid.space/foundations/sigma/">SIGMA</a></li><li><a href="https://urs.groupoid.space/foundations/exists/">EXISTS</a></li><li><a href="https://urs.groupoid.space/foundations/real/">REAL</a></li><li><a href="https://urs.groupoid.space/foundations/path/">PATH</a></li><li><a href="https://urs.groupoid.space/foundations/glue/">GLUE</a></li></ol></div><div class="macro__col"><h3 id="mltt"><b>SUP-SMTH</b></h3><ol><li><a href="https://urs.groupoid.space/foundations/graded/">GRADED</a></li><li><a href="https://urs.groupoid.space/foundations/flat/">FLAT</a></li><li><a href="https://urs.groupoid.space/foundations/sharp/">SHARP</a></li><li><a href="https://urs.groupoid.space/foundations/bose/">BOSE</a></li><li><a href="https://urs.groupoid.space/foundations/fermi/">FERMI</a></li><li><a href="https://urs.groupoid.space/foundations/tensor/">TENSOR</a></li><li><a href="https://urs.groupoid.space/foundations/smth/">SMTH</a></li><li><a href="https://urs.groupoid.space/foundations/supsmth/">SUPSMTH</a></li></ol></div><div class="macro__col"><h3 id="mltt"><b>DIFF-K</b></h3><ol><li><a href="https://urs.groupoid.space/foundations/spectra/">SPECTRA</a></li><li><a href="https://urs.groupoid.space/foundations/grpd/">GRPD</a></li><li><a href="https://urs.groupoid.space/foundations/form/">FORM</a></li><li><a href="https://urs.groupoid.space/foundations/ku/">KU</a></li><li><a href="https://urs.groupoid.space/foundations/diff/">DIFF</a></li><li><a href="https://urs.groupoid.space/foundations/diffku/">DIFFKU</a></li></ol></div><div class="macro__col"><h3 id="mltt"><b>QUANTUM</b></h3><ol><li><a href="https://urs.groupoid.space/foundations/conf/">CONF<sup>n</sup></a></li><li><a href="https://urs.groupoid.space/foundations/braid/">BRAID</a></li><li><a href="https://urs.groupoid.space/foundations/qubit/">QUBIT</a></li><li><a href="https://urs.groupoid.space/foundations/linear/">LINEAR</a></li></ol></div></div><br><p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.014ex;" xmlns="http://www.w3.org/2000/svg" width="12.923ex" height="1.586ex" role="img" focusable="false" viewBox="0 -695 5712 701" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1110-TEX-B-1D408" d="M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z"></path><path id="MJX-1110-TEX-B-1D427" d="M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 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</p><p>Urs operates as an OCaml framework, manipulating pure ASTs, similar to a Lisp-like
environment, pending a dedicated syntax.
</p><code>$ git clone [email protected]<span class="h__symbol">:</span>groupoid/urs
$ ocamlfind ocamlc -o urs -package z3 -linkpkg src/urs.ml
</code></section><section><h1>SYNTAX</h1><p style="text-align:center;"><mjx-container class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -29.681ex;" xmlns="http://www.w3.org/2000/svg" width="39.559ex" height="60.493ex" role="img" focusable="false" viewBox="0 -13618.9 17485.2 26737.8" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1111-TEX-N-55" d="M128 622Q121 629 117 631T101 634T58 637H25V683H36Q57 680 180 680Q315 680 324 683H335V637H302Q262 636 251 634T233 622L232 418V291Q232 189 240 145T280 67Q325 24 389 24Q454 24 506 64T571 183Q575 206 575 410V598Q569 608 565 613T541 627T489 637H472V683H481Q496 680 598 680T715 683H724V637H707Q634 633 622 598L621 399Q620 194 617 180Q617 179 615 171Q595 83 531 31T389 -22Q304 -22 226 33T130 192Q129 201 128 412V622Z"></path><path id="MJX-1111-TEX-N-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 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</p><h1>SEMANTICS</h1><br><h2>Homotopy</h2><p>A foundational framework where types are ∞-groupoids,
and propositions emerge as homotopical structures.
Equality is a space of paths, extensible to higher
dimensions, weaving homotopy theory into type-theoretic reasoning.
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it blends topological invariance with constructive logic.
This system enables synthetic reasoning about continuity,
deformation, and equivalence, treating mathematical spaces
as dynamic objects rich with paths, homotopies, and higher
coherences—a language for capturing the shape of mathematical truth.
</p><h2>Graded Cohesion</h2><p>A cohesive, graded extension of type theory infused with supergeometry,
where types inhabit universes indexed by parity, symmetry, and smoothness.
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it equips types with tensor structures, group actions, and super-modality
operators to model differential cohesion, infinitesimal geometry,
and graded symmetries. This system unifies the logic of supermanifolds,
derived geometry, and higher smooth stacks, enabling type-theoretic
reasoning about spaces enriched by quantum and differential symmetry.
</p><h2>Differential K-Theory</h2><p>A homotopy-theoretic framework augmented with differential structure,
where types encode spectra, groupoids, and refined cohomological data.
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differential K-theory to provide a synthetic language for characteristic
classes, index theory, and topological invariants. By integrating stable
homotopy, smooth geometry, and categorical semantics, it articulates the
interplay of curvature, bundles, and higher cohomological operations
within a unified type system of paths and forms.</p><h2>Qunatum Linearity</h2><p>A type-theoretic foundation for quantum systems, internalizing configuration spaces,
braids, and higher categorical symmetries within a resource-sensitive linear logic.
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braided
monoidal categories, quantum group actions, and path spaces of indistinguishable particles.
This system supports quantum computation and field theory semantics, using
linear types to model entanglement, interference, and non-classical information flow,
ensuring quantum processes respect spatial and algebraic phase coherence.
</p></section><section><h1>Bibliography</h1><br></section><section><h2>TED-K</h2><p></p><p style="text-align:center;"><br>
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