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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Simplicial Type Theory</title>
<link rel="stylesheet" href="index.css?v=init">
</head>
<body>
<nav class="sticky">
<div class="nav-container">
<a href="https://groupoid.space" class="logo"><img width="50" style="margin-bottom:-15px;"
src="https://avatars.githubusercontent.com/u/17128096?s=200&v=4"></a>
<ul class="nav-links">
<li><a href="https://henk.groupoid.space">Henk</a></li>
<li><a href="https://frank.groupoid.space">Frank</a></li>
<li><a href="https://christine.groupoid.space">Christine</a></li>
<li><a href="https://per.groupoid.space">Per</a></li>
<li><a href="https://homotopy.dev">Anders</a></li>
<li>Dan</li>
<li><a href="https://urs.groupoid.space">Urs</a></li>
</ul>
</div>
</nav>
<header>
<div class="hero-content">
<h1>Dan</h1>
<p class="hero-subtitle"><b>General Algebra Processing</b> (Dan), <b>1-Categories</b> (Mike),
<nobr>and <b>∞-Categories</b> (Ulrik) </nobr> in a unified simplicial environment by 5HT.
</p>
</div>
</header>
<main>
<section>
<h2>Abstract</h2>
<div style="max-width:800px;margin:0 auto;text-align: center;">
<p>We present a domain-specific language (DSL) ecosystem linking the operational, computer-algebraic
core (<b>Dan</b>) with the theoretical proof assistants <b>Mike</b> (Directed Type Theory) and
<b>Ulrik</b> (Simplicial Type Theory). Designed as a fast type checker with a focus on algebraic
purity, the system employs a Lean/Anders-like sequent syntax to define k-dimensional simplices,
categories, and spectra via explicit contexts, vertex lists, and face relations.
</p>
<p>By balancing algebraic simplicity with practical efficiency, the system avoids complex path-filling
(e.g. <code>PathP</code>) at the surface, enabling linear-time combinatorial checks at the
operational layer. High-level objects are elaborated and translated directly to type theory, where
metatheorems (such as limits, adjunctions, or Yoneda mappings) are proven and verified.</p>
</div>
</section>
<section id="cosmic-cube-section">
<h2>Cosmic Cube</h2>
<p style="text-align:center;max-width:760px;margin:0 auto 1.5rem;">
A conceptual framework organizing higher category theory, homotopy theory and type theory
along three axes: <b>Strictness</b> (∇/∆), <b>Groupoidality</b> (𝟐/ℕ), and <b>Stability</b> (𝟏/ℍ).
Each vertex corresponds to a distinct type-theoretic or categorical structure.
</p>
<div class="cosmic-cube-wrapper">
<div class="cube-scene" id="cubeScene" style="margin-top:10px">
<div class="cube-3d" id="cube3d">
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<div class="cube-face face-front"></div>
<div class="cube-face face-back"></div>
<div class="cube-face face-left"></div>
<div class="cube-face face-right"></div>
<div class="cube-face face-top"></div>
<div class="cube-face face-bottom"></div>
<!-- 8 vertices with labels -->
<!-- Vertex positions: (front/back) × (top/bottom) × (left/right) at ±90px -->
<!-- Front-Bottom-Left: (∆, 𝟐, 𝟏) = MLTT -->
<div class="cube-vertex" style="transform: translate3d(-90px, 90px, 90px);">
<div class="vertex-label">(∆, 𝟐, 𝟏)</div>
</div>
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<div class="cube-vertex" style="transform: translate3d(90px, 90px, 90px);">
<div class="vertex-label">(∆, 𝟐, ℍ)</div>
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<div class="vertex-label">(∆, ℕ, 𝟏)</div>
</div>
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<div class="cube-vertex" style="transform: translate3d(90px, -90px, 90px);">
<div class="vertex-label">(∆, ℕ, ℍ)</div>
</div>
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<div class="cube-vertex" style="transform: translate3d(-90px, 90px, -90px);">
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<div class="cube-vertex" style="transform: translate3d(90px, 90px, -90px);">
<div class="vertex-label">(∇, 𝟐, ℍ)</div>
</div>
<!-- Back-Top-Left: (∇, ℕ, 𝟏) = Modal HoTT -->
<div class="cube-vertex" style="transform: translate3d(-90px, -90px, -90px);">
<div class="vertex-label">(∇, ℕ, 𝟏)</div>
</div>
<!-- Back-Top-Right: (∇, ℕ, ℍ) = Modal Linear HoTT -->
<div class="cube-vertex" style="transform: translate3d(90px, -90px, -90px);">
<div class="vertex-label">(∇, ℕ, ℍ)</div>
</div>
</div>
</div>
<br>
<!-- Axis Legend -->
<div class="cube-axes-legend">
<div class="axis-item">
<span class="axis-symbol axis-x">∆ ↔ ∇</span>
<span class="axis-desc">Strictness</span>
</div>
<div class="axis-item">
<span class="axis-symbol axis-y">𝟐 ↔ ℕ</span>
<span class="axis-desc">Groupoidality</span>
</div>
<div class="axis-item">
<span class="axis-symbol axis-z">𝟏 ↔ ℍ</span>
<span class="axis-desc">Stability</span>
</div>
</div>
</div>
<!-- Vertex table -->
<div class="cube-vertex-table">
<table>
<thead>
<tr>
<th>Configuration</th>
<th>Structure</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr><td class="vtx-config">(∆, 𝟐, 𝟏)</td><td class="vtx-name">ПΣ (MLTT-75)</td><td>Dependently typed λ-calculus — Per</td></tr>
<tr><td class="vtx-config">(∆, 𝟐, ℍ)</td><td class="vtx-name">Linear ПΣ</td><td>λ-calculus, resource-sensitive computation — Urs</td></tr>
<tr><td class="vtx-config">(∆, ℕ, 𝟏)</td><td class="vtx-name">HoTT</td><td>Homotopy Type Theory — Anders</td></tr>
<tr><td class="vtx-config">(∆, ℕ, ℍ)</td><td class="vtx-name">Linear HoTT</td><td>Linear Homotopy Type Theory — Urs</td></tr>
<tr><td class="vtx-config">(∇, 𝟐, 𝟏)</td><td class="vtx-name">Modal MLTT</td><td>Modal Dependently Typed λ-calculus – Anders</td></tr>
<tr><td class="vtx-config">(∇, 𝟐, ℍ)</td><td class="vtx-name">∞-toposes</td><td>∞-toposes, Quantum Field Theory (dirtt, stt) — Dan</td></tr>
<tr><td class="vtx-config">(∇, ℕ, 𝟏)</td><td class="vtx-name">Modal HoTT</td><td>Synthetic Differential Geometry — Anders</td></tr>
<tr><td class="vtx-config">(∇, ℕ, ℍ)</td><td class="vtx-name">Modal Linear HoTT</td><td>Modal Linear Homotopy Type Theory — Urs</td></tr>
</tbody>
</table>
</div>
</section>
<style>
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<section>
<h2>The Three Languages</h2>
<p style="text-align: center; margin-bottom: 2rem;">A tiered approach to simplicial category theory,
bridging active computation with formal proof verification.</p>
<div class="product-grid">
<div class="product-card">
<h3>Dan</h3>
<p style="font-weight: 600; color: var(--accent); margin-top:-0.5rem; margin-bottom: 1rem;">General
Algebra Processing (GAP)</p>
<p><b>Dan</b> is a statically checked, presentation-oriented language. It allows programmers and
mathematicians to define combinatorial cells, groups, rings, and simplicial complexes, and
checks their structural soundness.</p>
<p>Dan maps directly into theoretical engines, proving metatheorems that hold "for free" across all
models.</p>
</div>
<div class="product-card">
<h3>Mike</h3>
<p style="font-weight: 600; color: var(--accent); margin-top:-0.5rem; margin-bottom: 1rem;">Directed
Type Theory (dirtt)</p>
<p><b>Mike</b> is a proof assistant based on Directed Type Theory (dirtt) with linear polarities
(+/-) and quadraticality constraints. It is tailored for formal 1-category theory.</p>
<p>It represents hom-spaces directly, handling limits/colimits as ends and coends, and maps linear
constructs such as tensors ⊗ and linear functions ⊸ to the simplicial core.</p>
</div>
<div class="product-card">
<h3>Ulrik</h3>
<p style="font-weight: 600; color: var(--accent); margin-top:-0.5rem; margin-bottom: 1rem;">
Simplicial Type Theory (STT)</p>
<p><b>Ulrik</b> is a proof assistant based on Riehl-Shulman simplicial type theory (STT). It
features a directed interval 𝕀, Segal precategories, and Rezk categories where identity paths
and isomorphisms coincide.</p>
<p>It implements modal operators (opposite category, twisted arrow categories) and forms the core
checker for synthetic ∞-category theory in compact modal manner. Eventually we will type check "Higher Topos Theory".</p>
</div>
</div>
</section>
<section>
<h2>The Lift: Semantical Transport</h2>
<div style="max-width:800px;margin:0 auto;text-align: center;">
<p>The <code>lift</code> compiler operates as a semantical translation bridge, mapping discrete
<b>operational presentations</b> in <code>Dan</code> to <b>synthetic category theory</b> in
<code>Ulrik</code> and
<code>Mike</code>. Similar to how <code>coq-of-ocaml</code> translates program semantics into proof
assistant definitions,
<code>lift</code> lifts presentation-based algebraic systems into categorical objects and sheaves,
establishing a
direct link between discrete computer algebra and formal simplicial type theory.
</p>
</div>
<div class="lift-pipeline">
<div class="lift-step">
<h4>1. Signature Ingestion</h4>
<p style="font-size:0.9rem;">Captures algebraic generator sets, boundary face maps (∂ᵢ), and
presentation equations from the operational domain.</p>
</div>
<div class="lift-arrow">➔</div>
<div class="lift-step">
<h4>2. Semantical Lifting</h4>
<p style="font-size:0.9rem;">Translates signatures into category-theoretic structures (functors,
categories, sheaves) via semantic mapping.</p>
</div>
<div class="lift-arrow">➔</div>
<div class="lift-step">
<h4>3. Elaboration to STT</h4>
<p style="font-size:0.9rem;">Maps composition equations to identity paths (<code>EId</code>) and
represents domains via the directed interval 𝕀.</p>
</div>
<div class="lift-arrow">➔</div>
<div class="lift-step">
<h4>4. Metatheory Absorption</h4>
<p style="font-size:0.9rem;">Absorbs abstract theorems (Yoneda, limits, adjunctions) proven in the
theoretical library for the lifted models.</p>
</div>
</div>
<div style="max-width:900px;margin:0 auto; margin-top:3rem;">
<h3 style="text-align: center; margin-bottom: 1.5rem;">Case Study: Möbius Strip Representation</h3>
<p style="text-align: center; margin-bottom: 2rem;">Comparison between local combinatorial complexes
(Dan) and global synthetic fiber bundles in HoTT (Anders).</p>
<div class="premium-table-wrapper">
<table class="premium-table">
<thead>
<tr>
<th width="200px;">Attribute</th>
<th>Simplicial HoTT / Dan Kan <code>Möbius</code></th>
<th>HoTT / Cubical <code>Möbius</code></th>
</tr>
</thead>
<tbody>
<tr>
<td><b>Framework</b></td>
<td>General Computer Algebra (GAP-like) / Simplicial Sets</td>
<td>Homotopy Type Theory (HoTT/Cubical)</td>
</tr>
<tr>
<td><b>Mathematical Type</b></td>
<td>A product of Simplex variables and boundary relations</td>
<td>A function/fibration: <code>S¹ → U</code></td>
</tr>
<tr>
<td><b>Homotopy Type</b></td>
<td>Contractible (∗) since it models a single local triangle</td>
<td>Homotopy equivalent to a circle (S¹)</td>
</tr>
<tr>
<td><b>Mechanism of Twist</b></td>
<td>Specified algebraically via boundary face compositions in coordinates</td>
<td>Specified via univalent path induction (<code>ua</code>) using a boolean negation
equivalence</td>
</tr>
<tr>
<td><b>Conceptual Role</b></td>
<td>A local 2-cell configuration (oriented triangular building block)</td>
<td>The global non-trivial fiber bundle (the double cover total space)</td>
</tr>
</tbody>
</table>
</div>
<p style="font-size: 0.95rem; text-align: center; color: var(--text-secondary); margin-top: 1rem;">
<b>Connection:</b> glues multiple local <code>Möbius</code> triangles with a twist to form a
simplicial set X. Its geometric realization |X| is homeomorphic to the total space of the HoTT
fibration ∑_{x : S¹} moebius(x).
</p>
</div>
</section>
<section>
<h2>Algebraic Presentations in Dan</h2>
<p style="text-align: center; margin-bottom: 2rem;">Sequent definitions declared in the computer algebra
core are checked in linear time.</p>
<div class="grid-2">
<div class="package-card">
<h3>Group presentation: ℤ/3ℤ</h3>
<hr>
<p>Presented by a single generator subject to relation a³ = e. Identity and generators are declared
in the context.</p>
<pre class="code-snippet" style="font-size: 0.85rem;">
def z3 : Group
:= П (e a : Simplex),
a³ = e
⊢ 1 (a | a³ = e)</pre>
</div>
<div class="package-card">
<h3>Monoid presentation: ℕ</h3>
<hr>
<p>Natural numbers defined algebraically as a monoid. Verification checks composition and boundary
coherence.</p>
<pre class="code-snippet" style="font-size: 0.85rem;">
def nat_monoid : Monoid
:= П (z s : Simplex),
s ∘ z = s, z ∘ s = s
⊢ 2 (z s | s ∘ z = s, z ∘ s = s)</pre>
</div>
<div class="package-card">
<h3>Simplicial Circle S¹</h3>
<hr>
<p>Defined by a vertex and edge mapping endpoints to the basepoint. The degeneracy constraint
collapses v to an edge.</p>
<pre class="code-snippet" style="font-size: 0.85rem;">
def circle : Simplicial
:= П (v e : Simplex),
∂₁₀ = v, ∂₁₁ = v, s₀ < v
⊢ 1 (v, e | ∂₁₀ ∂₁₁, s₀)</pre>
</div>
<div class="package-card">
<h3>Category with Group Symmetries</h3>
<hr>
<p>Nested algebraic structures in the context allow categories to inherit properties from internal
groups.</p>
<pre class="code-snippet" style="font-size: 0.85rem;">
def path_z2_category : Category
:= П (x y : Simplex),
(f g h : Simplex),
(z2 : Group(П (e a : Simplex),
a² = e ⊢ 1 (a | a² = e))),
f ∘ g = h
⊢ 2 (x y | f g h | f ∘ g = h)</pre>
</div>
</div>
</section>
<section>
<h2>The Middle Way</h2>
<p style="text-align: center; margin-bottom: 2rem;">How constructors and types map between the operational
and theoretical layers of the ecosystem.</p>
<div class="premium-table-wrapper">
<table class="premium-table">
<thead>
<tr>
<th>Language</th>
<th>Construct</th>
<th>AST Node</th>
<th>Classification</th>
<th>Target Mapping</th>
</tr>
</thead>
<tbody>
<tr>
<td><b>Dan (GAP)</b></td>
<td>Type Former</td>
<td><code>Category</code></td>
<td>Elaborated</td>
<td>Segal Precategory / Type with composition path</td>
</tr>
<tr>
<td></td>
<td>Type Former</td>
<td><code>Group</code></td>
<td>Elaborated</td>
<td>Classifying Space BG (single-object Segal type)</td>
</tr>
<tr>
<td></td>
<td>Constraint</td>
<td><code>Eq</code> / <code>Map</code></td>
<td>Elaborated</td>
<td>Strict equality <code>EId</code> or interval ordering <code>ELeq</code></td>
</tr>
<tr>
<td><b>Ulrik (STT)</b></td>
<td>Type Former</td>
<td><code>EPi</code> / <code>ESig</code></td>
<td>Primitive</td>
<td>Dependent function space Π and pair space Σ</td>
</tr>
<tr>
<td></td>
<td>Type Former</td>
<td><code>EIDir</code> / <code>ELeq</code></td>
<td>Primitive</td>
<td>Interval coordinate space 𝕀 and inequality relation</td>
</tr>
<tr>
<td></td>
<td>Type Former</td>
<td><code>hom(x, y)</code></td>
<td>Derived</td>
<td>Extension type {f : 𝕀 → A | f0 = x, f1 = y}</td>
</tr>
<tr>
<td><b>Mike (dirtt)</b></td>
<td>Type Former</td>
<td><code>MHom(cat, a, b)</code></td>
<td>Elaborated</td>
<td>Simplicial <code>hom</code> type</td>
</tr>
<tr>
<td></td>
<td>Type Former</td>
<td><code>MTensor(M, N)</code></td>
<td>Elaborated</td>
<td>Simplicial dependent pair space <code>ESig</code> (Σ)</td>
</tr>
<tr>
<td></td>
<td>Type Former</td>
<td><code>MEnd</code> / <code>MCoend</code></td>
<td>Elaborated</td>
<td>Simplicial Π (for Limits) and Σ (for Colimits)</td>
</tr>
</tbody>
</table>
</div>
</section>
<section class="section-standards">
<h2>Open Source</h2>
<center>
<a href="https://github.com/groupoid/ulrik">
<img style="width:150px;" src="https://n2o.dev/img/github.svg" alt="GitHub Logo">
</a>
<br>
<br>
<br>
<br>
<img src="https://homotopy.dev/img/1F1E9-1F1F0.svg" height=50 style="margin:-00px;padding:0px;">
<img src="https://homotopy.dev/img/1F1FA-1F1E6.svg" height=50 style="margin:-00px;padding:0px;">
</center>
</section>
</main>
<footer>
<div class="footer-columns">
<div>
<h4>Simplicial HoTT Prover</h4>
<p>Linear operational algebra core (<b>Dan</b>) and theoretical proof checkers (<b>Mike</b>,
<b>Ulrik</b>) developed in OCaml by <a rel="me" href="https://mathstodon.xyz/@5ht">5HT</a>.
</p>
</div>
<div>
<h4>Bibliography</h4>
<div class="grid-2" style="font-size: 0.9rem;">
<div>
<h5>Operational Core (Dan)</h5>
<ul>
<li><a href="https://github.com/groupoid/dan">Dan Kan: Simplicial HoTT Computer Algebra
System. [Groupoid Infinity]</a></li>
<li>Daniel Kan. Abstract Homotopy I & II. (1955-1956)</li>
<li>Daniel Kan. On c.s.s. Complexes. (1957)</li>
<li>Daniel Kan. A Combinatorial Definition of Homotopy Groups. (1958)</li>
<li>Graeme Segal. Classifying spaces and spectral sequences. (1968)</li>
<li>Charles Rezk. A model for the homotopy theory of homotopy theory. (2001)</li>
</ul>
</div>
<div>
<h5>Theoretical Backends (Mike, Ulrik)</h5>
<ul>
<li>Emily Riehl, Mike Shulman. <a href="https://arxiv.org/abs/1705.07442">A type theory for
synthetic ∞-categories. (2017)</a></li>
<li>Dan Licata, Andreas Nuyts, Patrick Schultz, Mike Shulman. <a
href="https://dan.groupoid.space/doc/dirtt.pdf">A Directed Type Theory for Formal
Category Theory. (2020)</a></li>
<li>Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz. <a
href="https://arxiv.org/pdf/2501.13229">Yoneda embedding in simplicial type theory.
(2026)</a></li>
<li><a href="https://pure.tue.nl/ws/portalfiles/portal/2039924/256169.pdf">AUTOMATH, a
language for mathematics. [N.G. de Bruijn]</a></li>
<li>Per Martin-Löf. <a href="https://5ht.co/mltt-80.pdf">Intuitionistic Type Theory</a>.
(1980)</li>
<li>Данiєль Гратцер, Джонатан Вайнбергер та Ульрiк Бухгольц. <a
href="https://groupoid.space/friends/ulrik/yoneda-simplicialtt-ua.pdf">Вкладення
Йонеди в сiмплiцiальнiй теорiї типiв</a>.</li>
<li>Ульрiк Бухгольц. <a
href="https://groupoid.space/friends/ulrik/higher-structures-ua.pdf">Вищi структури
в гомотопiчнiй теорiї типiв</a>.</li>
<li>Ульрiк Бухгольц, Флорiс ван Доорн та Егберт Райке. <a
href="https://groupoid.space/friends/ulrik/higher-groups-ua.pdf">Вищi групи в
гомотопiчнiй теорiї типiв</a>.</li>
</ul>
</div>
</div>
</div>
</div>
</footer>
</body>
</html>