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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Standard library of theories · Catlab.jl</title><meta name="title" content="Standard library of theories · Catlab.jl"/><meta property="og:title" content="Standard library of theories · Catlab.jl"/><meta property="twitter:title" content="Standard library of theories · Catlab.jl"/><meta name="description" content="Documentation for Catlab.jl."/><meta property="og:description" content="Documentation for Catlab.jl."/><meta property="twitter:description" content="Documentation for Catlab.jl."/><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link 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class="collapsed"><li><a class="tocitem" href="../../generated/sketches/partitions/">Partitions</a></li><li><a class="tocitem" href="../../generated/sketches/preorders/">Preorders</a></li><li><a class="tocitem" href="../../generated/sketches/meets/">Meets in Preorders</a></li><li><a class="tocitem" href="../../generated/sketches/smc/">Symmetric Monoidal Categories</a></li><li><a class="tocitem" href="../../generated/sketches/cat_elements/">The Category of Elements</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-2" type="checkbox"/><label class="tocitem" for="menuitem-2-2"><span class="docs-label">Graphs</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../../generated/graphs/graphs/">The Category of Graphs</a></li><li><a class="tocitem" href="../../generated/graphs/graphs_label/">Labeled Graphs</a></li><li><a class="tocitem" href="../../generated/graphs/subgraphs/">Algebra of subgraphs</a></li></ul></li><li><span 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href="../../generated/graphics/graphviz_wiring_diagrams/">Drawing wiring diagrams in Graphviz</a></li><li><a class="tocitem" href="../../generated/graphics/composejl_wiring_diagrams/">Drawing wiring diagrams in Compose.jl</a></li><li><a class="tocitem" href="../../generated/graphics/tikz_wiring_diagrams/">Drawing wiring diagrams in TikZ</a></li><li><a class="tocitem" href="../../generated/graphics/layouts_vs_drawings/">Layouts versus drawings of wiring diagrams</a></li></ul></li></ul></li><li><span class="tocitem">Modules</span><ul><li class="is-active"><a class="tocitem" href>Standard library of theories</a></li><li><a class="tocitem" href="../categorical_algebra/">Categorical algebra</a></li><li><a class="tocitem" href="../graphs/">Graphs</a></li><li><a class="tocitem" href="../wiring_diagrams/">Wiring diagrams</a></li><li><a class="tocitem" href="../graphics/">Graphics</a></li><li><a class="tocitem" href="../programs/">Programs</a></li></ul></li><li><span class="tocitem">Developer Docs</span><ul><li><a class="tocitem" href="../../devdocs/style/">Style Guide for AlgebraicJulia</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Modules</a></li><li class="is-active"><a href>Standard library of theories</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Standard library of theories</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/AlgebraicJulia/Catlab.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/main/docs/src/apis/theories.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Standard-library-of-theories"><a class="docs-heading-anchor" href="#Standard-library-of-theories">Standard library of theories</a><a id="Standard-library-of-theories-1"></a><a class="docs-heading-anchor-permalink" href="#Standard-library-of-theories" title="Permalink"></a></h1><p>Through the module <code>Catlab.Theories</code>, Catlab provides a standard library of <a href="https://algebraicjulia.github.io/GATlab.jl">generalized algebraic theories</a> for categories, monoidal categories, and other categorical structures. The theories correspond, in most cases, to standard definitions in category theory and they are used throughout Catlab and the AlgebraicJulia ecosystem to structure programs and provide a common interface for applied category theory. The module also provides default syntax systems for many of the theories.</p><p>Categorical structures for which theories are provided include:</p><ul><li>categories</li><li>monoidal and symmetric monoidal categories</li><li>cartesian and cocartesian categories</li><li>semiadditive categories/biproduct categories</li><li>hypergraph categories</li><li>bicategories of relations</li><li>categories with two monoidal products, such as distributive monoidal categories</li></ul><p>The contents of this module can be supplemented by the user, and it is even possible to use many parts of Catlab without using this module. The user is free to create new syntax systems for the theories defined here and also to define entirely new theories.</p><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Catlab.Theories" href="#Catlab.Theories"><code>Catlab.Theories</code></a> — <span class="docstring-category">Module</span></header><section><div><p>Catlab&#39;s standard library of generalized algebraic theories.</p><p>The focus is on categories and monoidal categories, but other related structures are also included.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Theories.jl#L1-L6">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Catlab.Theories.CategoryExpr" href="#Catlab.Theories.CategoryExpr"><code>Catlab.Theories.CategoryExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Base type for GAT expressions in categories or other categorical structures.</p><p>All symbolic expression types exported by <code>Catlab.Theories</code> are subtypes of this abstract type.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Theories.jl#L18-L23">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Catlab.Theories.HomExpr" href="#Catlab.Theories.HomExpr"><code>Catlab.Theories.HomExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Base type for morphism expressions in categorical structures.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Theories.jl#L29-L31">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Catlab.Theories.ObExpr" href="#Catlab.Theories.ObExpr"><code>Catlab.Theories.ObExpr</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Base type for object expressions in categorical structures.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Theories.jl#L25-L27">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.collect-Tuple{ObExpr}" href="#Base.collect-Tuple{ObExpr}"><code>Base.collect</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Collect generators of object in monoidal category as a vector.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Monoidal.jl#L60-L62">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Base.ndims-Tuple{ObExpr}" href="#Base.ndims-Tuple{ObExpr}"><code>Base.ndims</code></a> — <span class="docstring-category">Method</span></header><section><div><p>Number of &quot;dimensions&quot; of object in monoidal category.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/AlgebraicJulia/Catlab.jl/blob/8d35ef724f0f0864ecdccef173eec958329f43e5/src/theories/Monoidal.jl#L66-L68">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../generated/graphics/layouts_vs_drawings/">« Layouts versus drawings of wiring diagrams</a><a class="docs-footer-nextpage" href="../categorical_algebra/">Categorical algebra »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="auto">Automatic (OS)</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.2.1 on <span class="colophon-date" title="Monday 26 February 2024 21:53">Monday 26 February 2024</span>. 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