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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Fibrant object</span></span>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, specifically in <a href="Homotopy_theory" title="Homotopy theory">homotopy theory</a> in the context of a <a href="Model_category" title="Model category">model category</a> <i>M</i>, a <b>fibrant object</b> <i>A</i> of <i>M</i> is an <a href="Object_(category_theory)" class="mw-redirect" title="Object (category theory)">object</a> that has a <a href="Fibration" title="Fibration">fibration</a> to the <a href="Terminal_object" class="mw-redirect" title="Terminal object">terminal object</a> of the <a href="Category_(mathematics)" title="Category (mathematics)">category</a>.
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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<p>The fibrant objects of a <a href="Closed_model_category" class="mw-redirect" title="Closed model category">closed model category</a> are characterized by having a <a href="Right_lifting_property" class="mw-redirect" title="Right lifting property">right lifting property</a> with respect to any <a href="Cofibration" title="Cofibration">trivial cofibration</a> in the category. This property makes fibrant objects the "correct" objects on which to define <a href="Homotopy_group" title="Homotopy group">homotopy groups</a>. In the context of the theory of <a href="Simplicial_set" title="Simplicial set">simplicial sets</a>, the fibrant objects are known as <b>Kan complexes</b> after <a href="Daniel_Kan" title="Daniel Kan">Daniel Kan</a>. They are the <a href="Kan_fibration" title="Kan fibration">Kan fibrations</a> over a point.
</p><p>Dually is the notion of cofibrant object, defined to be an object <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">
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<mi>c</mi>
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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
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</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> such that the unique morphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing \to c}">
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<annotation encoding="application/x-tex">{\displaystyle \varnothing \to c}</annotation>
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</math></span><img src="./8a51621d64371d4b0397e76d244f9178268e026e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.429ex; height:2.009ex;" alt="{\displaystyle \varnothing \to c}" loading="lazy"></span> from the initial object to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">
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<mi>c</mi>
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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
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</math></span><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> is a cofibration.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>P.G. Goerss and J.F. Jardine, <i>Simplicial Homotopy Theory</i>, Progress in Math., Vol. 174, Birkhauser, Boston-Basel-Berlin, 1999. <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7643-6064-X</bdi>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><a rel="nofollow" class="external free" href="https://ncatlab.org/nlab/show/fibrant+object">https://ncatlab.org/nlab/show/fibrant+object</a></li></ul>
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