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<title>Strict initial object</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Strict initial object</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In the mathematical discipline of <a href="Category_theory" title="Category theory">category theory</a>, a <b>strict initial object</b> is an <a href="Initial_object" class="mw-redirect" title="Initial object">initial object</a> 0 of a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> <i>C</i> with the property that every <a href="Morphism" title="Morphism">morphism</a> in <i>C</i> with <a href="Codomain" title="Codomain">codomain</a> 0 is an <a href="Isomorphism_(category_theory)" class="mw-redirect" title="Isomorphism (category theory)">isomorphism</a>. In a <a href="Cartesian_closed_category" title="Cartesian closed category">Cartesian closed category</a>, every initial object is strict.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Also, if <i>C</i> is a <a href="Distributive_category" title="Distributive category">distributive</a> or <a href="Extensive_category" title="Extensive category">extensive category</a>, then the initial object 0 of <i>C</i> is strict.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFMcLarty1992" class="citation book cs1">McLarty, Colin (4 June 1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V8cON1x39bIC"><i>Elementary Categories, Elementary Toposes</i></a>. <a href="Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0191589497</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">13 February</span> 2017</span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFCarboniLackWalters1993" class="citation journal cs1">Carboni, Aurelio; Lack, Stephen; Walters, R.F.C. (3 February 1993). "Introduction to extensive and distributive categories". <i><a href="Journal_of_Pure_and_Applied_Algebra" title="Journal of Pure and Applied Algebra">Journal of Pure and Applied Algebra</a></i>. <b>84</b> (2): <span class="nowrap">145–</span>158. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-4049%2893%2990035-R">10.1016/0022-4049(93)90035-R</a>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/strict+initial+object">Strict initial object</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li></ul>
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This article is issued from <a class="external text" title="Last edited on 2023-12-02" href="https://en.wikipedia.org/wiki/?title=Strict_initial_object&oldid=1188019515">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
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