16 lines
1 KiB
HTML
16 lines
1 KiB
HTML
<h2>Functor</h2>
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<p> A functor is a mapping between two categories.
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Let \(\C\) and \(\D\) be two categories.
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A <em>functor</em> <span class="yellow">\(\F\)</span> from <span class="blue">\(\C\)</span> to <span class="green">\(\D\)</span>:</p>
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<ul>
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<li> Associate objects: <span class="backblue">\(A\in\ob{\C}\)</span> to <span class="backgreen">\(\F(A)\in\ob{\D}\)</span> </li>
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<li> Associate morphisms: <span class="backblue">\(f:A\to B\)</span> to <span class="backgreen">\(\F(f) : \F(A) \to \F(B)\)</span>
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such that
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<ul>
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<li>\( \F (\)<span class="backblue blue">\(\id_X\)</span>\()= \)<span class="backgreen"><span class="green">\(\id\)</span>\(\vphantom{\id}_{\F(}\)<span class="blue">\(\vphantom{\id}_X\)</span>\(\vphantom{\id}_{)} \)</span>,</li>
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<li>\( \F (\)<span class="backblue blue">\(g∘f\)</span>\()= \)<span class="backgreen">\( \F(\)<span class="blue">\(g\)</span>\() \)<span class="green">\(\circ\)</span>\( \F(\)<span class="blue">\(f\)</span>\() \)</span></li>
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</ul>
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</li>
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</ul>
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