Open Problem Garden - A construction of direct product in the category of continuous maps between endo-funcoids - Comments https://www.openproblemgarden.org/op/a_construction_of_direct_product_in_the_category_of_continuous_maps_between_endo_funcoids Comments for "A construction of direct product in the category of continuous maps between endo-funcoids" en A construction of direct product in the category of continuous maps between endo-funcoids https://www.openproblemgarden.org/op/a_construction_of_direct_product_in_the_category_of_continuous_maps_between_endo_funcoids <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids.</p> <p>Remind from <a href="http://www.mathematics21.org/algebraic-general-topology.html">my book</a> that morphisms <img class="teximage" src="/files/tex/0490c5611c35f2acda709159d3ff9b2ac7231a73.png" alt="$ f: A\rightarrow B $" /> of this category are defined by the formula <img class="teximage" src="/files/tex/7e25329442aab16509064fae7eef5f55713d9129.png" alt="$ f\circ A\sqsubseteq B\circ f $" /> (here and below by abuse of notation I equate functions with corresponding principal funcoids).</p> <p>Let <img class="teximage" src="/files/tex/dce884f1b54c3c901f338dead96e520309c8a80b.png" alt="$ F_0, F_1 $" /> are endofuncoids,</p> <p>We define <img class="teximage" src="/files/tex/13abc7a67602834ad8b717393e386f08c6b173b9.png" alt="$ F_0\times F_1 = \bigsqcup \left\{ \Phi \in \mathsf{FCD} \,|\, \pi_0 \circ \Phi \sqsubseteq F_0 \circ \pi_0 \wedge \pi_1 \circ \Phi \sqsubseteq F \circ \pi_1 \right\} $" /></p> <p>(here <img class="teximage" src="/files/tex/6386b0ca70709cfe33f8268b6215852603f17604.png" alt="$ \pi_0 $" /> and <img class="teximage" src="/files/tex/372933243be749b190e43e454685361b874e1024.png" alt="$ \pi_1 $" /> are cartesian projections).</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; The above defines categorical direct product (in the above mentioned category, with products of morphisms the same as in <strong>Set</strong>).</div> </tr></td> </table> </td> </tr> </table> Porton, Victor categorical product direct product Topology https://www.openproblemgarden.org/op/a_construction_of_direct_product_in_the_category_of_continuous_maps_between_endo_funcoids#comment Sat, 31 Aug 2013 17:02:35 +0200 porton 56532 at https://www.openproblemgarden.org