<div dir="ltr"><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px">As part of the series of seminars <i>&quot;Insalate di Matematica&quot;</i>, Francesca Guffanti, PhD student at the University of Milan, will deliver an online talk. Here the details:</div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><u>Date and time</u>: <b>6th May </b><b>2021, 11:00 am </b><i>(be careful with the change of starting time)</i></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><u>Title</u>: <span style="background-color:rgb(255,217,102)"><i>&quot;An introduction to Lawvere&#39;s doctrines</i></span><i><span style="background-color:rgb(255,217,102)">&quot;</span><br><span style="color:rgb(34,34,34);background-color:rgb(241,194,50)"></span></i></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto"><u style="color:rgb(49,49,49);word-spacing:1px">Abstract</u><font color="#313131"><span style="word-spacing:1px">: </span></font><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">Doctrines</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">were</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">introduced</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> by </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">Lawvere</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> in 1969 </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">as</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">mathematical</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">structures</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">capable</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">of</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">synthesizing</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">structural</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">properties</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">of</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> a </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">logical</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> system, </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">giving</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> a </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">possible</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">way</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">to</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">study</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">logic</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">within</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">category</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">theory</span><span style="font-family:arial,sans-serif;color:rgb(0,0,0)">. </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">A </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">doctrine</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">is</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> a </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">presheaf</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> on a small </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">category</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> (</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">with</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> finite </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">products</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">) </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">taking</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">values</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> in </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">partially</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">ordered</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">sets</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">. </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">To</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">any</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">given</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">first</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">order</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">theory</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">one</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">can</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> associate a </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">doctrine</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">by </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">attaching</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">to</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">each</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> finite list </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">of</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">variables</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">poset</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">of </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">formulae</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">that</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">can</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> be </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">written</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">using</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">those</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> free </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">variables</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">, </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">ordered</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> by </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">provable</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">reciprocal</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">consequence</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">. </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">During</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> talk </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">we</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">will</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">discuss</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">definition</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">of</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> a </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">doctrine</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">, and </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">we</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">will</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">explain</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">how </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">main</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">logic</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">-</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">theoretical</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">notions</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">can</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> be </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">translated</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">into</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">the</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">doctrine</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">-</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">theoretical</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)"> </span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">language</span><span style="font-family:arial,sans-serif;white-space:pre-wrap;color:rgb(0,0,0)">.</span></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto"><u style="color:rgb(49,49,49);word-spacing:1px">Keywords</u><font color="#313131"><span style="word-spacing:1px">: </span></font><span style="color:rgb(0,0,0)">Lawvere&#39;s</span><span style="color:rgb(0,0,0)"> </span><span style="color:rgb(0,0,0)">doctrines</span><span style="color:rgb(0,0,0)">, </span><span style="color:rgb(0,0,0)">categorical</span><span style="color:rgb(0,0,0)"> </span><span style="color:rgb(0,0,0)">logic</span><span style="color:rgb(0,0,0)">, </span><span style="color:rgb(0,0,0)">adjunctions</span>.</div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><span style="color:rgb(0,0,0);white-space:pre-wrap"><br></span></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><span style="color:rgb(0,0,0);white-space:pre-wrap"><br></span></div><div dir="auto" style="word-spacing:1px"><font color="#313131">The event will be held via Google Meet platform at the following link: </font><span style="font-family:Roboto,Arial,sans-serif;letter-spacing:0.3px;white-space:nowrap"><font color="#000000"></font></span><span style="font-family:Roboto,Arial,sans-serif;letter-spacing:0.3px;white-space:nowrap;background-color:rgb(255,255,255)"><font color="#000000"><a href="http://meet.google.com/vxb-zhux-veg">meet.google.com/vxb-zhux-veg</a></font></span></div><div dir="auto"><br></div><div dir="auto"><i><b>** We inform you that the talk will be recorded and uploaded on our website. If you join the talk <u>after the starting time</u>, we kindly ask you to ensure that <u>your microphone and webcam are turned off</u> **</b></i><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px">You can find the poster of the event in the attachment. We are looking forward to seeing you!</div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px">For further information, please visit our website: <a href="https://sites.google.com/view/insalate-di-matematica" target="_blank">https://sites.google.com/view/insalate-di-matematica</a></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br></div><div dir="auto" style="color:rgb(49,49,49);word-spacing:1px"><br>The organizers: Luigi Appolloni, Federico Bernini, Francesca Cottini, Luca Di Gravina, Ludovico Marini</div></div>