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Sep 23, 2012 at 6:47 comment added Jonathan Chiche This is the name Joyal calls it in his notes on quasi-categories, see for instance p. 26 of math.uchicago.edu/~may/IMA/JOYAL/JoyalDec08.pdf. There are some other notes by Joyal which deal with that construction in slightly more details, if I am not mistaken, but I cannot recall whether he states a universal property or not. Anyway, he uses it to define slices of quasi-categories.
Sep 23, 2012 at 3:38 comment added Mozibur Ullah @tanaka:thats the name they give it in [nlab](ncatlab.org/nlab/show/join+of+categories), but as Todd points out below its a special case of a collage or cograph. Out of interest what have you been calling it?
Sep 22, 2012 at 19:51 comment added Hiro Lee Tanaka Could you give a citation as to where this is called the join? I'd like to know because this is not what I've been calling the join of two categories--I hope I'm not in trouble!
Sep 22, 2012 at 19:37 vote accept Mozibur Ullah
Sep 22, 2012 at 19:32 answer added Todd Trimble timeline score: 11
Sep 22, 2012 at 18:52 history asked Mozibur Ullah CC BY-SA 3.0