The Gladiator background (an Entertainer variant, from PHB p. 130-131) gives a strange weapon (like a trident or net) as starting equipment, but doesn't explicitly specify that it gives proficiency with said weapon.
Is it implied that it does? (Or if not, should it, considering that a seasoned gladiator is expected to be particularly skilled with his weapon of choice?)
The Entertainer Background (PHB, 130) give you proficiency in one type of musical instrument AND gives you said musical instrument.
The Gladiator variant (PHB, 131) states:
You can replace the musical instrument in your equipment package with an inexpensive but unusual weapon, such as a trident or net.
While it doesn't state specifically to swap out the music instrument proficiency as well, it seems clear that you are swapping out the musical instrument and knowledge of how to play it for a weapon (and possibly knowledge of how to use it.)
The language is clear that you are only to swap out the equipment - and nothing is stated on the changing of the proficiency.
But talk with your DM, as custom backgrounds are also in the PHB (Page 126) and this is a pretty reasonable request (and a neat way to get a weapon proficiency you may not have.)
The added proficiency that Backgrounds give you are all Tools - no background gives you additional weapon proficiencies (not even soldier.) While it seems to make sense to give the proficiency, weapon proficiency is a hard thing to come by, and it is very reasonable to not allow a background to give a weapon proficiency.
