There are 5 Trinity Sanctums to find in Lies of P. To unlock them you will need a trinity key which you get from solving riddles on the ringing telephones throughout the game.
Key 1 - Workshop Union Entrance (To the right before the bridge leading to the factory) I answered monster which the riddle master said was wrong but I still got the key. Not entirely sure if it matters.
Door 1 - Near Veniginis work control room stargazer.
Key 2 - Malum District (In front of you after climbing a ladder) Answer: Candle
Door 2 - St. Frangelico Cathedral (You'll reach a spot where you are dodging electric balls. At the end to the right is a ladder you can take, follow the path to the door.)
Key 3 - Grand Exhibition (You'll see the phone a long the path shortly after killing an elite who bursts through a gate.) Answer - Egg. After the phone go down the next staircase and turn the statue on the right twice to get the key.
Door 3 - Estella Opera House (before going into the boss room jump down where the flaming chandelier is and you should see it.)
Key 4 - Krat Central Station (When you go back to krat central station the phone is a long the waterfront not far from this stargazer.) Answer - Ergo
Door 4 - Krat Central Station Lobby (Fast travel to the stargazer go up the first flight of stairs and its on the left)
Key 5 - Relic of Trismegistus Entrance (near the boss instead of jumping down stay on the path upwards and you'll see the phone) Answer - No
Door 5 - Arche Abbey Outer Hall (Deep into the path you'll work your way back inside the castle and there will be multiple flame thrower enemies. The door is all the way at the bottom.)
When you reach the final Trinity door make sure to exhaust all dialogue options with the riddle master. He will give you a special relic you can take to Venigni which will unlock the achievement for talking to him.