The Arbiter of Ash is the first Atlas pinnacle boss.
You can access this fight by obtaining the three Crisis Fragments and taking them to the Burning Monolith.
Here is a guide to defeating this boss.
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Witness To History
At the top of the Arbiter atlas passive tree is 
Witness to History. This does not do what players unfamiliar with the Arbiter fight believe.
It's much better.
It does not add a 20% chance to get a 'drop anywhere' unique item such as 
Wanderlust to the drop pool. It instead grants more of the Arbiter's exclusive unique items. 1.2 per run, not 1.
Having this allocated requires the Arbiter to be attempted at difficulty 2 or higher. Because this grants 20% more of the most important loot, all large-scale samples of boss loot from the Arbiter are from higher difficulties, and little is known about difficulty 0 or 1 drop rates.
Unique Item Overview
The Arbiter can drop the following items:
Morior Invictus - This item always spawns with four rune sockets (a fifth can be corrupted onto it). As it has 7 modifiers and two are chosen from it, there are 21 possible versions of this item. They are not of equal value.
This item appears to be the most common at all difficulties.
Tip: You can sell Morior Invictus to other players unidentified, if you would rather get sure currency than take a gamble hoping to ID an exceptionally popular outcome.
Solus Ipse - This is second most common at all difficulties. It's not equipped often, and fairly cheap.
Sine Aequo - This item is very rare, but despite its rarity is not frequently desired by players. In trade leagues it is cheap even when close to perfect rolls.
Ab Aeterno - This item is between Solus Ipse and Sine Aequo in rarity.
Sacred Flame - This item can only appear on Difficulty 4.
Prism of Belief - This item can only appear on Difficulty 4, and has an additional two layers of RNG - the skill rolled has a huge impact on how desireable the jewel is, as does the level boost. Because this item always drops corrupted, you cannot apply a Divine Orb to it.
Tip: As with Morior Invictus, you can sell this item to other players unidentified, if you would rather get sure currency than take a gamble hoping to ID an exceptionally good outcome.
Drop rates are estimates based on runs compiled by the Discord community Prohibited Library.
Drop Rates: Difficulty 0 and 1
Unfortunately, little is known about these difficulties, due to people focusing their efforts upon difficulties 2 and higher, where Witness to History is able to be used.
Many players suspect that each 12 runs average 6 Morior Invictus, 3 Solus Ipse, 2 Ab Aeterno and 1 Sine Aequo, but the evidence for this is insufficient to confirm this.
Drop Rates: Difficulty 2 and 3
Hundreds of runs have been logged at these difficulties, and the evidence for a 6:3:2:1 ratio here is stronger.
However, because of Witness to History, you only require 10 runs to get 12 Arbiter-exclusive items.
Estimated drop rates:
Morior Invictus 60%
Solus Ipse 30%
Ab Aeterno 20%
Sine Aequo 10%
This adds to 120%, as you get 1.2 items average per run.
Tip: In a trade league, don't consider running these without Witness to History - the people who are using that node will outbid you on the Arbiter access fragments. Difficulty 0 and 1 exist to unlock your Arbiter points. Difficulty 2 and up require Witness.
Drop Rates: Difficulty 4
At difficulty 4, two new items are added to the pool. These replace about one third of the items you would otherwise get from the encounter and there does not appear to be a bias towards losing specific ones.
Estimated drop rates:
Morior Invictus 40%
Solus Ipse 20%
Ab Aeterno 13.33%
Sine Aequo 6.67%
Sacred Flame 13.33%
Prism of Belief 26.67%
Again, this adds to 120% as there is an assumption you are using Witness to History.
A Very Lucrative Boss
In trade leagues, if you can become very good at this fight on difficulty 4, and sell the RNG unique items unidentified, the unique items alone will more than pay for your entry tokens (the set of three Crisis Fragments), allowing you to acquire a large number of the other special rewards - rare Precursor Tablets, high level gems that have a chance to be level 20, or other options.