The Arbiter of Ash fight and how you access it changed in patch 0.3.0.
The old version had five difficulties. Now, there are two. The standard Arbiter, which is the hardest of the 'base' Pinnacle bosses but still designed to be quite accessible, and an 'Uber' version that is much more challenging.
No matter which version you attempt, you will need to path to the Burning Monolith on the Atlas, which will always be located very close to the origin.
Crisis and Calamity Fragments
Crisis Fragments
A set of single-use items is used to access your preferred version of the fight.
If you'd prefer to tackle the 'base' version of the fight, you'll need to provide the following three items:
These are the default loot for defeating the Citadel bosses, dropping from Geonor in the Iron Citadel, Jamanra in the Copper Citadel and Doryani in the Stone Citadel respectively.
VERY IMPORTANT: You can get more than one Fragment per Citadel. The more waystone drop rate you stack on the waystone that grants you access to the Citadel, the more average fragments you will drop.
Calamity Fragments
These are used instead if you'd rather attempt the Uber version of the Arbiter of Ash.
As of patch 0.3.0c, this is not proven to improve the loot of the fight, but most players suspect that it does. This improvement is suspected to include biasing drops so that you are less likely to receive common Arbiter drops and more likely to get rarer ones.
If you die during the Uber Arbiter, your instance is 'downgraded' to a regular Arbiter fight and you can try again.
These Calamity Fragments replace the drops of Crisis Fragments you would otherwise have received if your Atlas has 10 points allocated on the boss killing tree.
You do not need to allocate the exact ten points shown, any ten will do.
This requires you to defeat the Citadel bosses in their hardest forms, as practically every point on the boss tree makes them stronger. In exchange for this additional risk, you get additional rewards - better fragments.
Unlocking Ten Boss Tree Points
You unlock these points two at a time.
First Two Points:
Slay a map boss in an area level of 79 or higher while the waystone has at least 6 modifiers.
Any tier 15 waystone will reach level 79.
Points 3 and 4:
As for the first two points, but you require level 80 or higher.
This requires a tier 16 waystone (corruption exclusive tier), or a tier 15 waystone that has been boosted by Irradiation (from RNG, a Precursor Tablet or Organised Forces) or by certain Atlas Corruption effects.
Points 5 and 6:
Slay an Anomaly Map Boss in an area level 79 or higher with at least 6 modifiers.
Points 7 and 8:
Slay an Citadel Map Boss in an area level 79 or higher with at least 6 modifiers. This will award one of the basic Crisis fragments.
Points 9 and 10:
Slay the Arbiter of Ash in either its basic or Uber version.