In the story there are 4 Host Computers: Yggdrasil, Nexus, Homeostasis and Heterostasis.
Two rule on the Digital worlds on Servers of the Network in local realities while the remaining two are above any known server and are Multiversal Hosts. Altogether they work in apparent harmony and have the Key to Infinity.
YGGDRASIL
[Note: parts of the following articles were taken from Digimon Wikia.]
Yggdrasill is a 9000 Digimon. It is the mysterious host computer who rules over all of the “Digital World” on one Server of the Network. Its primary form is that of the Server Tree, an enormous, sacred tree.
Bagramon (from reality 06) replaced its lost eyes and right half of its body, taken from it by God (Homeostasis), with wood cut from the sacred tree. It has created an avatar in the form of “Yggdrasill_7D6“, whose body has many unidentified details.
In some realities, a Digimon known as Shakamon is rumored to be the being closest to Yggdrasill in the Digital World.
In reality 09, 3 digital Angels used to be the closest to Yggdrasil: Clavis Angemon, Akarimon and Holy Angemon.
Clavis Angemon was Yggdrasil´s first Guardian and served him for Millenniums until he met death by a Lucemon´s hands.
Akarimon, who used to be his apprentice, was not allowed to leave the Digiworld with Magnus Takariyama (they fell in love and Magnus wanted to bring her back home as his bride) after the evil angel´s defeat and Akarimon was forced to honor her oath as Clavis Angemon´s successor. This Yggdrasil´s attitude provoked the ire of a digital angel known as Shukumeimon (who was reincarnated as Magnus himself) and critiqued the deity for its lack of emotions and a narrow minded concept of law and order. Before leaving Shukumeimon challenged the Host on assuming a mortal form and live among humans and learn to deal with elements such as emotions. Shukumeimon´s words haunted the Host Computer for a long, long time.
Akarimon she stayed by Yggdrasil´s side for a Millennia until she finally met a digital angel she thought worthy of being her replacement, Holy Angemon. Once the Ultimate angel passed all her tests and proved his worthiness, Akarimon left the Digiworld of reality 09 for good and Holy Angemon became Yggdrasil´s closest guardian.
Shukumeimon´s words echoed through Yggdrasil from several realities and each would eventually separate part of their conscience and assume one or more human identities:
In reality 09 Yggdrasil split its conscience into two human forms: Norun and Gabriel Mikihara. In reality 012 it fused its conscience with a comatose boy ´s soul and Alexander Mc Coy became the local Yggdrasil. Alice Mc Coy from reality 03 went through an identical situation and unbeknownst to the tamers she wields Yggdrasil. Realities 05 and 07 Relena Norstein voluntarily accepted to become Yggdrasil´s vessel. The Yggdrasil from reality 08 , in the last minute, before being given the final blow by Ominmon X that would reset the entire digiworld, managed to send part of his codes to the neighboring human world. His codes traveled through the net and ended taking possession of a google boy called Takeshi who was just playing a videogame at that time.
Saturn Angemon , one of the villains from the story, is also hunting down Yggdrasil…
HOMEOSTASIS
It is a digital lifeform that in many realities is assumed is the true God of the Digital World but according to Shukumeimon is Yggdrasil´s counterpart thus both share the same status “albeit with different concepts but equally faulty”
It seldom shows itself and is more inclined to take possession over a human or digimon when it wants to communicate.
In reality 01 it possessed 8 year old Hikari Yagami in order to communicate with the DigiDestineds and end a quarrel between Taichi and Yamatto. It revealed events from the past in which Piedmon attempted to take the Digi-Eggs, Digivices, Tags, and Crests created by the Agents, Gennai managed escaped with the Digi-Eggs and Digivices except the ones that corresponded to the future Gatomon.
Years later in 2005, Homeostasis possessed Hikari once again to speak with the Digimon about the Human World, the Digital World, and the Reboot.
In reality 06 Yggdrasill had gone berserk and attempted to destroy the Human World in an attempt to protect the Digital World. Homeostasis was created to replace it as the host computer and God of the Digital World, and was intended to prioritize maintaining harmony across the two worlds.
Much later, Bagramon, one of its closest angels, questioned Homeostasis about the world’s justice, and eventually fell from grace entirely. Homeostasis scourged the rebellious angel’s body and cast it out of the Kernel.
At some point, Homeostasis made a prophecy regarding the “red and black dragon” who would destroy the Digital World, based on its conclusions from carefully observing the Digital World. This prophecy was pursued by the Royal Knights, who originally thought the dragon was a member of the Seven Great Demon Lords, but it was eventually revealed to be ZeedMillenniummon, created from AxeKnightmon and Millenniummon through Bagramon’s machinations.
In reality 09, uneasy about Yggdrasil´s activities , Homeostasis created a digital angel called Akarimon and sent her to Clavis Angemon . Clavis agreed to take her as his apprentice and kept her origins a secret from Yggdrasil. What Homeostasis did not know was that Clavis Angemon was actually a Digital Angel from Nexus and was ordered by the entity to watch over both Digigods.
In reality 04 , in ancient times, the Digital World was immersed in chaos, torn apart by a war between Human and Beast types Digimons.
Displeased by Yggdrasil´s apparent lack of interest on stopping the war Homeostasis chose one of his angels, Lucemon, to intervene and reinstate order and harmony. To ensure the success of the mission Homeostasis infused some of its holy powers on Lucemon´s data to make him more powerful than the Mega Digi-angels, . 
He descended to the Digital World and accomplished his mission. He ended the war and established himself as the new protector but eventually became corrupted by his own power ; from beloved protector he became into a tyrant due to his beliefs that the world would be a better place if he had absolute power and made all the decisions. His power made him unbeatable until the Ten Legendary Warriors rose up in rebellion and defeated him, sealing him away in the Dark Area.Homeostasis and Yggdrasil then summoned three Digital Angels to rule over the Digiworld : Seraphimon, Ophanimon and Cherubimon.
The first two represented Homeostasis´s affairs while Cherubimon represented Yggdrasil´s. Yet the relations between Yggdrasil and Homeostasis since then became strained, each blaming each other for the Lucemon incident and remained without talking to each other for thousands of years.
NEXUS
Nexus is a digital lifeform, a Host Computer or Digigod like Yggdrasil and Homeostasis. Unlike the other two who rule on the Digital worlds on a Server of the Network Nexus is above any server; it is actually a multiversal Host Computer as it rules over Time and the connection between realities.
Located in a superior plane, at the intersection of countless of servers, Nexus rarely interacts with the rest of the host computers.
When a visitor is occasionally allowed inside Nexus will meet a multicolored crystal scenario; the more he or she advances it will turn into a shiny Escher-esque labyrinth with an endless set of halls, stairs and doors . It´s usually given the impression of being an empty building but it is actually inhabited by countless of Digital Angels and deities , from Order and Chaos. They just rarely allow to be seen by external parties.
Shukumeimon is among its main and most powerful keepers.
Clavis Angemon, Mastemon and Tempus Angemon are also part of Nexus although they rarely stay there for long periods of time. Digimons like Alphamon, Bagramon and Chronomon sometimes are invoked by Nexus itself and will carry on special tasks.
If any of the Yggdrasil´s from any reality becomes corrupted and threatens to break the barriers to other realities then Shukumeimon will take action. Like Clavis Angemon and Tempus Angemon he´s subjected to certain protocols yet he´s allowed to use more extreme measures if he considers the threat too revolting…to the point of vanishing an entire timeline if he wants to.
The Hall of Mirrors is its most famous feature, a ceiling-less and floor-less room filled with floating mirrors everywhere; each mirror represents a universe and a mere scratch on the surface can cause serious alterations on a reality. A rebel angel of Time, Saturn Angemon, broke the holy oath by allowing Lucemon inside Nexus and led him straight to that room; Lucemon´s madness ended creating a disaster inside the Hall of Mirrors aftr he on purpose destroyed a couple of them in order to escape from Nova and Tempus Angemon.
After learning about Saturn Angemon and Lucemon´s intention of absorbing Nexus and make use of its powers to recreate the universes from scratch Shukumeimon moved Nexus elsewhere across the realities and challenged both good and bad sides to find it. Whoever manages to find and enter Nexus first Shukumeimon will declare that side the winner regardless of the motives.
But what Shukumeimon did not tell is that Nexus can take any form and alter its size. So it can be as huge as a planet or as tiny as a ring…
HETEREOSTASIS
It a multiversal Host Computer and Nexus´s counterpart.
Yet some its actions ended causing an unwelcomed effect through realities and brought more than one paradox reality on the plate.
In reality 06 it was Hetereostasis that convinced Bagramon to go against Homeostasis….
Mirei Mikagura´s Mastemon is an angel from Hetereostasis and so is Ryu Yamaki from reality 04´s Slash Angemon…
Digimon Infinity 