The TechnoCore is hard to define, especially since no one is really sure what it is. It's not as ridiculous as your toaster taking over the world, but it's close. It seems that "the AIs had peacefully seceded from human control more than three centuries ago" (5.223), but we're not so sure what that means. We do know that they control the All Thing, the evolution of the Internet, "which had evolved a form of autonomy and consciousness all its own" (2.239). (So, basically, like Reddit.)
Whatever happened, the AIs now exert powerful influence over government. It's like if the Watson computer earned a senate seat. Hey, that might not be a bad idea.
Anyway, in Hyperion, the AIs aren't represented by cute little abstract orbs on a flatscreen like Watson. They exist somewhere else entirely, in a cyberspace not unlike that of William Gibson in Neuromancer. When Brawne and her cyberpunk friend, BB Surbringer, travel into cyberspace, Brawne says, "[We] could feel the weight of the AIs' wrath above [us]—[we] were insects under elephants' feet" (5.1059).
Whatever they are, it's immense. It's powerful. And it doesn't exactly like us. The TechnoCore is one of Simmons's quietly terrifying creations: we created it. Like Frankenstein's monster, humans created artificial intelligence. Now it might be working against us. For now, they're working with us... as government aides. So, yeah, they're working against us.
They may be cooperating with the Hegemony Senate, but they also have dozens of tricks up their digital sleeves. One of these is the personality retrieval project, which implants personalities into cybrids. Fun, party-loving personalities like Ezra Pound and John Keats.
Another is the Ultimate Intelligence Project. This one must be more important, because it's capitalized. You see, the Core wants to know everything. They can predict 98.9995 percent of human events for up to 200 years, but that last 1.0005 percent is just killing them. It's like watching all of Lost except the last episode. (Not like that really cleared anything up.)
To achieve Ultimate Intelligence, the TechnoCore is doing all sorts of trippy stuff, like reconstructing Earth to scale. This is all because of three arguing factions within the TechnoCore (even AIs don't all get along): the Stables, the Volatiles, and the Ultimates. To keep it simple, the Stables like us, the Volatiles don't, and the Ultimates couldn't care either way. They just want to know everything, and Hyperion, that pesky planet weirder than anything David Lynch could ever think of, is standing in their way.
All of this is just part of "the terrible beauty of the datumplane" (5.1059) The phrase "terrible beauty" echoes Kassad's sentiment about the "terrible beauty of combat" (2.292), equating the TechnoCore with modern warfare. And that can't be a good thing. We're imagining a TechnoCore revolt to be about 100x worse than a computer crash. The toaster spraying burnt crumbs into our face would be the least of our worries.