I'm not an expert on the settings I was under the impression that arcologies were just Extra Thicc Valkyries and Valkyries were under the UN.
Sorta yes, sorta technically no. It's more a matter of consequence than cause. All known Valkyries are with the UN, but that is largely due to the simple fact that the UN is basically humanity's
only faction, with anyone not affiliated with the UN no longer around anymore and thus possession of Valkyrie Cores and compatible people defaulting to the UN; Iran, for example, had a significant Valkyrie contingent in the past per Shuri's backstory, but after screwing up by the numbers and
losing nearly all of them in addition to massive population in a badly mismanaged effort against an Antagonist offensive, what was left of Iran was assimilated into the UN. Anyone can pick up a Valkyrie Core and use it if they can just find it and happen to be compatible, but the UN has (at least almost) exclusive hold over the Valkyries through statistics and feasibility. The UN is a global organisation with the support network appropriate for such; if there's a Valkyrie Core out there accessible to humanity, it's pretty much guaranteed to be within UN-held territory populated by UN people, and even if an anomalous Core beats the odds, the new Valkyrie is probably going to just join the UN anyway. It's technically possible for someone not affiliated with the UN to stumble across one, but the chance of there being anyone outside of the UN
to do so is vanishingly small already, and such a person also happening to indeed find a Core
and be compatible with it
and not promptly die soon after anyway rapidly trends towards zero. Anna, however, by dint of narrative causality, is precisely that unlikely person, having found Durga in the wake of the collapse of broader civilisation in North America and managed to hold her own for years in isolation from the UN until social problems saw the end of her settlement where the Antagonists could not... and she then just joined up with the UN.
An Archology is essentially a Valkyrie who just delves a lot more into support than mainline combat and takes that stance to the extreme. Before the sweeping incursion of the Antagonists, the Impact—in which the meteor carrying the Valkyrie Cores to Earth made a solid attempt at replaying the demise of the dinosaurs resulted—saw humanity strained nearly to its breaking point. The discovery of the Valkyrie Cores, however, allowed the initial Valkyries to basically save a doomed world, integrating and in turn upgrading the kind of equipment suited for rebuilding humanity from the sorry wreck that it was after such devastation. Even something like a basic water purifier and electrical generator once assimilated by a Valkyrie and put to use could be an absolute godsend in a situation like that, to say nothing of the Valkyries' capacity to aid reconstruction efforts. The Impact was altogether dire, but humanity pulled through pretty well in the end thanks to the Valkyrie Cores, and things started looking up for mankind... which of course was when the cosmic smite got followed up by genocidal alien invasion. The "classic" Valkyrie emerged when the people bonded to Valkyrie Cores to employ the likes of fabricators and macro-scale hydroponics arrays ditched their old gear and started integrating weaponry, but some of those Valkyries were already heavily upgraded centers for providing for civilisation in a (now
doubly) post-apocalyptic environment, and the need for those developed shields against the outside world didn't go away, so the broad category of Valkyries ended up with the typical mecha masquerading as just power armour and Archologies as they are known now at opposite ends of a spectrum, as well as the likes of battleship Valkyries somewhere in between, too.
Since not all Valkyries
have to be UN, though, there can be some deviation from UN norms for the different circumstances. Anna was running something similar to a typical UN Archology on a smaller and less developed scale. However, for being the only Valkyrie to her faction and her hometown under siege within Antagonist territory, she had to manage the complicated dual roles of both managing the settlement and killing the enemy trying to wipe it off the map. This left her hometown as an odd arrangement in which the (literal) core element of her Archology-analogue had to be able to frequently separate and leave. If Anna did not have her "the monster" unhappy version background as the The Warrior character, circumstances would have been different, with Anna's protected people actually staying alive and her having the ability to basically make everyone around her able to laugh off any pathetic xeno scum's feeble attempts to hurt them, so Hometown in that alternate scenario might have turned out something more like a typical UN Archology only extra spicy, perhaps. Similarly for comparison, Hell's Hand as a stealth facility able to go unbothered by the enemy could afford to devote itself towards the civic management side of things, to end up something quite akin to the Archologies familiar to the UN even if it didn't have anyone to actively protect it.