Sometimes I wonder what kind of operations would actually be useful to do on such data in the real world.

I mean, the way the web currently works is that I trust some server to host my data. I can have this service auth an external consumer site and display data in an iframe, say, which the consumer site can't get at. This is good enough for displaying people their personalized info (name, friends) on various services (eg directions to their house in an iframe, for a user authenticated with my chosen provider).

But to go further, what if I don't want to trust any provider?

Then I could simply encrypt the data and store encrypted data with the provider (or providers for redundancy). The authentication could be replaced with visitors holding a key to decrypt the data (because I gave it to them) and I can switch to using some other key and effectively "unfriend" those who don't get my updated key.

But all this is good enough for displaying data and files I upload. Now, why would I want to do operations on those files "in the cloud" without trusting a provider? I am already trusting my friends with the data, since they can reshare it once it's displayed to them. So why not trust a provider? One of my friends can run the provider.

I guess the only scenario I see it being useful is if all my friends can only have limited access to the data and all manipulations on the data are collaborative, and that's where the homeomorphism comes in. Perhaps no one will be able to see the whole data and it's not really about data at all, but views of some Enigmatic process running on some network (like an autonomous corporation.) is that the use case?