Hi James, welcome to the Atlassian Community!
Crowd is entirely server-based. You host the application on the infrastructure of your choosing. This can be something like AWS or Azure, but most commonly organizations have deployed Crowd on their on-premise hardware. We don't provide a Cloud option for Crowd, instead we ask people using Atlassian Cloud to take a peek at Atlassian Access if they need to apply 2FA or password policies to accounts in their company.
Hope that helps and happy to answer more questions!
Cheers,
Daniel
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