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Added a reference to the paper “Generators and bases for algebras over a monad” (Stefan Zetzsche, Alexandra Silva, Matteo Sammartino) (2020)
Edit: There is nothing about applying the concept to bases of topological spaces in this paper and in fact no more to any of the examples of basis on this nlab page. They are more focused on examples from linear algebra. However, I guess it is probably possible to apply their ideas to these more topological examples. Maybe by using the concept of monoidal topology (e.g. compact Hausdorff spaces are algebras for the ultrafilter monad).
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