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I have a recursive algebraic datatype. I (somewhat arbitrarily) defined one function to compute distance between instances, and am trying to define a function to approximate a "vector" between instances. I assume there's no single way to define a metric space on top of a definition of an algebraic data type, much less a vector space, but was wondering what literature exists about treating recursive algebraic datatypes as existing in a topological space with meaningful comparisons.

The reason I want to treat it like a vector space is because I want to define interpolation between two structures in some reasonable manner.

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  • $\begingroup$ What do you need these structures for? That might direct possible choices in a useful way. They yearning for vectors makes it look like you're secretly doing machine learning. $\endgroup$
    – Andrej Bauer
    Jul 31, 2018 at 13:48
  • $\begingroup$ Haha, I do come from something of a machine learning background, so maybe that shows through! $\endgroup$
    – lightning
    Jul 31, 2018 at 16:56

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It is possible to construct recursive datatypes (algebraic datatypes are a special case) in a suitable category of complete metric spaces, see for instance P. America and J. Rutten's Solving Reflexive Domain Equations in a Category of Complete Metric Spaces, Journal of Computer and System Sciences 39, 343-375 (1989).

As for making vector spaces out of algebraic datatypes, I do not really know, but if you explain what you want them for, we might be able to suggest something.

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