Functional Programming

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A few notes on Programming Language Theory (PLT) and Functional Programming (FP) idioms.

Functors

Parametricity/Free Theorems

Recursion Schemes

Monads

Free Monads

Some mathematicals objects are free in the sense that there’s always of getting one from nothing: for instance, given a set $S$ one can build the free monoid $S^*$ the set of lists of elements of $S$ under concatenation. It costs no extra assumption on $S$ to construct it, and in some sense it is not very useful apart from this one property: any other monoid on $S$ may be obtained from it by a suitable evaluation of lists of elements of $S$.

This intuition can be made precise by seeing free objects as initial objects in a suitable category (the comma category), but the “intuitive” idea is that free objects provide a syntactic basis with just enough structure to meet some algebraic requirements (for instance monoid, group, vector space) from which other objects can be obtained e.g. by quotienting using an equivalence relation. For instance, one can obtain a monoid homomorphism $\NN^* \to (\NN, +)$ by quotienting $\NN^*$ by the smallest monoid congruence satisfying:

What about free monads, then? Since “monads are just monoids in the category of endofunctors”, we expect free monads to be free monoids in the category of endofunctors.