On Wed, Jun 22, 2011 at 11:46 PM, wren ng thornton
One of the big benefits I see to using category theory for dealing with programming languages comes from using CT as a generalized logic for equational reasoning. In particular, making use of the ideas of (co)limits and adjunctions in order to prove that something must behave in a particular way. This requires a bit of work to contort your thinking into the patterns of CT, but it means that once you can prove that something is a foo, then all the theorems about foos will apply.
Yes, I can understand that and this is exactly what I am looking for. Except that I am looking for it not to deal with programming languages but with systems (and the way we design). Of course, we can always say that each system is a language of its own (rather than *has* a language...) which is what Eric Evans coined with its "Ubiquitous language" term. But I find it difficult to connect that particular dots.
As an example of this approach, see my comments in the thread about anonymous sum types. For something to be a product means that a particular diagram commutes: namely, a product is the limit of the two point category (and coproducts are the colimit). One of the many things this tells us is that if we posit that A*A ~ A in a category that "has products", we must infer that the category is a meet-(pre)semilattice[1].
I certainly won't argue with you on this given my limited knowledge of what a meet-(pre)semilattice could be :-) But then, if I happen to understand what it is, how could that help me design and code "better" systems? I have the intuition it could help me but again I do not (yet) see how. Thanks a lot for your answer Arnaud