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I became curious about this, there is of course game semantics but that's not what you're looking for. I found a couple of interesting references about using categories to study game theory which I'm going to add to my reading list, Game Theory from the Category Theory Point of View [0] and Towards Compositional Game Theory [1].

[0] http://www.gtcenter.org/Archive/2015/Conf/Jimenez1880.pdf

[1] http://www.cs.ox.ac.uk/people/julian.hedges/papers/Thesis.pd...

Check out Chris Martens' work, like her thesis "Programming Interactive Worlds with Linear Logic":

http://www.cs.cmu.edu/~cmartens/thesis/

(or some of the shorter papers on her website)

Thx, I saw this on HN a while back but totally forgot it existed.

Can you explain this more? What do you mean by "category theoretical" game design?

You model things in the game using category theory.

I'm not sure what you mean by that. What about board games do you expect to fruitfully from using category theory insights?

Exposing the underlying transformation structure.

How?