From Comodels to Coalgebras: State and Arrays

We investigate the notion of a comodel of a (countable) Lawvere theory, an evident dual to the notion of model. By taking the forgetful functor from the category of comodels to Set, every (countable) Lawvere theory generates a comonad on Set. But while Lawvere theories are equivalent to finitary monads on Set, and that result extends to higher cardinality, no such result holds for comonads, and that is not only for size reasons: it is primarily because, while Set is cartesian closed, Set^o^p is not. So every monad with rank on Set generates a comonad on Set, but not conversely. Our leading example is given by the countable Lawvere theory for global state: its category of comodels is the category of arrays, yielding a precise relationship between global state and arrays. Restricting from arbitrary comonads to those comonads generated by Lawvere theories allows us to study new and interesting constructions, in particular that of tensor product.