Distributivity of categories of coalgebras (Q1884878)

Categories of coalgebras for an endofunctor (of the category of sets, or of something fairly similar) play an important role in the semantics of computation. One conceptual difficulty in working with such categories is that, whilst the forgetful functor to the underlying category creates colimits, it does not in general create limits; in particular, products in such categories tend to be hard to describe. In this paper, the authors investigate when finite products distribute over finite coproducts in a category of coalgebras, and the closely related question of when such a category is extensive. Unsurprisingly, their results turn out to be similar to those obtained by the reviewer, \textit{J.~Power, T.~Tsujishita, H.~Watanabe} and \textit{J.~Worrell} [Theor. Comput. Sci. 260, 87--117 (2001; Zbl 0973.68178)] on the question of when such a category is a topos.