Container Combinatorics: Monads and Lax Monoidal Functors
Abbott et al.’s containers are a “syntax” for a wide class of set functors in terms of shapes and positions. Containers whose “denotation” carries a comonad structure can be characterized as directed containers, or containers where a shape and a position in it determine another shape, intuitively a subshape of this shape rooted by this position. In this paper, we develop similar explicit characterizations for container functors with a monad structure and container functors with a lax monoidal functor structure as well as some variations. We argue that this type of characterizations make a tool, e.g., for enumerating the monad structures or lax monoidal functors that some set functor admits. Such explorations are of interest, e.g., in the semantics of effectful functional programming languages.
I am very grateful to Thorsten Altenkirch, Pierre-Louis Curien, Conor McBride, Niccolò Veltri for discussions. Paul-André Melliès pointed me to Aguiar’s work. The anonymous reviewers of TTCS 2017 provided very useful feedback. This work was supported by the Estonian Ministry of Education and Research institutional research grant IUT33-13.
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