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GADTs can be represented either as their Church encodings a la Atkey, or as fixpoints a la Johann and Polonsky. While a GADT represented as its Church encoding need not support a map function satisfying the functor laws, the fixpoint representation of a GADT must support such a map function even to be well-defined. The two representations of a GADT thus need not be the same in general. This observation forces a choice of representation of data types in languages supporting GADTs. In this paper we show that choosing whether to represent data types as their Church encodings or as fixpoints determines whether or not a language supporting GADTs can have parametric models. This choice thus has important consequences for how we can program with, and reason about, these advanced data types. 

Abstract Reproduction is a fundamental aspect of life that affects all levels of biology, from genomes and development to population dynamics and diversification. The first Tree of Sex database synthesized a vast diversity of reproductive strategies and their intriguing distribution throughout eukaryotes. A decade on, we are reviving this initiative and greatly expanding its scope to provide the most comprehensive integration of knowledge on eukaryotic reproduction to date. In this perspective, we first identify important gaps in our current knowledge of reproductive strategies across eukaryotes. We then highlight a selection of questions that will benefit most from this new Tree of Sex project, including those related to the evolution of sex, modes of sex determination, sex chromosomes, and the consequences of various reproductive strategies. Finally, we outline our vision for the new Tree of Sex database and the consortium that will create it (treeofsex.org). The new database will cover all Eukaryota and include a wide selection of biological traits. It will also incorporate genomic data types that were scarce or non-existent at the time of the first Tree of Sex initiative. The new database will be publicly accessible, stable, and self-sustaining, thus greatly improving the accessibility of reproductive knowledge to researchers across disciplines for years to come. Lastly, the consortium will persist after the database is created to serve as a collaborative framework for research, prioritizing ethical standards in the collection, use, and sharing of reproductive data. The new Tree of Sex consortium is open, and we encourage all who are interested in this topic to join us.