The behaviour of statistical relational representations across differently
sized domains has become a focal area of research from both a modelling and a
complexity viewpoint.Recently, projectivity of a family of distributions
emerged as a key property, ensuring that marginal probabilities are independent
of the domain size. However, the formalisation used currently assumes that the
domain is characterised only by its size. This contribution extends the notion
of projectivity from families of distributions indexed by domain size to
functors taking extensional data from a database. This makes projectivity
available for the large range of applications taking structured input. We
transfer key known results on projective families of distributions to the new
setting. This includes a characterisation of projective fragments in different
statistical relational formalisms as well as a general representation theorem
for projective families of distributions. Furthermore, we prove a
correspondence between projectivity and distributions on countably infinite
domains, which we use to unify and generalise earlier work on statistical
relational representations in infinite domains. Finally, we use the extended
notion of projectivity to define a further strengthening, which we call
$\sigma$-projectivity, and which allows the use of the same representation in
different modes while retaining projectivity.

本文研究统计关系表示在不同规模域中的行为，并通过将域的概念从仅大小表示扩展到从数据库中获取的外延数据的函子，为更广泛的应用提供了 projectivity。将关于 projective families 的关键结果转移到了新的设置中，从而证明了 projectivity 和无限域上的分布之间的对应关系，并使用扩展后的 projectivity 定义了更强大的 sigma-projectivity。