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We provide a categorical framework for mathematical objects for which thereis both a sort of "independent" and "dependent" composition. Namely we modelthem as duoidal categories in which both monoidal structures share a unit andthe first is symmetric. We construct the free such category and observe that itis a full subcategory of the category of finite posets. Indeed each algebraicexpression in the two monoidal operators corresponds to the poset built bytaking disjoint unions and joins of the singleton poset. We characterize these"expressible" posets as precisely those which contain no "zig-zags." We thenmove on to describe categories equipped with $n$-ary operations for each$n$-element finite poset; we refer to them as "dependence categories" sincethey allow for combinations of objects based on any network of dependenciesbetween them.  These structures model various sorts of dependence including the space-likeand time-like juxtaposition of weighted probability distributions inrelativistic spacetime, which we model using polynomial endofunctors on thecategory of sets, as well as the runtimes for multiple computer programs run inparallel and series, which we model using the tropical semiring structure onnonnegative real numbers. With these examples in mind, we conclude bydescribing ways in which morphisms in a partial monoidal category can be"decorated" in a coherent manner by objects in a dependence category, such aslabeling a network of parallel programs with their runtimes.

Duoidal Structures for Compositional Dependence