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