Definition. Dependence category [mde-2AK7]

A dependence category is given by

  1. A category \(\mathbb {C}\)
    2. ,
  2. for every \(n \in \mathbb {N}\) and poset \(p\) on \([n]\), a functor \(\boxtimes ^n_P : \mathbb {C}^n \to \mathbb {C}\)
    3. ,
  3. for every identity-on-objects inclusion of posets \(P \to Q\), a natural transformation \(\boxtimes ^n_P \Rightarrow \boxtimes ^n_Q\)
    4. ,
  4. such that this data is compatible with multicategorical composition, units and symmetries up to isomorphism
    5. ,
  5. and satisfies appropriate unitality, associativity and equivariance equations.

Succinctly this is a symmetric \(\mathbf {Cat}\)-algebra for the symmetric 2-multicategory \(\mathbf {Pos}\) of finite posets, whose (thin) category of \(n\)-ary operations is the category of poset structures on \([n]\) and identity-on-objects inclusions.