Completing Categorical Algebras

Let Σ be a ranked set. A categorical Σ-algebra, cΣa for short, is a small category C equipped with a functor σC : C n //C, for each σ ∈ Σn, n ≥ 0. A continuous categorical Σ-algebra is a cΣa which has an initial object and all colimits of ω-chains, i.e., functors N //C; each functor σC preserves colimits of ω-chains. (N is the linearly ordered set of the nonnegative integers considered as a category as usual.) We prove that for any cΣa C there is an ω-continuous cΣa C, unique up to equivalence, which forms a “free continuous completion” of C. We generalize the notion of inequation (and equation) and show the inequations or equations that hold in C also hold in C. We then find examples of this completion when – C is a cΣa of finite Σ-trees – C is an ordered Σ algebra – C is a cΣa of finite A-sychronization trees – C is a cΣa of finite words on A.