Let ∑ be a ranked set. A categorical ∑-algebra, c∑a for short, is a small category C equipped with a functor σC: Cn → C, for each σ ∈ ∑n, n ≥ 0. A continuous categorical ∑-algebra is a c∑a which has an initial object and all colimits ofω-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 c∑a C there is an ω-continuous c∑a Cω, 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 Cω.We then find examples of this completion when - C is a c∑a of finite∑-trees - C is an ordered ∑ algebra - C is a c∑a of finite A-sychronization trees - C is a c∑a of finite words on A.