On Coalgebras which are Algebras

The category CoalgΣ of coalgebras with respect to a (bounded) signature Σ is known to be locally finitely presentable (see [1]). We strenghten this result by showing that CoalgΣ even is a presheaf category. Moreover, we give a presentation of this category as the category of all algebras of some (many-sorted) signature (without any equations). Σ–coalgebras, i.e., coalgebras with respect to a polynomial endofunctor HΣ on Set, HΣ(X) = ∐ n<λ Σn ×Xn, with Σ = (Σn)n<λ, a family of sets, are known to be intimately related to tree structures. On the one hand the set TΣ of all Σ–labelled trees (see 1 below) is the underlying set of a terminal object in CoalgΣ, the category of Σ–coalgebras; on the other hand, each Σ– labelled tree t is a Σ–coalgebra At in its own right (see Definition 3 below). The structural importance of the family of tree coalgebras At, t ∈ TΣ, already emerged in [1] where this family was shown to be a strong generator of finitely presentables in CoalgΣ. We are going to show in this note that this family even is an absolute generator, i.e., that the hom-functors determined by its members even preserve all colimits, which then leads to a representation of CoalgΣ as a presheaf– category (see also [5], where completely different methods have been used to establish such a presentation). The particular structure of the full subcategory spanned by the tree coalgebras then even allows for a simple explicit description of this presheaf category as a category of unary algebras without equations. We start by briefly recalling some basic concepts. 1 A Σ–labelled tree is a partial function t : ω∗ → Σ whose domain of definition, Deft, has the following two properties: (i) Deft contains the empty word and is prefix–closed, i.e., if uv ∈ Deft then u ∈ Deft