An Algebraic Description of Locally Multipresentable Categories

Locally finitely presentable categories are known to be precisely the categories of models of essentially algebraic theories, i.e., categories of partial algebras whose domains of definition are determined by equations in total operations. Here we show an analogous description of locally finitely multipresentable categories. We also prove that locally finitely multipresentable categories are precisely categories of models of sketches with finite limit and countable coproduct specifications, and we present an example of a locally finitely multipresentable category not sketchable by a sketch with finite limit and finite colimit specifications. Introduction. We have shown in [AR1] how each locally finitely presentable category K in the sense of [GU] can be described by an essentially algebraic theory. This means that there exists a (finitary, many-sorted) signature Σ = Σt ∪Σp such that K is equivalent to the category of partial Σ-algebras A such that (i) each operation σA with σ ∈ Σt is total (i.e., everywhere defined) (ii) for each σ ∈ Σp a finite set Defσ of equations in signature Σt is given and σA(a1 . . . an) is defined iff all equations of Defσ are fulfilled in (a1, . . . , an). and (iii) a set of equations is given which all partial algebras of the given category satisfy. In the present paper we discuss locally finitely multipresentable categories, as introduced by Y. Diers [D]. Recall that a category is locally finitely multipresentable iff it has (a) connected limits (or, equivalently, multicolimits) Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 is gratefully acknowledged. Received by the Editors 11 May 1995, and in revised form 22 June 1996. Published on 5 July 1996. 1991 Mathematics Subject Classification: 18C99.