Categorical constructions of Free Algebras, colimits, and completions of partial Algebras

Abstract

1.1. Given a base category K and a functor F: K -, K, we shall consider the category A(F) of F-algebras and the category PA(F) of partial F-algebras (for definitions, see 1.2). The category A(F) has been studied in a lot of papers [2-7, 13, 171 see also [ll, 121 in connection with the categorical universal algebra and automata and control .theory. We shall deal with the following problems: (1) Existence and construction of free algebras in A(F). (2) Existence and construction of colimits in A(F). (3) Existence and construction of left adjoints to functors A(F) -j A(G) induced by transformations G += F. (4) Completions of partial algebras and other properties of the category PA(F). (5) Cocompleteness of the category of algebras for a triple (F, q, p) in K. The present paper is based on the thesis [15] of the second author which generalizing categorical constructions [l, 4, 7, 131 and classical constructions of universal algebra attempts to form a general theory of constructions of free algebras, colimits etc. Some results of [ 151 are improved and the approach is applied to partial algebras and to algebras for a triple. As a technical tool, we shall embed A(F) into the category A*(F) [15] of algebraized chains which possess convenient properties (Sections 2-4). The category PA(F) will be investigated by means of the embedding into the category GPA(F) of generalized partial algebras. Notice that categories A*(F) and GPA(F) form “completions” of A(F): they have free algebras and are cocomplete. The usefulness