Free adjunction of morphisms
The authors develop a method for the free adjunction of morphisms, subject to certain equations, to a category. In the first two sections of this paper the tools for the solution of this problem are introduced. \(\S 1\) starts with the definiton of a graph \({\mathcal G}\) as a class with two operations \(c,d:{\mathcal G}\to{\mathcal G}\) satisfying \(cc=dc=c\) and \(dd= cd=d\). The elements of \({\mathcal G}\) are called morphisms, the elements of \(c({\mathcal G})\) objects or identities. A composition graph \({\mathcal G}\) is a graph with a partial product such that, if \(fg\) is defined, \(f,g\in {\mathcal G}\), \(c(fg)=cf\), \(d(fg)=dg\) and \(df=cg\) follow. A graph morphism \(F:{\mathcal G}\to{\mathcal H}\) is called a functor, if \(FfFg\) is defined and \(FfFg=F(fg)\) provided \(fg\) is defined. Next, for a composition graph \({\mathcal A}\), the free category \({\mathcal A}^*\) over \({\mathcal A}\) with the canonical functor \(R_{\mathcal A}:{\mathcal A}\to {\mathcal A}^*\) is constructed and identitive composition graphs are introduced. If \({\mathcal A}\) is a composition graph and \({\mathcal W}({\mathcal A})\) the category of all words over \({\mathcal A}\), one puts \(g\leq f\), \(g,f\in {\mathcal W}({\mathcal A})\), if \(f=g\) or \(g\) is obtained from \(f\) by either composing two letters or by removing a letter that is an identity. \(\sqsubseteq\) denotes the transitive closure of \(<\). As \({\mathcal A}^*\) is the quotient of \({\mathcal W}({\mathcal A})\) with respect to the equivalence relation generated by \(\leq\), any equivalence class contains a minimal word with respect to \(\sqsubseteq\). Essential for the problem of freely adjoining morphisms (satisfying certain equations) to a category is to find a type of composition graph for which there is a unique minimal word (w.r.t. \(\sqsubseteq)\) in every equivalence class. This type is introduced as semicategory in \(\S 2\). A composition graph is called a semicategory if it is weakly identitive and associative (2.5). For a semicategory there is exactly one minimal word in each equivalence class (2.3) and \(R_{\mathcal A}: {\mathcal A}\to {\mathcal A}^*\) is an embedding. In \(\S 3\) these results are used to construct free adjunctions of morphisms satisfying certain equations to a category and to use these constructions for characterizing absolutely initial morphisms (3.1) and absolute monomorphisms (3.6). \(\S 4\) generalizes the results of \(\S 3\) to sources and cones. The paper contains very interesting and relevant examples and applications. This paper is an essential contribution to the adjunction problem and to the investigation of compatible equivalence relations in a category.
scientific article; zbMATH DE number 1553291