Free adjunction of morphisms
type: Item
instance of: scholarly article
title: Free adjunction of morphisms
zbMATH Open document ID: 0977.18001
DOI: 10.1023/A:1008651618062
author: Lutz Schröder
author: Horst Herrlich
published in: Applied Categorical Structures
publication date: 2002-01-17T00:00:00Z
review text: 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 identit…
reviewed by: Dieter Pumplün
Mathematics Subject Classification ID: 18A10
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18A22
zbMATH DE Number: 1553291
zbMATH Keywords: category of words
zbMATH Keywords: adjunction of morphisms
zbMATH Keywords: composition graph
zbMATH Keywords: free adjunctions
zbMATH Keywords: absolutely initial morphisms
zbMATH Keywords: absolute monomorphisms
zbMATH Keywords: compatible equivalence relations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1553291