Coinverters and categories of fractions for categories with structure (Q690384)

The process of forming categories of fractions is a well known construction in category theory. The authors introduce a more general notion, namely that of a coinverter \(p : A\to C\) of a 2-cell \(\alpha : s\to t : S\to A\) in a 2-category: \(p\) is universal among those 1-cells \(q : A\to B\) for which \(q\alpha\) is invertible. It is the object \(C\) which is sometimes loosely called the coinverter. In that sense coinverters in \({\mathfrak {Cat}}\) include categories of fractions. The coinverter of a reflexive 2-cell is, by an abuse of language, said to be reflexive. The main result refers to the 2-categories \({\mathfrak T}\)-\({\mathfrak {Alg}}_ s\) and \({\mathfrak T}\)-\({\mathfrak {Alg}}\) introduced by \textit{G. M. Kelly} and \textit{S. Lack} in a paper on strongly finitary 2-monades [Appl. Categ. Struct. 1, 85-94 (1993; Zbl 0787.18007)]: The forgetful functors into \({\mathfrak {Cat}}\) create reflexive coinverters. As an application the following result is mentioned. An equational structure on a category \({\mathfrak A}\) passes canonically to the category of fractions \({\mathfrak A}[\Sigma ^{-1}]\) provided that the set \(\Sigma\) of morphisms contains the identities and is closed under the operations involved.