Algebra objects and algebra families for finite limit theories
A functor \(F: \mathbb{C}\to\text{\textbf{Set}}\) is called familially representable if it is a coproduct of representable functors, a representing family for \(F\) being a family \((A_ i)_{i\in I}\) of objects of \(\mathbb{C}\) such that \(F\simeq\coprod_{i\in I}\Hom_{\mathbb{C}}(A_ i,-)\). The purpose of the paper is to describe the structure that the family \((A_ i)_{i\in I}\) inherits from the functor \(F\). When \(F\) has, for example, an algebraic structure, the family \((A_ i)_{i\in I}\) gets the structure of an ``algebra family''. Conversely, any ``algebra family'' of objects in \(\mathbb{C}\) provides a familially representable functor carrying an algebraic structure. For example, the free monoid functor \(M: \text{\textbf{Set}}\to\text{\textbf{Set}}\) is familially representable and its representing family has the structure of a ``comonoid family''. Other examples are given involving ``cocategory family'' and giving explicit constructions of categories such as the Moore construction of a category of paths in a topological space, and other free constructions.
scientific article; zbMATH DE number 166157