Lallement Functor is a Weak Right Multiadjoint

Abstract

For a plural signature \(\Sigma \) and with regard to the category \(\textsf {NPIAlg}(\Sigma )_{\textsf {s}}\), of naturally preordered idempotent \(\Sigma \)-algebras and surjective homomorphisms, we define a contravariant functor \(\textrm{Lsys}_{\Sigma }\) from \(\textsf {NPIAlg}(\Sigma )_{\textsf {s}}\) to \(\textsf {Cat}\), the category of categories, that assigns to \({\textbf {I}}\) in \(\textsf {NPIAlg}(\Sigma )_{\textsf {s}}\) the category \({\textbf {I}}\)-\(\textsf {LAlg}(\Sigma )\), of \({\textbf {I}}\)-semi-inductive Lallement systems of \(\Sigma \)-algebras, and a covariant functor \((\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )\) from \(\textsf {NPIAlg}(\Sigma )_{\textsf {s}}\) to \(\textsf {Cat}\), that assigns to \({\textbf {I}}\) in \(\textsf {NPIAlg}(\Sigma )_{\textsf {s}}\) the category \((\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, {\textbf {I}})\), of the coverings of \({\textbf {I}}\), i.e., the ordered pairs \(({\textbf {A}},f)\) in which \({\textbf {A}}\) is a \(\Sigma \)-algebra and a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories \(\int ^{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}\textrm{Lsys}_{\Sigma }\) and \(\int _{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )\); define a functor \(\mathfrak {L}_{\Sigma }\) from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.