Lallement functor is a weak right multiadjoint

Abstract

For a plural signature $\Sigma$ and with regard to the category $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$, of naturally preordered idempotent $\Sigma$-algebras and surjective homomorphisms, we define a contravariant functor $\mathrm{Lsys}_{\Sigma}$ from $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ to $\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{LAlg}(\Sigma)$, of $\mathbf{I}$-semi-inductive Lallement systems of $\Sigma$-algebras, and a covariant functor $(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}$ the category $(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $\Sigma$-algebra and $f\colon \mathbf{A}\longrightarrow \mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $\int^{\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}}\mathrm{Lsys}_{\Sigma}$ and $\int_{\mathsf{NPIAlg}(\Sigma)_{\mathsf{s}}}(\mathsf{Alg}(\Sigma)\,{\downarrow_{\mathsf{s}}}\, \cdot)$; 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 P{\l}onka functor and the Lallement functor.