Lallement functor is a weak right multiadjoint

Abstract

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

31 pages. arXiv admin note: text overlap with arXiv:2305.03581