# Adjoint relations for the category of local dcpos

Document Type : Research Paper

Authors

Shaanxi Normal University

Abstract

In this paper, we consider the forgetful functor from the category {\bf LDcpo} of local dcpos (respectively, {\bf Dcpo} of dcpos) to  the category {\bf Pos} of posets (respectively, {\bf LDcpo} of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree $S$-ldcpos over a local dcpo, where $S$ is a local dcpo monoid. The main results are:
(1) The forgetful functor $U$ : {\bf LDcpo} $\longrightarrow$ {\bf Pos} has a left adjoint, but does not have a right adjoint;
(2) The inclusion functor $I$ : {\bf Dcpo} $\longrightarrow$ {\bf LDcpo} has a left adjoint, but does not have a right adjoint;
(3) The forgetful functor $U$ : {\bf LDcpo}-$S$ $\longrightarrow$ {\bf LDcpo} has
both left and right adjoints;
(4) If $(S,\cdot,1)$ is a good ldcpo-monoid, then the forgetful functor $U$: {\bf LDcpo}-$S$ $\longrightarrow$ {\bf Pos}-$S$ has a left adjoint.

Highlights

Dedicated  to Bernhard Banaschewski on the occasion of his $90^{th}$ birthday

Keywords

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