Digraph functors which admit both left and right adjoints

TL;DR

This paper explores conditions under which certain digraph functors, which have adjoints, also admit right adjoints, and examines their connection to homomorphism dualities.

Contribution

It identifies necessary conditions for digraph functors with adjoints to admit right adjoints and provides examples and connections to homomorphism dualities.

Findings

Necessary conditions for right adjoints are established.

Examples where these conditions are satisfied are provided.

A link between right adjoints and homomorphism dualities is discussed.

Abstract

For our purposes, two functors {\Lambda} and {\Gamma} are said to be respectively left and right adjoints of each other if for any digraphs G and H, there exists a homomorphism of {\Lambda}(G) to H if and only if there exists a homomorphism of G to {\Gamma}(H). We investigate the right adjoints characterised by Pultr in [A. Pultr, The right adjoints into the categories of relational systems, in Reports of the Midwest Category Seminar, IV, volume 137 of Lecture Notes in Mathematics, pages 100-113, Berlin, 1970]. We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where these necessary conditions are satisfied, and the right adjoint indeed exists. Finally, we discuss a connection between these right adjoints and homomorphism dualities.