Order-enriched solid functors

TL;DR

This paper explores order-enriched solid functors, highlighting their properties, differences from ordinary functors, and providing examples including ordered algebraic structures and vector spaces.

Contribution

It differentiates order-enriched solid functors from their ordinary counterparts and examines their behavior with respect to weighted (co)limits.

Findings

Order-enriched solid functors transfer weighted (co)limits from codomains to domains.

Examples include functors of ordered algebras and ordered vector spaces.

Order-enriched notions differ significantly from ordinary functors in specific limit behaviors.

Abstract

Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnkov\'a's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by Anghel. Our focus in this paper is on differentiating the order-enriched notion from the ordinary one, mostly in terms of the functor's behaviour with respect to specific weighted (co)limits, and on the presentation of examples, which include functors of general varieties of ordered algebras and special ones, such as ordered vector spaces.