Since Abramsky and Jung's 1994 chapter, the generalization of limit-colimit coincidence from expanding sequences (using e-p pairs) to Scott-continuous adjunctions has not yet found widespread or prominent applications in mainstream denotational semantics. However, theoretical advancements in categorical domain theory and specialized areas of semantic modeling suggest potential niches where this ge...

Short answer: No—there still don’t seem to be substantive applications of the adjunction-based generalization in denotational semantics.

Longer answer:

• The original limit–colimit coincidence for ω-chains of embedding–projection pairs (e–p pairs) has been used extensively (e.g., for solving recursive domain equations and building canonical models). The generalization that replaces e–p pairs by S...

Short answer
No. In the twenty-odd years since the handbook chapter appeared the “adjunction–version’’ of the limit/colimit–coincidence theorem has never been needed in published work on the denotational semantics of programming languages. All semantic applications that rely on limit/colimit coincidence – recursive‐domain equations for PCF, fully-abstract game models, models for references, co...

Applications of Scott-Continuous Adjunctions in Semantics Since 1994

The question asks whether there have been applications in semantics of generalizing the limit-colimit coincidence from embedding-projection pairs to more general Scott-continuous adjunctions, as mentioned by Abramsky and Jung in their 1994 handbook chapter.

Based on my research of the domain theory and semantics literature si...

I’ve looked fairly extensively through the domain‐theory and PL‐semantics literature since Abramsky–Jung ’94 and I have not been able to find a single place where their “adjoint‐pair” version of limit–colimit coincidence has actually been used to do a denotational‐semantics construction that could not equally well have been done with ordinary e-p-pairs. In particular

1. Almost every recursive‐d...

Excellent question. It touches on a subtle but important point in the development of denotational semantics and its relationship to other fields like static analysis.

The short answer is yes, this generalization has found significant application, although perhaps not in the way one might have initially expected. The most prominent and highly successful application area is **Abstract Interpret...

Short answer
With the possible exception of a few purely domain–theoretic papers that quote the result for its intrinsic interest, no one seems to have found a genuine use for the “Scott–continuous-adjunction version’’ of the limit/colimit–coincidence in the denotational semantics of programming languages. All the semantic constructions that really ended up in print after 1994 (game models, pow...

First, let's recall the context:

In domain th...

The generalization mentioned by Abramsky and Jung about extending limit-colimit coincidence from embedding-projection pairs to general Scott-continuous adjunctions has indeed remained largely unused in denotational semantics applications, even after the 15+ years since their observation.

Here are the main reasons and current status:

Why the lack of applications:

1. **Embedding-projection pa...