Models and theories of λ-calculus

In this paper we briefly summarize the contents of Manzonetto’s PhD thesis [35] which concerns denotational semantics and equational/order theories of the pure untyped λ-calculus. The main research achievements include: (i) a general construction of λ-models from reflexive objects in (possibly non-well-pointed) categories; (ii) a Stonestyle representation theorem for combinatory algebras; (iii) a proof that no effective λ-model can have λβ or λβη as its equational theory (this can be seen as a partial answer to an open problem introduced by HonsellRonchi Della Rocca in 1984). These results, and others, have been published in three conference papers [36, 10, 15] and a journal paper [37]; a further journal paper has been submitted [9].