A Cellular Howe Theorem

Abstract

We introduce a categorical framework for operational semantics, in which we define substitution-closed bisimilarity, an abstract analogue of the open extension of Abramsky's applicative bisimilarity. We furthermore prove a congruence theorem for substitution-closed bisimilarity, following Howe's method. We finally demonstrate that the framework covers the call-by-name and call-by-value variants of -calculus in big-step style. As an intermediate result, we generalise the standard framework of Fiore et al. for syntax with variable binding to the skew-monoidal case.

Formal metatheory of second-order abstract syntax

“…In our setting of families and coalgebras, a more refined notion of strength is required (similar to the structural strength of Borthelle et al [2020]). A coalgebraic strength for 𝐹 is a transformation str : 𝐹 ( P , X ) ( P , 𝐹 X ) where P is a pointed -coalgebra and X is a sorted family.…”

“…Concerning (𝑖), moving onto families with skewmonoidal structure is not enough to avoid the need for quotienting (e.g. Borthelle et al [2020] achieve the freeness proof in a skew-monoidal setting only with the aid of a general lemma of Fiore and Saville [2017, Theorem 4.8] that relies on the presentation of initial algebras as colimits of 𝜔-chains, rather than as inductive data types); while, concerning (𝑖𝑖), the fact that initial algebras in presheaves can be lifted from initial algebras in families had to be given mathematical grounding. As an upshot, using families (indexed types), instead of presheaves, leads to a lightweight practical formalisation that suits the intrinsically-typed setting well.…”

“…Recent work by Borthelle et al [2020] proposed that the presheaf model be adapted to a setting of sorted families by axiomatising the renaming structure and its interaction with initial algebra semantics -ideas which have been explored informally by Allais et al [2017] and Kaiser et al [2018] as well. To note, however, is that their skew-monoidal [Szlachányi 2012] approach still depends on categorical theory and results that lead to an impractical formalisation: for example, the syntax of terms is presented by means of the colimit of an 𝜔-chain (and therefore represented by equivalence classes), rather than by an inductive data type.…”

Recursion and Sequentiality in Categories of Sheaves

“…We think of this colimit as the finite vertical natural numbers. In particular, there is a cocone over diagram (2) with apex ω given by L n 1…”

“…Although the focus of this paper is on a simple PCF-like language, a broader agenda is to combine this analysis of recursion and sequentiality with recent sheaf-based models for other phenomena, including concurrency (e.g. [2]), differentiable programming [42,17], probabilistic programming [15], quantum programming [26] and homotopy type theory [1]. The broader context, then, is to use sheaf-based constructions as a principled approach to building sophisticated models of increasingly elaborate languages.…”