Coherence for Skew-Monoidal Categories

I motivate a variation (due to K. Szlach\'{a}nyi) of monoidal categories called skew-monoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the well-known monoidal case. In my (to my knowledge new) version, it becomes a statement of uniqueness of normalizing rewrites. I present a proof of this coherence theorem and also formalize it fully in the dependently typed programming language Agda.Comment: In Proceedings MSFP 2014, arXiv:1406.153

Coinductive Big-Step Semantics for Concurrency

In a paper presented at SOS 2010, we developed a framework for big-step semantics for interactive input-output in combination with divergence, based on coinductive and mixed inductive-coinductive notions of resumptions, evaluation and termination-sensitive weak bisimilarity. In contrast to standard inductively defined big-step semantics, this framework handles divergence properly; in particular, runs that produce some observable effects and then diverge, are not "lost". Here we scale this approach for shared-variable concurrency on a simple example language. We develop the metatheory of our semantics in a constructive logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221

Directed Containers as Categories

Directed containers make explicit the additional structure of those containers whose set functor interpretation carries a comonad structure. The data and laws of a directed container resemble those of a monoid, while the data and laws of a directed container morphism those of a monoid morphism in the reverse direction. With some reorganization, a directed container is the same as a small category, but a directed container morphism is opcleavage-like. We draw some conclusions for comonads from this observation, considering in particular basic constructions and concepts like the opposite category and a groupoid.Comment: In Proceedings MSFP 2016, arXiv:1604.0038

Stateful Runners of Effectful Computations

AbstractWhat structure is required of a set so that computations in a given notion of computation can be run statefully with this set as the state set? For running nondeterministic computations statefully, a resolver structure is needed; for interactive I/O computations, a “responder-listener” structure is necessary; to be able to serve stateful computations, the set must carry the structure of a lens. We show that, in general, to be a stateful runner of computations for a monad corresponding to a Lawvere theory (defined as a set equipped with a monad morphism between the given monad and the state monad for this set) is the same as to be a comodel of the theory, i.e., a coalgebra of the corresponding comonad. We work out a number of instances of this observation and also compare runners to handlers

Antifounded Coinduction in Type Theory

Capretta, Vene and myself have previously [2] studied antifounded algebras as algebras supporting a form of coinduction in a category-theoretic setting, arising as the dual concept of wellfounded coalgebras. Wellfounded coalgebras are a category theorist’s take on coalgebras supporting induction, introduced by Taylor [4]. Informally, wellfoundedness of a coalgebra means that a subobject of its carrier, called the wellfounded part, is isomorphic to the carrier. Under mild conditions, wellfounded coalgebras are the same as recursive coalgebras in the sense of Osius [3], i.e., coalgebras supporting recursion. Categorical wellfounded coalgebras are intimately related to the type-theoretic method of Bove and Capretta [1] for justifying function definitions by recursionand proofs by induction on a coalgebra. The idea of the method is this: One defines the wellfounded part of the coalgebra in terms of an inductively defined predicate of the carrier, to then rewrite the given “general” recursive definition or inductive proof on the carrier as a “structural ” recursive definition or inductive proof on the wellfounded part (these are systematic constructions). Separately, one establishes (in some way) that the carrier is contained in the wellfounded part

The Sequent Calculus of Skew Monoidal Categories

International audienc

Variations on Noetherianness

In constructive mathematics, several nonequivalent notions of finiteness exist. In this paper, we continue the study of Noetherian sets in the dependently typed setting of the Agda programming language. We want to say that a set is Noetherian, if, when we are shown elements from it one after another, we will sooner or later have seen some element twice. This idea can be made precise in a number of ways. We explore the properties and connections of some of the possible encodings. In particular, we show that certain implementations imply decidable equality while others do not, and we construct counterexamples in the latter case. Additionally, we explore the relation between Noetherianness and other notions of finiteness.Comment: In Proceedings MSFP 2016, arXiv:1604.0038