Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as the Galois correspondence. In particular, the covering space corresponding to the trivial group is the universal covering, which is a "1-connected" variant of the original space, in the sense that it has the same homotopy groups, except for the first one which is trivial. In this article, we formalize this correspondence in homotopy type theory, a variant of Martin-Löf type theory in which types can be interpreted as spaces (up to homotopy). Along the way, we develop an n-dimensional generalization of covering spaces. Moreover, in order to demonstrate the applicability of our approach, we formally classify the covering of lens spaces and explain how to construct the Poincaré homology sphere.

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A series of works has established rewriting as an essential tool in order to prove coherence properties of algebraic structures, such as MacLane's coherence theorem for monoidal categories, based on the observation that, under reasonable assumptions, confluence diagrams for critical pairs provide the required coherence axioms. We are interested here in extending this approach simultaneously in two directions. Firstly, we want to take into account situations where coherence is partial, in the sense that it only applies to a subset of the structural morphisms. Secondly, we are interested in structures which are cartesian in the sense that variables can be duplicated or erased. We develop theorems and rewriting techniques in order to achieve this, first in the setting of abstract rewriting systems, and then extend them to term rewriting systems, suitably generalized to take coherence into account. As an illustration of our results, we explain how to recover the coherence theorems for monoidal and symmetric monoidal categories.

International audience

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Zm which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of Zm. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m = 2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.

General coherence theorems are constructed that yield explicit presentations of categorical and algebraic objects. The categorical structures involved are finitary discrete Lawvere 2-theories, though they are approached within the language of term rewriting theory. Two general coherence theorems are obtained. The first applies to terminating and confluent rewriting 2-theories. This result is exploited to construct systematic presentations for the higher Thompson groups and the Higman-Thompson groups. The presentations are categorically interesting as they arise from higher-arity analogues of the Stasheff/Mac Lane coherence axioms, which involve phenomena not present in the classical binary axioms. The second general coherence theorem holds for 2-theories that are not necessarily confluent or terminating and is used to construct a new proof of coherence for iterated monoidal categories, which arise as categorical models of iterated loop spaces and fail to be confluent.

International audience

Abstract Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which are adapted to mechanized computations and include Gray categories as particular cases. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squier’s theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, and Frobenius monoids.

The area of fault-tolerant distributed computability is concerned with the solvability of decision tasks in various computational models where the processes might crash. A very successful approach to prove impossibility results in this context is that of combinatorial topology, started by Herlihy and Shavit’s paper in 1999. They proved that, for wait-free protocols where the processes communicate through read/write registers, a task is solvable if and only if there exists some map between simplicial complexes satisfying some properties. This approach was then extended to many different contexts, where the processes have access to various synchronization and communication primitives. Usually, in those cases, the existence of a simplicial map from the protocol complex to the output complex is taken as the definition of what it means to solve a task. In particular, no proof is provided of the fact that this abstract topological definition agrees with a more concrete operational definition of task solvability. In this paper, we bridge this gap by proving a version of Herlihy and Shavit’s theorem that applies to any kind of object. First, we start with a very general way of specifying concurrent objects, and we define what it means to implement an object B in a computational model where the processes are allowed to communicate through shared objects A_1, ..., A_k. Then, we derive the notion of a decision task as a special case of concurrent object. Finally, we prove an analogue of Herlihy and Shavit’s theorem in this context. In particular, our version of the theorem subsumes all the uses of the combinatorial topology approach that we are aware of.

The notion of pasting diagram is central in the study of strict ω -categories: it encodes a collection of morphisms for which the composition is defined unambiguously. As such, we expect that a pasting diagram itself describes an ω-category which is freely generated by the cells constituting it. In practice, it seems very difficult to characterize this notion in full generality and various definitions have been proposed with the aim of being reasonably easy to compute with, and including common examples (e.g. cubes or orientals). One of the most tractable such structure is parity complexes, which uses sets of cells in order to represent the boundaries of a cell. In this work, we first show that parity complexes do not satisfy the aforementioned freeness property by providing a mechanized proof in Agda. Then, we propose a new formalism that satisfies the freeness property and which can be seen as a corrected version of parity complexes.

Opetopes are algebraic descriptions of shapes corresponding to compositions in higher dimensions. As such, they offer an approach to higher-dimensional algebraic structures, and in particular, to the definition of weak ω-categories, which was the original motivation for their introduction by Baez and Dolan. They are classically defined inductively (as free operads in Leinster's approach, or as zoom complexes in the formalism of Kock et al.), using abstract constructions making them difficult to manipulate with a computer. Here, we present a purely syntactic description of opetopes and opetopic sets as a sequent calculus. Our main result is that well-typed opetopes in our sense are in bijection with opetopes as defined in the more traditional approaches. We expect that the resulting structures can serve as natural foundations for mechanized tools based on opetopes.

Linearizability is a widely accepted notion of correctness for concurrent objects. Recent research has investigated redefining linearizability for particular hardware weak memory models, in particular for TSO. In this paper, we provide an overview of this research and show that such redefinitions of linearizability are not required: under an interpretation of specification behaviour which abstracts from weak memory effects, the standard definition of linearizability is sound and complete on all hardware weak memory models. We prove our result with respect to a definition of object refinement which takes a weak memory model as a parameter. The main consequence of our findings is that we can leverage the range of existing techniques and tools for standard linearizability when verifying concurrent objects running on hardware weak memory models.

We identify a general principle of distributed computing: one cannot force two processes running in parallel to see each other. This principle is formally stated in the context of asynchronous processes communicating through shared objects, using trace-based semantics. We prove that it holds in a reasonable computational model, and then study the class of concurrent specifications which satisfy this property. This allows us to derive a Galois connection theorem for different variants of linearizability.

We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie's definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck's approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, called a coherator. The coherator encodes all operations required to be present in an ω-category: both the compositions of pasting schemes as well as their coherences. Our main contribution is to provide a canonical type-theoretical characterization of pasting schemes as contexts which can be derived from inference rules. Finally, we present an implementation of a corresponding proof system.

International audience

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by the means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation (in the spirit of rewriting modulo), which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one is based on restricting to objects which are normal forms, one consists in turning equational generators into identities (i.e. considering a quotient category), and one consists in formally adding inverses to equational generators (i.e. localizing the category). We show that, under suitable coherence conditions on the presentation, the three constructions coincide, thus generalizing celebrated results on presentations of groups. We illustrate our techniques on a non-trivial example, and hint at a generalization for 2-categories.

One of the most used algorithm in rewriting theory is the Knuth-Bendix completion procedure which starts from a terminating rewriting system and iteratively adds rules to it, trying to produce an equivalent convergent rewriting system. It is in particular used to study presentations of monoids, since normal forms of the rewriting system provide canonical representatives of words modulo the congruence generated by the rules. Here, we are interested in extending this procedure in order to retrieve information about the low-dimensional homotopy properties of a monoid. We therefore consider the notion of coherent presentation, which is a generalization of rewriting systems that keeps track of the cells generated by confluence diagrams. We extend the Knuth-Bendix completion procedure to this setting, resulting in a homotopical completion procedure. It is based on a generalization of Tietze transformations, which are operations that can be iteratively applied to relate any two presentations of the same monoid. We also explain how these transformations can be used to remove useless generators, rules, or confluence diagrams in a coherent presentation, thus leading to a homotopical reduction procedure. Finally, we apply these techniques to the study of some examples coming from representation theory, to compute minimal coherent presentations for them: braid, plactic and Chinese monoids.

Hybrid systems are a widely used model to represent and reason about control-command systems. In an industrial context, these are often implemented in Simulink and their validity is checked by performing many numerical simulations in order to test their behavior with various possible inputs. In this article, we present a tool named HySon which performs set-based simulation of hybrid systems with uncertain parameters, expressed in Simulink. Our tool handles advanced features such as non-linear operations, zero-crossing events or discrete sampling. It is based on well-known efficient numerical algorithms that were adapted to handle set-based domains. We demonstrate the performance of our method on various examples.

Game semantics describe the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterise definable strategies, that is, strategies that actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. In this paper we present an original methodology to achieve this task, which requires a combination of advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model using generators and relations: these strategies can be generated from a finite set of atomic strategies, and the equality between strategies admits a finite axiomatisation, and this equational structure corresponds to a polarised variation of the bialgebra notion. The work described in this paper thus forms a bridge between algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanised analysis of causality in programming languages.

Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories, which can be seen as a generalization of term rewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2-categories.

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order prepositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages.