Search

We study generic representations of general linear groups over a finite ring R with coefficients in a field k in which the cardinality of R is invertible, that is functors from finitely-generated projective R-modules to k-vector spaces. We obtain especially a classification of such simple representations, what allows to prove a conjecture of Djament-Touzé-Vespa on dimensions taken by such a functor.; On étudie les représentations génériques des groupes linéaires sur un anneau fini R à coefficients dans un corps k dans lequel le cardinal de R est inversible, c'est-à-dire les foncteurs depuis les R-modules projectifs de type fini vers les k-espaces vectoriels. On obtient en particulier une classification de ces représentations simples qui permet de démontrer une conjecture de Djament-Touzé-Vespa sur les dimensions prises par un tel foncteur.

We propose a construction of the monoidal envelope of $\infty$-operads in the model of Segal dendroidal spaces, and use it to define cocartesian fibrations of such. We achieve this by viewing the dendroidal category as a "plus construction" of the category of pointed finite sets, and work in the more general language of algebraic patterns for Segal conditions. Finally, we rephrase Lurie's definition of cartesian structures as exhibiting the categorical fibrations coming from envelopes, and deduce a straightening/unstraightening equivalence for dendroidal spaces., Comment: 24 pages. Preliminary version, drafty in places. Comments welcome

Large-scale software verification relies critically on the use of compositional languages, semantic models, specifications, and verification techniques. Recent work on certified abstraction layers synthesizes game semantics, the refinement calculus, and algebraic effects to enable the composition of heterogeneous components into larger certified systems. However, in existing models of certified abstraction layers, compositionality is restricted by the lack of encapsulation of state. In this paper, we present a novel game model for certified abstraction layers where the semantics of layer interfaces and implementations are defined solely based on their observable behaviors. Our key idea is to leverage Reddy's pioneer work on modeling the semantics of imperative languages not as functions on global states but as objects with their observable behaviors. We show that a layer interface can be modeled as an object type (i.e., a layer signature) plus an object strategy. A layer implementation is then essentially a regular map, in the sense of Reddy, from an object with the underlay signature to that with the overlay signature. A layer implementation is certified when its composition with the underlay object strategy implements the overlay object strategy. We also describe an extension that allows for non-determinism in layer interfaces. After formulating layer implementations as regular maps between object spaces, we move to concurrency and design a notion of concurrent object space, where sequential traces may be identified modulo permutation of independent operations. We show how to express protected shared object concurrency, and a ticket lock implementation, in a simple model based on regular maps between concurrent object spaces.

59 pages; We prove that Mann-Robalo's construction of the brane action extends to general coherent ∞-operads, with possibly multiple colors and non-contractible spaces of unary operations. This requires to establish two results regarding spaces of extensions that were left unproven in the aforementioned construction. First, we show that Lurie's and Mann-Robalo's models for such spaces are equivalent. Second, we prove that the space of extensions in the sense of Lurie is not in general equivalent to the homotopy fiber of the associated forgetful morphism, but rather to its homotopy quotient by the ∞-groupoid of unary operations, correcting an oversight in existing literature. As an application, we obtain that the ∞-operads of B-framed little disks are coherent and therefore yield new operations on spaces of branes of perfect derived stacks.; On montre que la construction par Mann-Robalo de l'action de membranes s'étend aux ∞-opérades cohérentes générales, colorées et d'espaces d'opérations unaires non-contractiles. Cela requiert d'établir deux résultats concernant les espaces d'extensions laissés sans preuve dans la construction précédente. Le premier établit l'équivalence entre les modèles de Lurie et de Mann-Robalo pour ces espaces. Au travers du second résultat, on montre que l'espaces des extensions au sens de Lurie n'est en général pas équivalent à la fibre homotopique du morphisme d'oubli associé, mais au contraire s'identifie à son quotient homotopique par l'action naturelle de l'∞-groupoïde des opérations unaires, corrigeant ainsi une erreur de la littérature existante. Finalement, on applique les résultats précédents pour démontrer la cohérence des ∞-opérades de petits disques à repères B-tordus, ce qui donne des nouvelles opérations sur les espaces de membranes des champs dérivés parfaits.

In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets., Comment: This text is still in draft form. A new version should quickly replace this one

28 pages; International audience; This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.

For any object x in a category C it is possible to define the category of Beck modules over x as the category Ab(C/x) of abelian group objects in the category C/x. We can deduce from this construction, at least for any locally presentable category, the notion of cotangent module or module of differentials $\Omega$x of x in Ab(C/x).In the case of the category Algk of commutative k-algebras over a ring k, the category of Beck modules Ab(Algk/A) over a k-algebra A is equivalent to the category ModA of A-modules and the cotangent module is equal to the module of K{\"a}hler differentials of A.The aim of this article is to prove for any locally presentable category some results which generalize the classical properties of modules of K{\"a}hler differentials., Comment: in French language

Twisted separable functors generalize the separable functors of Nastasescu, Van den Bergh and Van Oystaeyen, and provide a convenient tool to compare various projective dimensions. We discuss when an adjoint functor is twisted separable, obtaining a version of Rafael's Theorem in the twisted case. As an application, we show that if $R$ is Hopf-Galois object over a Hopf algebra $A$, then their Hochschild cohomological dimension coincide, provided that the cohomological dimension of $A$ is finite and that $R$ has a unital twisted trace with respect to a semi-colinear automorphism.

In this thesis, we develop a new model for the category of dg-categories. Following Rezk’s example in the case of complete Segal spaces, we define dg-Segal spaces: functors between free dg-categories of finite type and simplicial spaces, with certain added properties. We define also complete dg-Segal spaces, and we prove that there exists an equivalence between the homotopy category of dg-categories and the homotopy category of functors defined above with a model structure making the complete dg-Segal spaces into its fibrant objects.; Dans cette thèse, on développe un nouveau modèle pour la catégorie des dg-catégories. En suivant l’exemple de Rezk dans le cadre des espaces de Segal complets, on définit des espaces de dg-Segal : des foncteurs entre les dg-catégories libres de type fini et les espaces simpliciaux, avec certaines propriétés. On définit aussi des espaces de dg-Segal complets, et on montre qu’il existe une équivalence entre la catégorie homotopique des dg-catégories et la catégorie homotopique des foncteurs décrits ci-dessus avec une structure de modèles faisant des espaces de dg-Segal complets leurs objets fibrants.

Precategories generalize both the notions of strict $n$-category and sesquicategory: their definition is essentially the same as the one of strict $n$-categories, excepting that we do not require the various interchange laws to hold. Those have been proposed as a framework in which one can express semi-strict definitions of weak higher categories: in dimension 3, Gray categories are an instance of them and have been shown to be equivalent to tricategories, and definitions of semi-strict tetracategories have been proposed, and used as the basis of proof assistants such as Globular. In this article, we are mostly interested in free precategories. Those can be presented by generators and relations, using an appropriate variation on the notion of polygraph (aka computad), and earlier works have shown that the theory of rewriting can be generalized to this setting, enjoying most of the fundamental constructions and properties which can be found in the traditional theory, contrarily to polygraphs for strict categories. We further study here why this is the case, by providing several results which show that precategories and their associated polygraphs bear properties which ensure that we have a good syntax for those. In particular, we show that the category of polygraphs for precategories form a presheaf category., Comment: 34 pages

Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched categories, enriched functors, and enriched cofunctors, whose horizontal and vertical 2-categories have 2-cells given by enriched natural transformations between functors and cofunctors, respectively. Enriched lenses are defined as a compatible enriched functor and enriched cofunctor pair; weighted lenses, which were introduced by Perrone, are precisely lenses enriched in weighted sets. Several other examples are also studied in detail., 16 pages

International audience; The language of Algebraic Geometry combines two complementary and dependent levels of discourse: on the geometric side, schemes define spaces of the same cohesive nature as manifolds ; on the vectorial side, every scheme X comes equipped with a symmetric monoidal category of quasicoherent modules, which may be seen as generalised vector bundles on the scheme X. In this paper, we use the functor of points approach to Algebraic Geometry developed by Grothendieck in the 1970s to establish that every covariant presheaf X on the category of commutative rings-and in particular every scheme Xcomes equipped "above it" with a symmetric monoidal closed category PshModX of presheaves of modules. This category PshModX defines moreover a model of intuitionistic linear logic, whose exponential modality is obtained by glueing together in an appropriate way the Sweedler dual construction on ring algebras. The purpose of this work is to establish on firm mathematical foundations the idea that linear logic should be understood as a logic of generalised vector bundles, in the same way as dependent type theory is understood today as a logic of spaces up to homotopy.

We give an explicit correspondence between stated skein algebras, which are defined via explicit relations on stated tangles in [Costantino F., L\^e T.T.Q., arXiv:1907.11400], and internal skein algebras, which are defined as internal endomorphism algebras in free cocompletions of skein categories in [Ben-Zvi D., Brochier A., Jordan D., J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or in [Gunningham S., Jordan D., Safronov P., arXiv:1908.05233]. Stated skein algebras are defined on surfaces with multiple boundary edges and we generalise internal skein algebras in this context. Now, one needs to distinguish between left and right boundary edges, and we explain this phenomenon on stated skein algebras using a half-twist. We prove excision properties of multi-edges internal skein algebras using excision properties of skein categories, and agreeing with excision properties of stated skein algebras when $\mathcal{V} = \mathcal{U}_{q^2}(\mathfrak{sl}_2)\text{-}{\rm mod}^{\rm fin}$. Our proofs are mostly based on skein theory and we do not require the reader to be familiar with the formalism of higher categories.

A correspondence functor is a functor from the category of finite sets and correspondences to the category of k k -modules, where k k is a commutative ring. We determine exactly which simple correspondence functors are projective. We also determine which simple modules are projective for the algebra of all relations on a finite set. Moreover, we analyze the occurrence of such simple projective functors inside the correspondence functor F F associated with a finite lattice and we deduce a direct sum decomposition of F F .

In earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced by Street and Burroni for strict categories, to all algebraic globular higher categories. In this work, we refine this perspective and introduce new constructions and properties for this class of higher categories. In particular, we define the notion of cellular extension and its associated free construction, from which we obtain another definition of polygraph and the adjunction between globular algebras and polygraphs. We moreover introduce two criteria allowing one to use most of the constructions of this article without having to describe explicitly the underlying globular monad.; Dans des travaux antérieurs, Batanin a montré qu'une classe importante de définitions de catégories supérieures pouvaient être appréhendées simplement en les voyant comme des monades sur les ensembles globulaires. Cela lui a permis de généraliser la notion de polygraphe, initialement introduite par Street et Burroni pour les catégories strictes, à toutes les catégories supérieures globulaires algébriques. Dans ce travail, on raffine ce point de vue et on introduit de nouvelles constructions et propriétés pour cette classe de catégories supérieures. En particulier, on définit la notion d'extension cellulaire et la construction libre qui lui est associée, et on obtient à partir de celle-là une autre définition des polygraphes et l'adjonction entre les algèbres globulaires et les polygraphes. On introduit de plus deux critères permettant l'utilisation de la plupart des constructions de cet article sans avoir à décrire explicitement la monade globulaire sous-jacente.

In this short paper, using category theory, we argue that logical rules can be seen as fractions and logics as limit sketches.

In this paper, we initiate the generalisation of the operadic calculus which governs the properties of homotopy algebras to a properadic calculus which governs the properties of homotopy gebras over a properad. In this first article of a series, we generalise the seminal notion of infini-morphisms and the ubiquitous homotopy transfer theorem. As an application, we recover the homotopy properties of involutive Lie bialgebras developed by Cieliebak--Fukaya--Latschev and we produce new explicit formulas., Submitted version

Coalgebras for an endofunctor provide a category-theoretic framework for modeling a wide range of state-based systems of various types. We provide an iterative construction of the reachable part of a given pointed coalgebra that is inspired by and resembles the standard breadth-first search procedure to compute the reachable part of a graph. We also study coalgebras in Kleisli categories: for a functor extending a functor on the base category, we show that the reachable part of a given pointed coalgebra can be computed in that base category.

This paper studies the homotopy theory of algebras and homotopy algebras over an operad. It provides an exhaustive description of their higher homotopical properties using the more general notion of morphisms called infinity-morphisms. The method consists in using the operadic calculus to endow the category of coalgebras over the Koszul dual cooperad or the bar construction with a new type of model category structure, Quillen equivalent to that of algebras. We provide an explicit homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy category, and we endow the category of homotopy algebras with an infinity-category structure., 33 pages. Proof of Theorem 2.1-(2) improved

We construct a Quillen equivalence between marked simplicial sets and (marked) Segal precategories enriched in marked simplicial sets. From this, we can deduce that Verity's $n$-complicial sets are a model of $(\infty,n)$-categories in the sense of Barwick and Schommer-Pries., Comment: This text is a working paper. All comments, both mathematical and redactional, are welcome!

Global transformations provide a categorical framework for capturing synchronous rewriting systems, generalizing cellular automata to dynamical systems over dynamic spaces. Originally developed for addressing deterministic dynamical systems, the presented work raises the question of non-determinism. While a usual approach is to develop a general non-deterministic setting where deterministic systems can be retrieved as a specific case, we show here that by choosing the right parametrization, global transformations can already be used to handle non-determinism. Context-free Lindenmayer systems, already shown to be captured by global transformation in the deterministic case, are used to illustrate the approach. From this concrete example, the formal obstructions are exhibited, leading to a solution involving a 2-categorical monad and its associated Kleisli construction., LIPIcs, Vol. 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), pages 49:1-49:13

It is known that the quantale of sup-preserving maps from a complete lattice to itself is a Frobenius quantale if and only if the lattice is completely distributive. Since completely distributive lattices are the nuclear objects in the autonomous category of complete lattices and sup-preserving maps, we study the above statement in a categorical setting. We introduce the notion of Frobenius structure in an arbitrary autonomous category, generalizing that of Frobenius quantale. We prove that the monoid of endomorphisms of a nuclear object has a Frobenius structure. If the environment category is star-autonomous and has epi-mono factorizations, a variant of this theorem allows to develop an abstract phase semantics and to generalise the previous statement. Conversely, we argue that, in a star-autonomous category where the monoidal unit is a dualizing object, if the monoid of endomorphisms of an object has a Frobenius structure and the monoidal unit embeds into this object as a retract, then the object is nuclear., LIPIcs, Vol. 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023), pages 18:1-18:20

In 1900 David Hilbert published his famous list of 23 problems. The sixth of them-the axiomatization of Physics-remains partially unsolved. In this work we will give a gentle introduction and a brief review to one of the most recent and formal approaches to this problem, based on synthetic higher categorical languages. This approach, developed by Baez, Schreiber, Sati, Fiorenza, Freed, Lurie and many others, provides a formalization to the notion of classical field theory in terms of twisted differential cohomologies in cohesive (∞, 1)-topos. Furthermore, following the Atiyah-Witten functorial style of topological quantum field theories, it provides a nonperturbative quantization for classical field theories whose underlying cohomology satisfies orientability and duality conditions. We follow a pedagogical and almost non-technical style trying to minimize the mathematical prerequisites. No categorical background is required.

In this paper, we study by rewriting methods the presentations of the super plactic monoid of type A which is related to the representations of the general linear Lie superalgebra. We construct a convergent presentation of this monoid whose generators are super columns and whose rules are defined by insertions on super tableaux over a signed alphabet. We extend this presentation into a coherent one whose syzygies are defined as relations among insertion algorithms. Finally, we reduce this coherent presentation to a Tietze equivalent one over the initial signed alphabet. Such coherent presentations are used for representations of super plactic monoids by describing their actions on categories., 20 pages. arXiv admin note: text overlap with arXiv:1609.01460, arXiv:2105.07819

Accepted to CPP 2022; International audience; In previous work (“From signatures to monads in UniMath”), we described a category-theoretic construction of abstract syntax from a signature, mechanized in the UniMath library based on the Coq proof assistant. In the present work, we describe what was necessary to generalize that work to account for simply-typed languages. First, some definitions had to be generalized to account for the natural appearance of non-endofunctors in the simply typed case. As it turns out, in many cases our mechanized results carried over to the generalized definitions without any code change. Second, an existing mechanized library on ω-cocontinuous functors had to be extended by constructions and theorems necessary for constructing multi-sorted syntax. Third, the theoretical framework for the semantical signatures had to be generalized from a monoidal to a bi-categorical setting, again to account for non-endofunctors arising in the typed case. This uses actions of endofunctors on functors with given source, and the corresponding notion of strong functors between actions, all formalized in UniMath using a recently developed library of bicategory theory. We explain what needed to be done to plug all of these ingredients together, modularly. The main result of our work is a general construction that, when fed with a signature for a simply-typed language, returns an implementation of that language together with suitable boilerplate code, in particular, a certified monadic substitution operation.

The process of replacing an arbitrary Boolean function by a bijective one, a fundamental tool in reversible computing and in cryptography, is interpreted algebraically as a particular instance of a certain group homomorphism from the X-fold cartesian power of a group G into the automorphism group of the free G-set over the set X. It is shown that this construction not only can be generalized from groups to monoids but, more generally, to internal categories in arbitrary finitely complete categories where it becomes a cartesian isomorphism between certain discrete fibrations.

PhD Thesis, 155 pages, in English (with English and French introductions); The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary monads on certain presheaf categories, generalising a well-known result due to Lawvere, B\'enabou and Linton for ordinary multisorted algebraic theories. We use this to recognise dependently typed algebraic theories for a number of classes of algebraic structures, such as small categories, n-categories, strict and weak omega-categories, planar coloured operads and opetopic sets. We then show that every locally finitely presentable category is the category of models of some dependently typed algebraic theory. Thus, with respect to their Set-models, these theories are just as expressive as GATs, essentially algebraic theories and finite limit sketches. However, dependently typed algebraic theories admit a good definition of homotopy-models in spaces, via a left Bousfield localisation of a global model structure on simplicial presheaves. Some cases, such as certain "idempotent opetopic theories", have a rigidification theorem relating homotopy-models and (strict) simplicial models. The second part of this dissertation concerns localisations of presentable $(\infty,1)$-categories. We give a definition of "pre-modulator", and show that every accessible orthogonal factorisation system on a presentable $(\infty,1)$-category can be generated from a pre-modulator by iterating a plus-construction resembling that of sheafification. We give definitions of "modulator" and "left-exact modulator", and prove that they correspond to those factorisation systems that are modalities and left-exact modalities respectively. Thus every left-exact localisation of an $\infty$-topos is obtained by iterating the plus-construction associated to a left-exact modulator.; Dans la première partie de cette thèse, nous proposons une définition de « théorie algébrique à types/sortes dépendants » qui généralise les théories algébriques ordinaires de Lawvere--Bénabou. Les théories algébriques à sortes dépendantes, en notre sens, forment une sous-classe stricte des « théories algébriques généralisées » de Cartmell. Nous démontrons un théorème de classification pour les théories algébriques à sortes dépendantes, et nous utilisons ce théorème pour montrer l'existence de plusieurs de ces théories --- parmi elles, les théories des petites catégories, des n-catégories, des omega-catégories strictes et faibles, des opérades planaires colorées, et des ensembles opétopiques. Nous étudions le cas opétopique en détail. Nous montrons également une équivalence de Morita entre les théories algébriques à sortes dépendantes et les théories essentiellement algébriques, et nous concluons que chaque catégorie localement finiment présentable admet une description comme catégorie des modèles d'une théorie algébrique à sortes dépendantes. Nous proposons également les définitions des modèles homotopiques strictes et faibles d'une théorie algébrique à sortes dépendantes, et nous montrons un théorème de rigidification dans un cas particulier. La deuxième partie de cette dissertation concerne les localisations refléctives accessibles des infini-catégories localement présentables. Nous donnons une définition de « pré-modulateur » et montrons une correspondance entre les pré-modulateurs et les systèmes de factorisations accessibles sur une infini-catégorie localement présentable. Nous montrons également que chaque tel système de factorisation est engendré à partir d'un pré-modulateur par itération transfinie d'une « construction plus ». Nous proposons les définitions de « modulateur » et de « modulateur exact à gauche » et nous montrons des correspondances avec les modalités et les modalités exactes à gauche respectivement.

The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary monads on certain presheaf categories, generalising a well-known result due to Lawvere, B\'enabou and Linton for ordinary multisorted algebraic theories. We use this to recognise dependently typed algebraic theories for a number of classes of algebraic structures, such as small categories, n-categories, strict and weak omega-categories, planar coloured operads and opetopic sets. We then show that every locally finitely presentable category is the category of models of some dependently typed algebraic theory. Thus, with respect to their Set-models, these theories are just as expressive as GATs, essentially algebraic theories and finite limit sketches. However, dependently typed algebraic theories admit a good definition of homotopy-models in spaces, via a left Bousfield localisation of a global model structure on simplicial presheaves. Some cases, such as certain "idempotent opetopic theories", have a rigidification theorem relating homotopy-models and (strict) simplicial models. The second part of this dissertation concerns localisations of presentable $(\infty,1)$-categories. We give a definition of "pre-modulator", and show that every accessible orthogonal factorisation system on a presentable $(\infty,1)$-category can be generated from a pre-modulator by iterating a plus-construction resembling that of sheafification. We give definitions of "modulator" and "left-exact modulator", and prove that they correspond to those factorisation systems that are modalities and left-exact modalities respectively. Thus every left-exact localisation of an $\infty$-topos is obtained by iterating the plus-construction associated to a left-exact modulator., Dans la première partie de cette thèse, nous proposons une définition de « théorie algébrique à types/sortes dépendants » qui généralise les théories algébriques ordinaires de Lawvere--Bénabou. Les théories algébriques à sortes dépendantes, en notre sens, forment une sous-classe stricte des « théories algébriques généralisées » de Cartmell. Nous démontrons un théorème de classification pour les théories algébriques à sortes dépendantes, et nous utilisons ce théorème pour montrer l'existence de plusieurs de ces théories --- parmi elles, les théories des petites catégories, des n-catégories, des omega-catégories strictes et faibles, des opérades planaires colorées, et des ensembles opétopiques. Nous étudions le cas opétopique en détail. Nous montrons également une équivalence de Morita entre les théories algébriques à sortes dépendantes et les théories essentiellement algébriques, et nous concluons que chaque catégorie localement finiment présentable admet une description comme catégorie des modèles d'une théorie algébrique à sortes dépendantes. Nous proposons également les définitions des modèles homotopiques strictes et faibles d'une théorie algébrique à sortes dépendantes, et nous montrons un théorème de rigidification dans un cas particulier. La deuxième partie de cette dissertation concerne les localisations refléctives accessibles des infini-catégories localement présentables. Nous donnons une définition de « pré-modulateur » et montrons une correspondance entre les pré-modulateurs et les systèmes de factorisations accessibles sur une infini-catégorie localement présentable. Nous montrons également que chaque tel système de factorisation est engendré à partir d'un pré-modulateur par itération transfinie d'une « construction plus ». Nous proposons les définitions de « modulateur » et de « modulateur exact à gauche » et nous montrons des correspondances avec les modalités et les modalités exactes à gauche respectivement.

We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$ is an arbitrary (generalized) ring. The case $\Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).

Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows., Comment: 39 pages; Erratum to Compositionality 3, 3, 2021, in which the notion of reparametrization category was incorrectly axiomatized

Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets of variables. They can be organized into a cartesian bicategory, which unfortunately fails to be closed for essentially two reasons, which we address here by suitably modifying the model. Firstly, a naive closure is too large to be well-defined, which can be overcome by restricting to polynomials which are finitary. Secondly, the resulting putative closure fails to properly take the 2-categorical structure in account. We advocate here that this can be addressed by considering polynomials in groupoids, instead of sets. For those, the constructions involved into composition have to be performed up to homotopy, which is conveniently handled in the setting of homotopy type theory: we use it here to formally perform the constructions required to build our cartesian bicategory, in Agda. Notably, this requires us introducing an axiomatization in a small universe of the type of finite types, as an appropriate higher inductive type of natural numbers and bijections., Comment: In Proceedings MFPS 2021, arXiv:2112.13746

International audience; The clusters considered in this paper are seen as morphisms between small arbitrary diagrams in a given locally small category C. They have initially been introduced to extend to all small diagrams the results for filtered diagrams, by exhibiting a very basic presentation of the formula used in the definition of the category Ind(C) of ind-objects in C. They constitute a category Clu (C) which contains Ind(C). We study these clusters, their construction and composition. Thus we provide any user with the means to generate clusters and perform calculations with them. So we can give a simple proof of the fact that Clu (C) is a strict free cocompletion of C for all small diagrams, determined up to isomorphism. We compare it to some other cocompletion problems.

International audience; Global transformations form a categorical framework adapting graph transformations to describe fully synchronous rule systems on a given data structure. In this work we focus on data structures that can be captured as presheaves and study the computational aspects of such synchronous rule systems. To obtain an online algorithm, a complete study of the sub-steps within each synchronous step is done at the semantic level. This leads to the definition of accretive rule systems and a local criterion to characterize these systems. Finally an online computation algorithm for theses systems is given.

Squaring the circle is one of the oldest challenges in mathematical geometry. In 1882, it was proven that π was transcendental, and the task of Squaring the Circle was considered impossible. When it was demonstrated that squaring the circles was not possible, this was before discovering quantum mechanics. Here, we consider Heisenberg's uncertainty principle and show that it is entirely possible to square the circle.

In this paper, we present a completely radical way to investigate the main problem of symbolic dynamics, the conjugacy problem, by proving that this problem actually relates to a natural question in category theory regarding the theory of traced bialgebras. As a consequence of this theory, we obtain a systematic way of obtaining new invariants for the conjugacy problem by looking at existing bialgebras in the literature.

This paper shows how to construct coherent presentations of a class of monoids, including left-cancellative noetherian monoids containing no nontrivial invertible element and admitting a Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of Deligne's original construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to a dual braid monoid, and to some monoids which are neither Artin-Tits nor Garside. For the Artin-Tits monoid of type $\widetilde{A}_{2}$, a finite coherent presentation is given, having a finite Garside family as a generating set.

Motivated by recent work on weak distributive laws and their applications to coalgebraic semantics, we investigate the algebraic nature of semialgebras for a monad. These are algebras for the underlying functor of the monad subject to the associativity axiom alone-the unit axiom from the definition of an Eilenberg-Moore algebras is dropped. We prove that if the underlying category has coproducts, then semialgebras for a monad M are in fact the Eilenberg-Moore algebras for a suitable monad structure on the functor id + M , which we call the semifree monad M^s. We also provide concrete algebraic presentations for semialgebras for the maybe monad, the semigroup monad and the finite distribution monad. A second contribution is characterizing the weak distributive laws of the form M T => T M as strong distributive laws M^s T => T M^s subject to an additional condition., In Proceedings MFPS 2021, arXiv:2112.13746

We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive., Comment: 21 pages, comments are welcome

A topological commutative ring is said to be rigid when for every set $X$, the topological dual of the $X$-fold topological product of the ring is isomorphic to the free module over $X$. Examples are fields with a ring topology, discrete rings, and normed algebras. Rigidity translates into a dual equivalence between categories of free modules and of "topologically-free" modules and, with a suitable topological tensor product for the latter, one proves that it lifts to an equivalence between monoids in this category (some suitably generalized topological algebras) and coalgebras. In particular, we provide a description of its relationship with the standard duality between algebras and coalgebras, namely finite duality., This version only deals with the monoidalily of the equivalence

We propose a new algebraic approach to graph transformation, called the Pullback-Pushout ( pbpo ) approach, where we combine smoothly the classical modifications to a host graph specified by a first part of a rule, defined as a span of graph morphisms, with the cloning of structures specified by a second span. The motivation behind this new approach is to support cloning of structures in an elegant and efficient way. After a formal definition of the proposed approach, we demonstrate that pbpo rewriting is a conservative extension of agree and the Sesqui-Pushout approaches. Contrary to agree , we show that the proposed pbpo transformation can easily be extended to cope with attributed graphs. In general, totally attributed graphs are not closed under pbpo transformation. We propose sufficient conditions which guarantee that the attribution of transformed graphs is total. Furthermore, a pbpo transformation can affect all parts of a host graph including non local parts (i.e., parts which are outside the image of the left-hand side of a rule). We propose and discuss some conditions which ensure a form of locality of pbpo transformations.

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup C l 1 ( Z P ) of S K 1 ( Z P ) , in terms of a genetic basis of P. We also introduce a deflation map C l 1 ( Z P ) → C l 1 ( Z ( P / N ) ) , for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of S K 1 ( Z P ) , when P is an elementary abelian p-group.

In a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define an adjoint bar-cobar (or "Koszul") duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of trees defining infinity-operads. We emphasize that our methods are completely elementary and explicit., Comment: 25 pages

In his recent and exploratory work on template games and linear logic, Mellies defines sequential and concurrent games as categories with positions as objects and trajectories as morphisms, labelled by a specific synchronization template. In the present paper, we bring the idea one dimension higher and advocate that template games should not be just defined as 1-dimensional categories but as 2-dimensional categories of positions, trajectories and reshufflings (or reschedulings) as 2-cells. In order to achieve the purpose, we take seriously the parallel between asynchrony in concurrency and the Gray tensor product of 2-categories. One technical difficulty on the way is that the category $\mathbb{S} = 2$-Cat of small 2-categories equipped with the Gray tensor product is monoidal, and not cartesian. This prompts us to extend the framework of template games originally formulated by Mellies in a category $\mathbb{S}$ with finite limits, and to upgrade it in the style of Aguiar’s work on quantum groups to the more general situation of a monoidal category $\mathbb{S}$ with coreflexive equalizers, preserved by the tensor product componentwise. We construct in this way an asynchronous template game semantics of multiplicative additive linear logic (MALL) where every formula and every proof is interpreted as a labelled 2-category equipped, respectively, with the structure of Gray comonoid for asynchronous template games, and of Gray bicomodule for asynchronous strategies.

International audience; We explore several different facets of the notion of stack, and how they relate to the applications of stacks in geometry.

The framework of global transformations aims at describing synchronous rewriting systems on a given data structure. In this work we focus on the data structure of graphs. Global transformations of graphs are defined and a local criterion is given for a rule system to extend to a graph global transformation. Finally we present an algorithm, with its correction, which computes online the global transformation of a finite graph in an accretive manner.

In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis-Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed., OASIcs, Vol. 90, 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021), pages 7:1-7:12

We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external information is taken into account. For the particular use of the authors, such an open functor is described by two components: a presheaf representing the possible external influences for each input, and a classical functor from the category of elements of this presheaf to the category of results. Considering the appropriate notion of composition then leads to a bicategory. This report describes this bicategory with as little auxiliary constructions as possible and gives all the details of all the proofs needed to establish the bicategory, as explicitly as possible. Subsequent reports will give other presentations of this bicategory and compare it to other existing constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories of pseudo-monads, and profunctors (or distributors).

52 pages, in French; We show that the Street nerve of a strict $\omega$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip $\mathcal{N}(C)$ with a structure of saturated complicial set where the $n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending the unique non-trivial $n$-cell of the domain to a weakly invertible cell of $C$.; On montre que le nerf de Street $\N(C)$ d'une $\omega$-catégorie stricte $C$ est un complexe de Kan (respectivement une quasi-catégorie) si et seulement si les $n$-cellules de $C$ pour $n\geq 1$ (respectivement $n> 1$) sont faiblement inversibles. De plus, on munit $\N(C)$ d'une structure d'ensemble complicial saturé où les $n$-simplexes marqués correspondent aux morphismes du $n^{i\grave{e}me}$ oriental vers $C$ envoyant l'unique $n$-cellule non-triviale du domaine sur une cellule faiblement inversible de $C$.

International audience; In this talk we discuss an approach to formalize the notion of "regularity", focusing specially on "regularity structures" on $C^k$-manifolds and on geometric objects, such as tensors,partial differential operators, connections on bundles,etc. We present an strategy that could be used to attack existence and multiplicity problems for these regularity structures.The talk is based on joint works arXiv:1908.04442 and arXiv:1910.03113 with Rodney Josué Biezuner, and some working in progress.

The skew monoidal categories of Szlach\'anyi are a weakening of monoidal categories where the three structural laws of left and right unitality and associativity are not required to be isomorphisms but merely transformations in a particular direction. In previous work, we showed that the free skew monoidal category on a set of generating objects can be concretely presented as a sequent calculus. This calculus enjoys cut elimination and admits focusing, i.e. a subsystem of canonical derivations, which solves the coherence problem for skew monoidal categories. In this paper, we develop sequent calculi for partially normal skew monoidal categories, which are skew monoidal categories with one or more structural laws invertible. Each normality condition leads to additional inference rules and equations on them. We prove cut elimination and we show that the calculi admit focusing. The result is a family of sequent calculi between those of skew monoidal categories and (fully normal) monoidal categories. On the level of derivability, these define 8 weakenings of the (unit,tensor) fragment of intuitionistic non-commutative linear logic., Comment: In Proceedings ACT 2020, arXiv:2101.07888