Academic literature on the topic 'Grothendieck fibration'

A major ambition of systems science is to uncover the building blocks of any biological network to decipher how cellular function emerges from their interactions. Here, we introduce a graph representation of the information flow in these networks as a set of input trees, one for each node, which contains all pathways along which information can be transmitted in the network. In this representation, we find remarkable symmetries in the input trees that deconstruct the network into functional building blocks called fibers. Nodes in a fiber have isomorphic input trees and thus process equivalent dynamics and synchronize their activity. Each fiber can then be collapsed into a single representative base node through an information-preserving transformation called “symmetry fibration,” introduced by Grothendieck in the context of algebraic geometry. We exemplify the symmetry fibrations in gene regulatory networks and then show that they universally apply across species and domains from biology to social and infrastructure networks. The building blocks are classified into topological classes of input trees characterized by integer branching ratios and fractal golden ratios of Fibonacci sequences representing cycles of information. Thus, symmetry fibrations describe how complex networks are built from the bottom up to process information through the synchronization of their constitutive building blocks.

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the "fibration condition" in the dependence of the Grothendieck class on the number of spin states and of the polynomial countability condition for these Potts model hypersurfaces. We then show that a period computation, formally similar to the parametric Feynman integrals of quantum field theory, arises by considering certain thermodynamic averages. One can show that these evaluate to combinations of multiple zeta values for Potts models on polygon polymer chains, while silicate tetrahedral chains provide a candidate for a possible occurrence of non-mixed Tate periods.

This paper extends the ‘lens’ concept for view updating in Computer Science beyond the categories of sets and ordered sets. It is first shown that a constant complement view updating strategy also corresponds to a lens for a categorical database model. A variation on the lens concept called a c-lens is introduced, and shown to correspond to the categorical notion of Grothendieck opfibration. This variant guarantees a universal solution to the view update problem for functorial update processes.

AbstractWe consider the 33 conjugacy classes of genus zero, torsion-free modular subgroups, computing ramification data and Grothendieck’s dessins d’enfants. In the particular case of the index 36 subgroups, the corresponding Calabi–Yau threefolds are identified, in analogy with the index 24 cases being associated to K3 surfaces. In a parallel vein, we study the 112 semi-stable elliptic fibrations over ${ \mathbb{P} }^{1} $ as extremal K3 surfaces with six singular fibres. In each case, a representative of the corresponding class of subgroups is identified by specifying a generating set for that representative.

Universal Causality is a mathematical framework based on higher-order category theory, which generalizes previous approaches based on directed graphs and regular categories. We present a hierarchical framework called UCLA (Universal Causality Layered Architecture), where at the top-most level, causal interventions are modeled as a higher-order category over simplicial sets and objects. Simplicial sets are contravariant functors from the category of ordinal numbers Δ into sets, and whose morphisms are order-preserving injections and surjections over finite ordered sets. Non-random interventions on causal structures are modeled as face operators that map n-simplices into lower-level simplices. At the second layer, causal models are defined as a category, for example defining the schema of a relational causal model or a symmetric monoidal category representation of DAG models. The third layer corresponds to the data layer in causal inference, where each causal object is mapped functorially into a set of instances using the category of sets and functions between sets. The fourth homotopy layer defines ways of abstractly characterizing causal models in terms of homotopy colimits, defined in terms of the nerve of a category, a functor that converts a causal (category) model into a simplicial object. Each functor between layers is characterized by a universal arrow, which define universal elements and representations through the Yoneda Lemma, and induces a Grothendieck category of elements that enables combining formal causal models with data instances, and is related to the notion of ground graphs in relational causal models. Causal inference between layers is defined as a lifting problem, a commutative diagram whose objects are categories, and whose morphisms are functors that are characterized as different types of fibrations. We illustrate UCLA using a variety of representations, including causal relational models, symmetric monoidal categorical variants of DAG models, and non-graphical representations, such as integer-valued multisets and separoids, and measure-theoretic and topological models.

Abstract We prove a correspondence between $\kappa$ -small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$ -compact maps in their underlying quasi-categories for suitably large regular cardinals $\kappa$ . We thus obtain a transition result between weakly universal small fibrations in the (type-theoretic) injective Dugger–Rezk-style standard presentations of model toposes and object classifiers in Grothendieck $\infty$ -toposes in the sense of Lurie.

We discuss several versions of the Family Signature Theorem: in rational cohomology using ideas of Meyer, in $KO[\tfrac {1}{2}]$ -theory using ideas of Sullivan, and finally in symmetric $L$ -theory using ideas of Ranicki. Employing recent developments in Grothendieck–Witt theory, we give a quite complete analysis of the resulting invariants. As an application we prove that the signature is multiplicative modulo 4 for fibrations of oriented Poincaré complexes, generalizing a result of Hambleton, Korzeniewski and Ranicki, and discuss the multiplicativity of the de Rham invariant.

Cette thèse est consacrée à l'étude des familles de catégories munies d'une structure homotopique. Les résultats principaux compris dans cette oeuvre sont : i. Une généralisation de la structure de modèles de Reedy, qui dans ce travail est construite pour les sections d'une famille convenable des catégories de modèles sur une catégorie de Reedy. À la différence des considérations précédentes, par exemple celles de Hirschowitz-Simpson, nous exigeons aussi peu de propriétés de la famille que possible, pour que notre résultat puisse être appliqué dans les situations où les foncteurs de transition ne sont pas linéaires. ii. Une extension du formalisme de Segal pour les structures algébriques, dans le territoire des catégories monoïdales sur une catégorie d'opérateurs au sens de Barwick. Pour ce faire, nous présentons les structures monoidales comme certaines opfibrations de Grotendieck, et introduisons les sections dérivées des opfibrations en utilisant les remplacements simpliciaux de Bousfield-Kan. Notre résultat concernant la structure de Reedy nous permet alors de travailler avec les sections dérivées. iii. Une preuve d'un certain résultat de la descente homotopique, qui donne des conditions suffisantes pour que le foncteur d'image inverse soit une équivalence entre catégories de sections dérivées au sens adapté. L'on montre ce résultat pour les foncteurs qui satisfont une propriété technique du genre ``Théorème A de Quillen'', les foncteurs que nous appelons résolutions. Un exemple d'une résolution est donné par un foncteur de la catégorie des arbres planaires stables de Kontsevich-Soibelman, au groupoïde fondamental stratifié de l'espace de Ran du $2$-disque
This thesis is devoted to the study of families of categories equipped with a homotopical structure. The principal results comprising this work are:i. A generalisation of the Reedy model structure, which, in this work, is constructed for sections of a suitable family of model categories over a Reedy category. Unlike previous considerations, such as Hirschowitz-Simpson, we require as little as possible from the family, so that our result may be applied in situations when the transition functors in the family are non-linear in nature. ii. An extension of Segal formalism for algebraic structures to the setting of monoidal categories over an operator category in the sense of Barwick. We do this by treating monoidal structures using the language of Grothendieck opfibrations, and introduce derived sections of the latter using the simplicial replacements of Bousfield-Kan. Our Reedy structure result then permits to work with derived sections. iii. A proof of a certain homotopy descent result, which gives sufficient conditions on when an inverse image functor is an equivalence between suitable categories of derived sections. We show this result for functors which satisfy a technical ``Quillen Theorem A''-type property, called resolutions. One example of a resolution is given by a functor from the category of planar marked trees of Kontsevich-Soibelman, to the stratified fundamental groupoid of the Ran space of the $2$-disc. An application of the homotopy descent result to this functor gives us a new proof of Deligne conjecture, providing an alternative to the use of operads

Cette thèse est consacrée à l'étude des interactions entre les structures homotopiques en théorie des catégories et les modèles catégoriques de la théorie des types de Martin-Löf. Le mémoire s'articule selon trois axes: les bifibrationos de Quillen, les catégories homotopiques des bifibrations de Quillen, et les tribus généralisées. Le premier axe définit une nouvelle notion de bifibration classifiant les pseudo foncteurs avec de bonnes propriétés depuis un catégorie de modèles et à valeurs dans la 2-catégorie des catégories de modèles et adjonctions de Quillen entre elles. En particulier on montre comment équipper d'une structure de modèle la construction de Grothendieck d'un tel pseudo foncteur. Le théorème principal de cette partie est une caractérisation des bonnes propriétés qu'un pseudo foncteur doit posséder pour supporter cette structure de catégorie de modèles sur sa construction de Grothendieck. En ce sens, on améliore les deux théorèmes précédemment existants dans la littérature qui ne donnent que des conditions suffisantes alors que nous donnons des conditions nécessaires et suffisantes. Le second axe se concentre sur le foncteur induit entre les catégories homotopiques des catégories de modèles mises en oeuvre dans une bifibration de Quillen. On y prouve que cette localization peut se faire en deux étapes au moyen d'un quotient homotopique à la Quillen itéré. De manière à rendre cette opération rigoureuse, on a besoin de travailler dans un cadre légèrement plus large que celui imaginé par Quillen : en se basant sur le travail d'Egger, on utilise des catégories de modèles sans nécessairement tous les (co)égalisateurs. Le chapitre de prérequis sert précisément à reconstruire la théorie basique des l'algèbre homotopique à la Quillen dans ce cadre élargi. Les structures mis à nu dans cette partie imposent de considérer des versions "homotopique" des poussés en avant et des tirés en arrière qu'on trouve habituellement dans les (op)fibrations de Grothendieck. C'est le point de départ pour le troisième axe, dans lequel on définit une nouvelle structure, appelée tribu relative, qui permet d'axiomatiser des versions homotopiques de la notion de flèche cartésienne et cocartésienne. Cela est obtenu en réinterprétant les (op)fibrations de Grothendieck en termes de problèmes de relèvement. L'outil principal dans cette partie est une version relative des systèmes de factorisation stricts ou faibles usuels. Cela nous permet en particulier d'expérimenter un nouveau demodèle de la théorie des types dépendants intentionnelle dans lequelles types identités sont donnés par l'exact analogue homotopique du prédicat d'égalité dans les hyperdoctrines de Lawvere
This thesis is concerned with the study of the interplay between homotopical structures and categorical model of Martin-Löf's dependent type theory. The memoir revolves around three big topics: Quillen bifibrations, homotopy categories of Quillen bifibrations, and generalized tribes. The first axis defines a new notion of bifibrations, that classifies correctly behaved pseudo functors from a model category to the 2-category of model categories and Quillen adjunctions between them. In particular it endows the Grothendieck construction of such a pseudo functor with a model structure. The main theorem of this section acts as a charaterization of the well-behaved pseudo functors that tolerates this "model Gothendieck construction". In that respect, we improve the two previously known theorems on the subject in the litterature that only give sufficient conditions by designing necessary and sufficient conditions. The second axis deals with the functors induced between the homotopy categories of the model categories involved in a Quillen bifibration. We prove that this localization can be performed in two steps, by means of Quillen's construction of the homotopy category in an iterated fashion. To that extent we need a slightly larger framework for model categories than the one originally given by Quillen: following Egger's intuitions we chose not to require the existence of equalizers and coequalizers in our model categories. The background chapter makes sure that every usual fact of basichomotopical algebra holds also in that more general framework. The structures that are highlighted in that chapter call for the design of notions of "homotopical pushforward" and "homotopical pullback". This is achieved by the last axis: we design a structure, called relative tribe, that allows for a homotopical version of cocartesian morphisms by reinterpreting Grothendieck (op)fibrations in terms of lifting problems. The crucial tool in this last chapter is given by a relative version of orthogonal and weak factorization systems. This allows for a tentative design of a new model of intentional type theory where the identity types are given by the exact homotopical counterpart of the usual definition of the equality predicate in Lawvere's hyperdoctrines

Thesis (MSc)--Stellenbosch University, 2014.
ENGLISH ABSTRACT: In this thesis we develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. We also make some first steps towards a possible link between this theory and the theory of databases in computer science. Both of these theories are based around the study of Grothendieck bifibrations and their generalisations. The main results in this thesis concern correspondences between certain structures on a category which are relevant to the study of categories of non-abelian group-like structures, and functors over that category. An investigation of these correspondences leads to a system of dual axioms on a functor, which can be considered as a solution to the proposal of Mac Lane in his 1950 paper "Duality for Groups" that a self-dual setting for formulating and proving results for groups be found. The part of the thesis concerned with the theory of databases is based on a recent approach by Johnson and Rosebrugh to views of databases and the view update problem.
AFRIKAANSE OPSOMMING: In hierdie tesis word ’n self-duale kategoriese benadering tot verskeie onderwerpe in nie-abelse algebra ontwikkel, wat gebaseer is op die vervanging van die raamwerk van ’n kategorie met dié van ’n kategorie saam met ’n funktor tot die kategorie. Ons neem ook enkele eerste stappe in die rigting van ’n skakel tussen hierdie teorie and die teorie van databasisse in rekenaarwetenskap. Beide hierdie teorieë is gebaseer op die studie van Grothendieck bifibrasies en hul veralgemenings. Die hoof resultate in hierdie tesis het betrekking tot ooreenkomste tussen sekere strukture op ’n kategorie wat relevant tot die studie van nie-abelse groep-agtige strukture is, en funktore oor daardie kategorie. ’n Verdere ondersoek van hierdie ooreemkomste lei tot ’n sisteem van duale aksiomas op ’n funktor, wat beskou kan word as ’n oplossing tot die voorstel van Mac Lane in sy 1950 artikel “Duality for Groups” dat ’n self-duale konteks gevind word waarin resultate vir groepe geformuleer en bewys kan word. Die deel van hierdie tesis wat met die teorie van databasisse te doen het is gebaseer op ’n onlangse benadering deur Johnson en Rosebrugh tot aansigte van databasisse en die opdatering van hierdie aansigte.

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

In this chapter, Grothendieck fibrations are defined, and the Grothendieck Fibration Theorem is proved. After discussing some basic definitions, properties, and examples of fibrations, this chapter constructs a 2-monad and proves in detail that its pseudo algebras are precisely cloven fibrations. Moreover, the strict algebras of this 2-monad are shown to be precisely split fibrations.