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Добірка наукової літератури з теми "Grothendieck category"

Автор: Grafiati

Опубліковано: 11 березня 2023

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Статті в журналах з теми "Grothendieck category"

Gorbunov, V. G. "Generalized Grothendieck category." Siberian Mathematical Journal 28, no. 5 (1988): 734–39. http://dx.doi.org/10.1007/bf00969313.

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Năstăsescu, C., and B. Torrecillas. "Atomical Grothendieck categories." International Journal of Mathematics and Mathematical Sciences 2003, no. 71 (2003): 4501–9. http://dx.doi.org/10.1155/s0161171203209418.

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Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories.

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Puig, Lluis. "Ordinary Grothendieck Groups of a Frobenius P-Category." Algebra Colloquium 18, no. 01 (March 2011): 1–76. http://dx.doi.org/10.1142/s1005386711000022.

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In [7] we have introduced the Frobenius categories [Formula: see text] over a finite p-group P, and we have associated to [Formula: see text] — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from [Formula: see text], determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from [Formula: see text], determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in [Formula: see text] of a suitable set of representatives of the [Formula: see text]-classes of elements of P.

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Coghetto, Roland. "Non-Trivial Universes and Sequences of Universes." Formalized Mathematics 30, no. 1 (April 1, 2022): 53–66. http://dx.doi.org/10.2478/forma-2022-0005.

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Summary Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCE-AUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay2. The following result links the universes U 0 (FinSETS) and U 1 (SETS): Grothendieck Universe ω = Grothendieck Universe U 0 = U 1 {\rm{Grothendieck}}\,{\rm{Universe}}\,\omega = {\rm{Grothendieck}}\,{\rm{Universe}}\,{{\bf{U}}_0} = {{\bf{U}}_1} Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, ncatlab.org [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category theory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].

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ARA, DIMITRI, DENIS-CHARLES CISINSKI, and IEKE MOERDIJK. "The dendroidal category is a test category." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 01 (April 26, 2018): 107–21. http://dx.doi.org/10.1017/s030500411800021x.

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AbstractWe prove that the category of trees Ω is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that this model category structure, up to a change of cofibrations, can be obtained as an explicit left Bousfield localisation of the operadic model category structure.

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Barot, M., D. Kussin, and H. Lenzing. "The Grothendieck group of a cluster category." Journal of Pure and Applied Algebra 212, no. 1 (January 2008): 33–46. http://dx.doi.org/10.1016/j.jpaa.2007.04.007.

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Bergh, Petter Andreas, and Marius Thaule. "The Grothendieck group of an -angulated category." Journal of Pure and Applied Algebra 218, no. 2 (February 2014): 354–66. http://dx.doi.org/10.1016/j.jpaa.2013.06.007.

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Bozicevic, M. "Grothendieck group of an equivariant derived category." International Mathematical Forum 2 (2007): 3219–31. http://dx.doi.org/10.12988/imf.2007.07296.

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ESTRADA, SERGIO, JAMES GILLESPIE, and SINEM ODABAŞI. "Pure exact structures and the pure derived category of a scheme." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 2 (November 23, 2016): 251–64. http://dx.doi.org/10.1017/s0305004116000980.

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AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

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Savoji, Fatemeh, and Reza Sazeedeh. "Local cohomology in Grothendieck categories." Journal of Algebra and Its Applications 19, no. 11 (November 14, 2019): 2050222. http://dx.doi.org/10.1142/s0219498820502229.

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Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

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Більше джерел

Дисертації з теми "Grothendieck category"

McBride, Aaron. "Grothendieck Group Decategorifications and Derived Abelian Categories." Thesis, Université d'Ottawa / University of Ottawa, 2015. http://hdl.handle.net/10393/33000.

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The Grothendieck group is an interesting invariant of an exact category. It induces a decategorication from the category of essentially small exact categories (whose morphisms are exact functors) to the category of abelian groups. Similarly, the triangulated Grothendieck group induces a decategorication from the category of essentially small triangulated categories (whose morphisms are triangulated functors) to the category of abelian groups. In the case of an essentially small abelian category, its Grothendieck group and the triangulated Grothendieck group of its bounded derived category are isomorphic as groups via a natural map. Because of this, homological algebra and derived functors become useful in surprising ways. This thesis is an expository work that provides an overview of the theory of Grothendieck groups with respect to these decategorications.

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Bittmann, Léa. "Quantum Grothendieck rings, cluster algebras and quantum affine category O." Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC024.

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L'objectif de cette thèse est de construire et d'étudier une structure d'anneau de Grothendieck quantique pour une catégorie O de représentations de la sous-algèbre de Borel Uq(b) d'une algèbre affine quantique Uq(g). On s'intéresse dans un premier lieu à la construction de modules standards asymptotiques pour la catégorie O, qui sont des analogues des modules standards existant dans la catégorie des représentations de dimension finie de Uq(^g). Une construction complète de ces modules est proposée dans le cas où l'algèbre de Lie simple sous-jacente g est sl2. Ensuite, nous définissons un tore quantique qui étend le tore quantique contenant l'anneau de Grothendieck quantique de la catégorie des représentations de dimension finie.Nous utilisons pour cela des notions liées aux algèbres amassées quantiques. Dans le même esprit, nous proposons une construction d'une structure d'algèbre amassée quantique sur l'anneau de Grothendieck quantique Kt(Cz) d'une sous-catégorie monoïdale Cz de la catégorie des représentations de dimension finie. Puis, nous définissons un anneau de Grothendieck quantique Kt(O+Z) d'une sous catégorie O+Z de la catégorie O, comme une algèbre amassée quantique. Nous établissons ensuite que cet anneau de Grothendieck quantique contient celui de la catégorie des représentations de dimension finie. Ce résultat est montré directement en type A, puis en tout type simplement lacé en utilisant la structure d'algèbre amassée quantique de Kt(CZ).Enfin, nous définissons des (q,t)-caractères pour des représentations simples de dimension infinie remarquables de la catégorie O. Ceci nous permet d'écrire des versions t-déformées de relations importantes dans l'anneau de Grothendieck classique de la catégorie O+Z qui ont des liens avec les systèmes intégrables quantiques associés
The aim of this thesis is to construct and study some quantum Grothendieck ring structure for the category O of representations of the Borel subalgebra Uq(^b) of a quantum affine algebra Uq(^g). First of all, we focus on the construction of asymptotical standard modules, analogs in the context of the category O of the standard modules in the category of finite-dimensional Uq(^g)-modules. A construction of these modules is given in the case where the underlying simple Lie algebra g is sl2. Next, we define a new quantum torus, which extends the quantum torus containing the quantum Grothendieck ring of the category of finite-dimensional modules. In order todo this, we use notions linked to quantum cluster algebras. In the same spirit, we build a quantum cluster algebra structure on the quantum Grothendieck ring of a monoidal subcategory CZ of the category of finite-dimensional representations. With this quantum torus, we de_ne the quantum Grothendieck ring Kt(O+Z) of a subcategory O+Z of the category O as a quantum cluster algebra. Then, we prove that this quantum Grothendieck ring contains that of the category of finite-dimensional representation. This result is first shown directly in type A, and then in all simply-laced types using the quantum cluster algebra structure of Kt(CZ). Finally, we define (q,t)-characters for some remarkable infinite-dimensional simple representations in the category O+Z. This enables us to write t-deformed analogs of important relations in the classical Grothendieck ring of the category O, which are related to the corresponding quantum integrable systems

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Perera, Simon. "Grothendieck rings of theories of modules." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/grothendieck-rings-of-theories-of-modules(897cbbd9-77b6-47fb-8cf8-d15c7432e61b).html.

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We consider right modules over a ring, as models of a first order theory. We explorethe definable sets and the definable bijections between them. We employ the notionsof Euler characteristic and Grothendieck ring for a first order structure, introduced byJ. Krajicek and T. Scanlon in [24]. The Grothendieck ring is an algebraic structurethat captures certain properties of a model and its category of definable sets.If M is a module over a product of rings A and B, then M has a decomposition into a direct sum of an A-module and a B-module. Theorem 3.5.1 states that then the Grothendieck ring of M is the tensor product of the Grothendieck rings of the summands.Theorem 4.3.1 states that the Grothendieck ring of every infinite module over afield or skew field is isomorphic to Z[X].Proposition 5.2.4 states that for an elementary extension of models of anytheory, the elementary embedding induces an embedding of the corresponding Grothendieck rings. Theorem 5.3.1 is that for an elementary embedding of modules, we have the stronger result that the embedding induces an isomorphism of Grothendieck rings.We define a model-theoretic Grothendieck ring of the category Mod-R and explorethe relationship between this ring and the Grothendieck rings of general right R-modules. The category of pp-imaginaries, shown by K. Burke in [7] to be equivalentto the subcategory of finitely presented functors in (mod-R; Ab), provides a functorial approach to studying the generators of theGrothendieck rings of R-modules. It is shown in Theorem 6.3.5 that whenever R andS are Morita equivalent rings, the rings Grothendieck rings of the module categories Mod-R and Mod-S are isomorphic.Combining results from previous chapters, we derive Theorem 7.2.1 saying that theGrothendieck ring of any module over a semisimple ring is isomorphic to a polynomialring Z[X1,...,Xn] for some n.

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Cagne, Pierre. "Towards a homotopical algebra of dependent types." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC063/document.

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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

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Weighill, Thomas. "Bifibrational duality in non-abelian algebra and the theory of databases." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/96125.

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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.

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Murfet, Daniel Saul. "The mock homotopy category of projectives and Grothendieck duality." Phd thesis, 2007. http://hdl.handle.net/1885/151476.

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Ahmadi, Amir. "Axiomatic approach to cellular algebras." Thesis, 2020. http://hdl.handle.net/1866/23949.

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Les algèbres cellulaires furent introduite par J.J. Graham et G.I. Lehrer en 1996. Elles forment une famille d’algèbres associatives de dimension finie définies en termes de « données cellulaires » satisfaisant certains axiomes. Ces données cellulaires, lorsqu’elles sont identifiées pour une certaine algèbre, permettent une construction explicite de tous ses modules simples, à isomorphisme près, et de leurs couvertures projectives. Dans ce mémoire, nous définissons ces algèbres cellulaires en introduisant progressivement chacun des éléments constitutifs d’une façon axiomatique. Deux autres familles d’algèbres associatives sont discutées, à savoir les algèbres quasihéréditaires et celles dont les modules forment une catégorie de plus haut poids. Ces familles furent introduites durant la même période de temps, au tournant des années quatre-vingtdix. La relation entre ces deux familles ainsi que celle entre elles et les algèbres cellulaires sont prouvées.
Cellular algebras were introduced by J.J. Graham and G.I. Lehrer in 1996. They are a class of finite-dimensional associative algebras defined in terms of a “cellular datum” satisfying some axioms. This cellular datum, when made explicit for a given associative algebra, allows for the explicit construction of all its simple modules, up to isomorphism, and of their projective covers. In this work, we define these cellular algebras by introducing each building block of the cellular datum in a fairly axiomatic fashion. Two other families of associative algebras are discussed, namely the quasi-hereditary algebras and those whose modules form a highest weight category. These families were introduced at about the same period. The relationships between these two, and between them and the cellular ones, are made explicit.

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Книги з теми "Grothendieck category"

A, Grothendieck, and Cartier P, eds. The Grothendieck festschrift: A collection of articles written in honor of the 60th birthday of Alexander Grothendieck. Boston: Birkhauser, 2006.

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M, Katz Nicholas, Manin Yuri I, Illusie Luc, Laumon Gérard, and Ribet Kenneth A, eds. The Grothendieck Festschrift Volume III: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. Boston, MA: Birkhäuser Boston, 2007.

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service), SpringerLink (Online, ed. Frobenius categories versus Brauer blocks: The Grothendieck group of the Frobenius category of a Brauer block. Basel, Switzerland: Birkhäuser, 2009.

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Robson, J. C. (James Christopher), 1940-, ed. Hereditary noetherian prime rings and idealizers. Providence, R.I: American Mathematical Society, 2011.

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Conference on Hopf Algebras and Tensor Categories (2011 University of Almeria). Hopf algebras and tensor categories: International conference, July 4-8, 2011, University of Almería, Almería, Spain. Edited by Andruskiewitsch Nicolás 1958-, Cuadra Juan 1975-, and Torrecillas B. (Blas) 1958-. Providence, Rhode Island: American Mathematical Society, 2013.

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(Editor), Pierre Cartier, Luc Illusie (Editor), Nicholas M. Katz (Editor), Gérard Laumon (Editor), Yuri I. Manin (Editor), and Kenneth A. Ribet (Editor), eds. The Grothendieck Festschrift Volume I: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck (Modern Birkhäuser Classics). Birkhäuser Boston, 2006.

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(Editor), Pierre Cartier, Luc Illusie (Editor), Nicholas M. Katz (Editor), Gérard Laumon (Editor), Yuri I. Manin (Editor), and Kenneth A. Ribet (Editor), eds. The Grothendieck Festschrift Volume II: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck (Modern Birkhäuser Classics). Birkhäuser Boston, 2006.

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(Editor), Pierre Cartier, Luc Illusie (Editor), Nicholas M. Katz (Editor), Gérard Laumon (Editor), Yuri I. Manin (Editor), and Kenneth A. Ribet (Editor), eds. The Grothendieck Festschrift Volume III: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck (Modern Birkhäuser Classics). Birkhäuser Boston, 2006.

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Caramello, Olivia. Quotients of a theory of presheaf type. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198758914.003.0010.

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In this chapter the quotients of a given theory of presheaf type are investigated by means of Grothendieck topologies that can be naturally attached to them, establishing a ‘semantic’ representation for the classifying topos of such a quotient as a subtopos of the classifying topos of the given theory of presheaf type. It is also shown that the models of such a quotient can be characterized among the models of the theory of presheaf type as those which satisfy a key property of homogeneity with respect to a Grothendieck topology associated with the quotient. A number of sufficient conditions for the quotient of a theory of presheaf type to be again of presheaf type are also identified: these include a finality property of the category of models of the quotient with respect to the category of models of the theory and a rigidity property of the Grothendieck topology associated with the quotient.

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Johnson, Niles, and Donald Yau. 2-Dimensional Categories. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198871378.001.0001.

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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.

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Johnson, Niles, and Donald Yau. "The Grothendieck Construction." In 2-Dimensional Categories, 371–438. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198871378.003.0010.

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This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

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"The Grothendieck groups of a Frobenius P-category." In Frobenius Categories versus Brauer Blocks, 211–39. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-7643-9998-6_15.

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"Category of representations and constructions of Grothendieck groups and rings." In Representation Theory and Higher Algebraic K-Theory, 31–50. Chapman and Hall/CRC, 2016. http://dx.doi.org/10.1201/b13624-7.

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McLarty, Colin. "Saunders Mac Lane: From Principia Mathematica through Göttingen to the Working Theory of Structures." In The Prehistory of Mathematical Structuralism, 215–38. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190641221.003.0009.

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Saunders Mac Lane heard David Hilbert’s weekly lectures on philosophy and utterly believed Hilbert’s declaration that mathematics will know no limits. He studied algebra with Emmy Noether, and both mathematics and philosophy with Hermann Weyl. As a young algebraist he created today’s standard working method for mathematical structure: category theory, with topologist Samuel Eilenberg. As one step, they created the now standard definition of “isomorphism.” They originally saw categories as just a working tool. But in the 1950s, Mac Lane saw Alexander Grothendieck and others radically extend the range of the theory, and in the 1960s, he took up William Lawvere’s idea of categorical foundations. The essay relates all of this to current philosophical structuralism, especially concerning isomorphisms and automorphisms of structures. It concludes by comparing Mac Lane’s motives for structuralist working mathematics with current philosophical motives for structuralist ontology.

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Palmgren, Erik. "On universes in type theory." In Twenty Five Years of Constructive Type Theory. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780198501275.003.0012.

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The notion of a universe of types was introduced into constructive type theory by Martin-Löf (1975). According to the propositions-as-types principle inherent in type theory, the notion plays two roles. The first is as a collection of sets or types closed under certain type constructions. The second is as a set of constructively given infinitary formulas. In this paper we discuss the notion of universe in type theory and suggest and study some useful extensions. We assume familiarity with type theory as presented in, for example, Martin-Löf (1984). Universes have been effective in expanding the realm of constructivism. One example is constructive category theory where type universes take the roles of Grothendieck universes of sets, in handling large categories. A more profound example is Aczel’s (1986) type-theoretic interpretation of constructive set theory (CZF). It is done by coding ϵ-diagrams into well-order types, with branching over an arbitrary type of the universe. The latter generality is crucial for interpreting the separation axiom. The introduction of universes and well-orders (W-types) in conjunction gives a great proof-theoretic strength. This has provided constructive justification of strong subsystems of second-order arithmetic studied by proof-theorists (see Griffor and Rathjen (1994) and Setzer (1993), and for some early results, see Palmgren (1992)). At present, it appears that the most easily justifiable way to increase the proof-theoretic strength of type theory is to introduce ever more powerful universe constructions. We will give two such extensions in this paper. Besides contributing to the understanding of subsystems of second-order arithmetic and pushing the limits of inductive definability, such constructions provide intuitionistic analogues of large cardinals (Rathjen et al, in press). A third new use of universes is to facilitate the incorporation of classical reasoning into constructive type theory. We introduce a universe of classical propositions and prove a conservation result for ‘Π-formulas’. Extracting programs from classical proofs is then tractable within type theory. The next section gives an introduction to the notion of universe. The central part of the paper is section 3 where we introduce a universe forming operator and a super universe closed under this operator.

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