Статті в журналах: "Grothendieck category"

1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

CRIVEI, SEPTIMIU, CONSTANTIN NĂSTĂSESCU, and BLAS TORRECILLAS. "ON THE OSOFSKY–SMITH THEOREM." Glasgow Mathematical Journal 52, A (June 24, 2010): 61–67. http://dx.doi.org/10.1017/s0017089510000169.

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AbstractWe recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

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12

DEITMAR, ANTON. "CONGRUENCE SCHEMES." International Journal of Mathematics 24, no. 02 (February 2013): 1350009. http://dx.doi.org/10.1142/s0129167x13500092.

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13

Foxby, Hans-Bjørn, and Esben Bistrup Halvorsen. "Grothendieck groups for categories of complexes." Journal of K-Theory 3, no. 1 (January 9, 2008): 165–203. http://dx.doi.org/10.1017/is008001002jkt023.

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AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

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14

Puig, Lluis. "Categorizations of Limits of Grothendieck Groups over a Frobenius P-Category." Algebra Colloquium 29, no. 04 (December 2022): 541–94. http://dx.doi.org/10.1142/s1005386722000402.

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In [Frobenius Categories Versus Brauer Blocks, Progress in Math., 274] and in [Ordinary Grothendieck groups of a Frobenius [Formula: see text]-category, Algebra Colloq. 18 (2011) 1–76], we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics [Formula: see text] and zero, obtained from a folded Frobenius [Formula: see text]-category[Formula: see text], which covers the case of the Frobenius[Formula: see text]-categories associated with blocks; moreover, in [Beyond a question of Markus Linckelmann, arxiv.org/abs/1507.04278] we show that a folded Frobenius [Formula: see text]-categoryis actually equivalent to the choice of a regular central [Formula: see text]-extension [Formula: see text] of [Formula: see text]. Here, taking advantage of the existence of a perfect [Formula: see text] -locality [Formula: see text], we exhibit those inverse limits as the true Grothendieck groups of the categories of [Formula: see text]-and [Formula: see text]-modules for a suitable [Formula: see text]-group [Formula: see text] associated to the [Formula: see text]-category [Formula: see text] obtained from [Formula: see text] and [Formula: see text]. It depends on a vanishing cohomology result, given with more generality in the Appendix.

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15

Canonaco, Alberto, and Paolo Stellari. "Uniqueness of dg enhancements for the derived category of a Grothendieck category." Journal of the European Mathematical Society 20, no. 11 (July 20, 2018): 2607–41. http://dx.doi.org/10.4171/jems/820.

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16

Balodi, Mamta, Abhishek Banerjee, and Samarpita Ray. "Cohomology of Modules Over -categories and Co--categories." Canadian Journal of Mathematics 72, no. 5 (August 6, 2019): 1352–85. http://dx.doi.org/10.4153/s0008414x19000403.

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AbstractLet $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category ${\mathcal{C}}$ as modules over the smash extension ${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category ${\mathcal{D}}$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $\text{Hom}$ objects and higher derived functors of coinvariants.

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17

Garcia, J., and D. Simson. "On rings whose flat modules form a Grothendieck category." Colloquium Mathematicum 73, no. 1 (1997): 115–41. http://dx.doi.org/10.4064/cm-73-1-115-141.

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18

Herzog, I. "The Ziegler Spectrum of a Locally Coherent Grothendieck Category." Proceedings of the London Mathematical Society 74, no. 3 (May 1997): 503–58. http://dx.doi.org/10.1112/s002461159700018x.

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19

AHMADI, KAIVAN, and REZA SAZEEDEH. "HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY." Bulletin of the Australian Mathematical Society 94, no. 3 (September 26, 2016): 421–30. http://dx.doi.org/10.1017/s0004972716000563.

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Let ${\mathcal{A}}$ be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of ${\mathcal{A}}$ and the lattice of subsets of $\text{ASpec}\,{\mathcal{A}}$ in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of ${\mathcal{A}}$ and closed subsets of $\text{ASpec}\,{\mathcal{A}}$. If ${\mathcal{A}}$ is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space $\text{ASpec}\,{\mathcal{A}}$ is Alexandroff.

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20

Hosseini, Esmaeil. "On λ-pure acyclic complexes in a Grothendieck category". Journal of Algebra 525 (травень 2019): 245–58. http://dx.doi.org/10.1016/j.jalgebra.2019.01.034.

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21

Neeman, Amnon. "The homotopy category of flat modules, and Grothendieck duality." Inventiones mathematicae 174, no. 2 (August 20, 2008): 255–308. http://dx.doi.org/10.1007/s00222-008-0131-0.

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22

Wang, Na. "The realizations of Lie algebra gl(∞) and tau functions in homotopy category." International Journal of Modern Physics A 31, no. 18 (June 29, 2016): 1650105. http://dx.doi.org/10.1142/s0217751x16501050.

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In this paper, we realize the Lie algebra [Formula: see text] in one homotopy category, and we find that tau functions can be obtained from complexes in this homotopy category by taking Grothendieck group. We start with the realization of Fermions in a homotopy category over a Heisenberg category which is constructed by Khovanov.

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23

Lazarev, Oleg. "Symplectic flexibility and the Grothendieck group of the Fukaya category." Journal of Topology 15, no. 1 (March 2022): 204–37. http://dx.doi.org/10.1112/topo.12217.

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24

TAMBARA, Daisuke. "The Grothendieck ring of linear representations of a finite category." Hokkaido Mathematical Journal 19, no. 2 (June 1990): 325–37. http://dx.doi.org/10.14492/hokmj/1381517365.

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25

T., Pirasgvili. "Category of eilenberg - maclane fibrations and cohomology of grothendieck constructions." Communications in Algebra 21, no. 1 (January 1993): 309–41. http://dx.doi.org/10.1080/00927879208824563.

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26

Riccardo, Colpi, Gregorio Enrico, and Orsatti Adalberto. "The basic ring of a locally artinian commutative grothendieck category." Communications in Algebra 19, no. 5 (January 1991): 1449–56. http://dx.doi.org/10.1080/00927879108824213.

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27

Albu, Toma, and John Van Den Berg. "An indecomposable nonlocally finitely generated Grothendieck category with simple objects." Journal of Algebra 321, no. 5 (March 2009): 1538–45. http://dx.doi.org/10.1016/j.jalgebra.2008.12.007.

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28

Virili, Simone. "On the relation between length functions and exact Sylvester rank functions." Topological Algebra and its Applications 7, no. 1 (December 31, 2019): 69–74. http://dx.doi.org/10.1515/taa-2019-0006.

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AbstractInspired by the work of Crawley-Boevey on additive functions in locally finitely presented Grothendieck categories, we describe a natural way to extend a given exact Sylvester rank function on the category of finitely presented left modules over a given ring R, to the category of all left R-modules.

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29

ABRASHKIN, VICTOR A. "ON A LOCAL ANALOGUE OF THE GROTHENDIECK CONJECTURE." International Journal of Mathematics 11, no. 02 (March 2000): 133–75. http://dx.doi.org/10.1142/s0129167x0000009x.

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We prove that the functor from the category of all complete discrete valuation fields with finite residue fields of characteristic ≠2 to the category of profinite filtered groups given by taking the Galois group of corresponding field together with its filtration by higher ramification subgroups is fully faithful. If [K; ℚp]<∞ we also study the opportunity to recover K from the knowledge of the filtered group ΓK(p)/ΓK(p)(a), where a>0, ΓK(p) is the absolute Galois group of the maximal p-extension of K and filtration is induced by ramification filtration.

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30

Bahiraei, Payam. "Cotorsion pairs and adjoint functors in the homotopy category of N-complexes." Journal of Algebra and Its Applications 19, no. 12 (December 11, 2019): 2050236. http://dx.doi.org/10.1142/s0219498820502369.

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In this paper, we first construct some complete cotorsion pairs on the category [Formula: see text] of unbounded [Formula: see text]-complexes of Grothendieck category [Formula: see text], from two given cotorsion pairs in [Formula: see text]. Next, as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an [Formula: see text]-complex version of the results that were shown by Neeman in the category of ordinary complexes.

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31

Biglari, Shahram. "On lambda operations on mixed motives." Journal of K-Theory 12, no. 2 (May 23, 2013): 381–404. http://dx.doi.org/10.1017/is013005008jkt231.

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AbstractWe study the natural λ-ring structure on the Grothendieck ring of the triangulated category of mixed motives. Basic properties of a natural notion of characteristic-like series are developed in the context of equivariant objects.

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32

El Kaoutit, L., and J. Vercruysse. "Cohomology for Bicomodules. Separable and Maschke functors." Journal of K-Theory 3, no. 1 (November 30, 2007): 123–52. http://dx.doi.org/10.1017/is007011017jkt017.

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AbstractWe introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].

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33

Kanda, Ryo. "Extension groups between atoms and objects in locally noetherian Grothendieck category." Journal of Algebra 422 (January 2015): 53–77. http://dx.doi.org/10.1016/j.jalgebra.2014.09.009.

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34

Hartmann, Elisa. "Coarse cohomology with twisted coefficients." Mathematica Slovaca 70, no. 6 (December 16, 2020): 1413–44. http://dx.doi.org/10.1515/ms-2017-0440.

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AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

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35

SMITH, S. PAUL. "COMPUTATION OF THE GROTHENDIECK AND PICARD GROUPS OF A TORIC DM STACK BY USING A HOMOGENEOUS COORDINATE RING FOR." Glasgow Mathematical Journal 53, no. 1 (September 1, 2010): 97–113. http://dx.doi.org/10.1017/s001708951000056x.

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AbstractWe compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

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36

Ebrahimi, M. M., and M. Mahmoudi. "CATEGORY OF M-ALGEBRAS AND INTERNAL HOMOMORPHISMS." Asian-European Journal of Mathematics 03, no. 02 (June 2010): 307–22. http://dx.doi.org/10.1142/s1793557110000210.

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Recall that for (universal) algebras A and B in a category [Formula: see text], [Formula: see text] is usually an external object, in the sense that it is just a set and is neither an algebra nor an (internal) object of the base category [Formula: see text]. Ebrahimi introduced an object [A, B] inside [Formula: see text] to be the best counterpart of Hom(A, B), for universal algebras A, B in a Grothendieck topos [Formula: see text]. In this paper, taking a monoid M and an equational category of algebras [Formula: see text], we introduce the category [Formula: see text] of universal algebras in the category [Formula: see text], of sets with an action of a monoid M, together with members of [A, B], called internal homomorphisms, as the M-set of homomorphisms from A to B. We study some algebraic and categoric ingredients, monomorphisms, epimorphisms, limits and colimits, of the category [Formula: see text]. Among other things, we show that in this category equalizers do not exist in general while all colimits exist.

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37

Achar, Pramod N., Simon Riche, and Cristian Vay. "Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups." Canadian Journal of Mathematics 72, no. 1 (January 9, 2019): 1–55. http://dx.doi.org/10.4153/cjm-2018-034-0.

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AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

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38

Jardine, J. F. "Intermediate Model Structures for Simplicial Presheaves." Canadian Mathematical Bulletin 49, no. 3 (September 1, 2006): 407–13. http://dx.doi.org/10.4153/cmb-2006-040-8.

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AbstractThis note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.

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39

Antipov, M. A. "The Grothendieck group of the stable category of symmetric special biserial algebras." Journal of Mathematical Sciences 136, no. 3 (July 2006): 3833–36. http://dx.doi.org/10.1007/s10958-006-0204-9.

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40

Hébert, David. "Structure de poids à la Bondarko sur les motifs de Beilinson." Compositio Mathematica 147, no. 5 (July 29, 2011): 1447–62. http://dx.doi.org/10.1112/s0010437x11005422.

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AbstractBondarko defines and studies the notion of weight structure and he shows that there exists a weight structure over the category of Voevodsky motives with rational coefficients (over a field of characteristic 0). In this paper we extend this weight structure to the category of Beilinson motives (for any scheme of finite type over a base scheme which is excellent of dimension at most two) introduced and studied by Cisinsky and Déglise. We also check the weight exactness of the Grothendieck operations.

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41

VICKERS, STEVEN. "Topical categories of domains." Mathematical Structures in Computer Science 9, no. 5 (October 1999): 569–616. http://dx.doi.org/10.1017/s0960129599002741.

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This paper shows how it is possible to express many techniques of categorical domain theory in the general context of topical categories (where ‘topical’ means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as ‘generalized topological spaces’ to be treated in a transparently spatial way, and also shows the constructivity of the arguments. The theory of strongly algebraic (SFP) domains is given as a case study in which the topical category is Cartesian closed.

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42

Caenepeel, S., and T. Guédénon. "On the cohomology of comodules over smash coproducts." Journal of Algebra and Its Applications 18, no. 10 (August 6, 2019): 1950189. http://dx.doi.org/10.1142/s0219498819501895.

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We consider the category of comodules over a smash coproduct coalgebra [Formula: see text]. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from [Formula: see text]-colinear, [Formula: see text]-colinear and rational [Formula: see text]-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

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43

Konovalov, V. A. "СLASSIFIER OF BIG DATA OBJECTS OF THE SOCIO-ECONOMIC SYSTEM". Vestnik komp'iuternykh i informatsionnykh tekhnologii, № 212 (лютий 2022): 32–39. http://dx.doi.org/10.14489/vkit.2022.02.pp.032-039.

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The well-known models that provide the synthesis of object classifiers are analyzed: the Grothendieck topos model, Kripke’s intuitionistic model and the model of the normal Markov algorithm, in which similar approaches to the representation of objects are highlighted. The differences in the classifiers used in them are highlighted. A categorical-theoretical model of topos has been developed, in which sets are represented by n-categories. A generalized classifier has been developed that includes the properties of the Grothendieck, Kripke and Markov classifiers. An N-scheme is synthesized to replace the g-scheme of the normal Markov algorithm. A model of n-category topos has been developed and theoretically substantiated, in which the compositions of morphisms of an object are specified by an N-scheme. The alphabet of the n-category associator is synthesized, which provides the representation of non-associative compositions of morphisms, interacting objects, taking into account the assumptions laid down in the Kripke model. A database has been developed that implements the synthesized topos model. It is noted that such a database can be implemented as a cascade of unordered containers with the computation of hash functions to quickly find and retrieve keys and values. For some keys, such a base, it is allowed to duplicate a key in containers. The database provides a synthesis of self-learning on the input sampling of artificial intelligence data and supported by control lists of objects of selection, exclusion and deletion. The normal Markov algorithm is adapted to the theoretical model of the Grothendieck topos, taking into account the assumptions of the Kripke model, by replacing the g-scheme with the synthesized N-scheme.

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44

Luna-Torres, Joaquín. "Filters and compactness on small categories and locales." Open Journal of Mathematical Sciences 6, no. 1 (March 3, 2022): 1–13. http://dx.doi.org/10.30538/oms2022.0174.

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In analogy with the classical theory of filters, for finitely complete or small categories, we provide the concepts of filter, $\mathfrak{G}$-neighborhood (short for "Grothendieck-neighborhood") and cover-neighborhood of points of such categories, to study convergence, cluster point, closure of sieves and compactness on objects of that kind of categories. Finally, we study all these concepts in the category $\mathbf{Loc}$ of locales.

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45

García, Josefa M., and Pascual Jara. "Gradual and Fuzzy Modules: Functor Categories." Mathematics 10, no. 22 (November 15, 2022): 4272. http://dx.doi.org/10.3390/math10224272.

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The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod−P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P=(0,1], to build a torsionfree class J in Mod−P, and a hereditary torsion theory in Mod−P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G,F), of decreasing gradual submodules of M, where G belongs to J, satisfying G=Fd, and ∪αF(α) is a disjoint union of F(1) and F(α)\G(α), where α is running in (0,1].

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46

Kuznetsov, Alexander, and Alexander Perry. "Derived categories of Gushel–Mukai varieties." Compositio Mathematica 154, no. 7 (May 25, 2018): 1362–406. http://dx.doi.org/10.1112/s0010437x18007091.

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We study the derived categories of coherent sheaves on Gushel–Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel–Mukai fourfold in more detail. Namely, we show this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel–Mukai varieties, which was one of the main motivations for this work.

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47

Jardine, J. F. "Stable Homotopy Theory of Simplicial Presheaves." Canadian Journal of Mathematics 39, no. 3 (June 1, 1987): 733–47. http://dx.doi.org/10.4153/cjm-1987-035-8.

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Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre(C) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre(C)stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [1], and makes direct use of their Theorem A.7.This result gives a precise description of the associated stable homotopy category Ho(S Pre(C))stab, according to well known results of Quillen [6]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.

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48

Asadollahi, Javad, Rasool Hafezi, and Razieh Vahed. "Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor." Canadian Journal of Mathematics 67, no. 1 (February 1, 2015): 28–54. http://dx.doi.org/10.4153/cjm-2014-018-7.

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AbstractWe study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

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49

McLarty, Colin. "Elementary axioms for canonical points of toposes." Journal of Symbolic Logic 52, no. 1 (March 1987): 202–4. http://dx.doi.org/10.2307/2273873.

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Two elementary extensions of the topos axioms are given, each implying the topos has a local geometric morphism to a category of sets. The stronger one realizes sets as precisely the decidables of the topos, so there is a simple internal description of the range of validity of the law of excluded middle in the topos. It also has a natural geometric meaning. Models of the extensions in Grothendieck toposes are described.

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50

Barlev, Jonathan. "-modules on spaces of rational maps." Compositio Mathematica 150, no. 5 (March 31, 2014): 835–76. http://dx.doi.org/10.1112/s0010437x13007707.

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AbstractLet$X$be an algebraic curve. We study the problem of parametrizing geometric structures over$X$which are only generically defined. For example, parametrizing generically defined maps (rational maps) from$X$to a fixed target scheme$Y$. There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of$D$-modules ‘on’$B(K)\backslash G(\mathbb{A})/G(\mathbb{O})$, and we combine results about this category coming from the different presentations.

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