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On Modules Over Motivic Ring Spectra [article]

Elden Elmanto, Håkon Kolderup
2019 arXiv   pre-print
In this note, we provide an axiomatic framework that characterizes the stable ∞-categories that are module categories over a motivic spectrum.  ...  As an application, this gives an alternative approach to Röndigs and Ø stvæ r's theorem relating Voevodsky's motives with modules over motivic cohomology, and to Garkusha's extension of Röndigs and Ø stvæ  ...  γ C * : SH τ (C) → SH τ (Sm S ) is conservative and preserves colimits.  ... 
arXiv:1708.05651v4 fatcat:m4zosayijvcgvk5c4g6pcwvuv4

Motivic Tambara Functors [article]

Tom Bachmann
2018 arXiv   pre-print
In this article we provide an explicit description of NAlg(HI_0(k)) as the category of sheaves with generalized transfers and \'etale norms, and explain how this is closely related to the classical notion  ...  Let k be a field and denote by SH(k) the motivic stable homotopy category. Recall its full subcategory HI_0(k) of effective homotopy modules.  ...  The top composite is N f,V1 (using that R r C ∼ = R f V 1 , as established above, and r * R r C ∼ = f * R f V 1 , by transitivity of base change) and the bottom composite is N f,V2 , so commutativity is  ... 
arXiv:1807.02981v1 fatcat:4uba7qqrijhplmk3prow5hy6wa

Revisiting derived crystalline cohomology [article]

Zhouhang Mao
2021 arXiv   pre-print
We also develop a non-completed animated analogue of prisms and prismatic envelopes.  ...  This allows us to generalize classical results to non-flat and non-finitely-generated situations.  ...  We also thank Denis Nardin for discussions about ∞-categories and in particular, of simplicial homotopy theory in ∞-categories, and Yu Min for several discussions.  ... 
arXiv:2107.02921v1 fatcat:jgqqpj5cbnczxdk3a66shbiho4

Motivic Tambara functors

Tom Bachmann
2020 Mathematische Zeitschrift  
Let k be a field and denote by SH(k) the motivic stable homotopy category. Recall its full subcategory SH(k) eff♥ (Bachmann in J Topol 10(4):  ...  I would like to thank Marc Hoyois for teaching me essentially everything I know about ∞-categories, extensive discussions on normed spectra, and several discussions regarding the results in this article  ...  Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long  ... 
doi:10.1007/s00209-020-02581-x fatcat:23j4pca35zdphcmi7cztnivtly

Logical foundations of CafeOBJ

Răzvan Diaconescu, Kokichi Futatsugi
2002 Theoretical Computer Science  
Moreover, the design of CafeOBJ emerged from its logical foundations, and institution concepts played a crucial rôle in structuring the language design.  ...  concurrent speciÿcation and rewriting logic.  ...  Acknowledgements We thank the editors and both anonymous referees for their detailed and careful suggestions and comments which helped improving the presentation of the paper.  ... 
doi:10.1016/s0304-3975(01)00361-9 fatcat:c4hwlhey65fgdahyamgg6fiwsu

Derived Algebraic Geometry II: Noncommutative Algebra [article]

Jacob Lurie
2007 arXiv   pre-print
We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits in each variable), and thereby recover the classical smash-product  ...  We also develop an infinity-categorical theory of monads, and prove a version of the Barr-Beck theorem.  ...  (iii) Composition in C ⊗ is determined by composition of order preserving maps, composition in C, and the associativity and unit constraints of the monoidal structure on C.  ... 
arXiv:math/0702299v5 fatcat:6tdcjvtqtjctvlqxahu7kvrtcu

Modulated bicategories

Aurelio Carboni, Scott Johnson, Ross Street, Dominic Verity
1994 Journal of Pure and Applied Algebra  
In other words, when are all fibrations in LX? flat? Novel axioms on X are provided for this, and we call a bicategory S' modulated when sP' 1s such a X.  ...  Here, the choice of the conservative arrows, leads to our notion of faithfully conservative bicategory X in which two-sided discrete fibrations become the arrows of a bicategory 9 = DFib(X).  ...  The finite colimits of & provide local finite colimits and global finite bicategorical coproducts. A monad R on the ordered object (A,<) is precisely a transitive relation on A containing 5.  ... 
doi:10.1016/0022-4049(94)90009-4 fatcat:c4dcfnro65a6zk63of5tu3gifu

Algebraization and Tannaka duality [article]

Bhargav Bhatt
2014 arXiv   pre-print
Our goal in this paper is to identify certain naturally occurring colimits of schemes and algebraic spaces. To do so, we use (and prove) some new Tannaka duality theorems for maps of algebraic spaces.  ...  In particular, C admits all limits and colimits.  ...  To show preservation under filtered colimits, as i : C → Vect(X) preserves filtered colimits, it is enough to check that the composite QCoh(X) → Vect(X) preserves filtered colimits; this is a consequence  ... 
arXiv:1404.7483v1 fatcat:27wb3xkxgnc7zf5ij7omi7hoj4

Monadicity of localization for Lie super-algebras 𝔤𝔩(m, n) [article]

Slava Pimenov
2021 arXiv   pre-print
We show that the right localization is monadic in a suitable sense, which identifies the coderived category of 𝔤-modules with the ind-completion of compactly generated W-modules for some algebra W in  ...  We study the localization functor from the category of representation of Lie super-algebra 𝔤 = 𝔤𝔩(m, n) into monodromic D-modules on the flag manifold X = G/B.  ...  Since filtered colimits in the categories of A-modules andÃ-modules are calculated at the level of underlying vector spaces we see that both functors preserve filtered colimits and hence send coacyclic  ... 
arXiv:2110.00802v1 fatcat:77eafwrxjjgyddzyqeuluqbpuy

Chiral Koszul duality

John Francis, Dennis Gaitsgory
2011 Selecta Mathematica, New Series  
We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in [BD1], to higher-dimensional varieties.  ...  This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras.  ...  The general theory of monads, 7 implies: Since the functor Bar O , being a left adjoint, commutes with colimits, and since oblv O commutes with colimits and is conservative, we obtain that the functor  ... 
doi:10.1007/s00029-011-0065-z fatcat:imliquiasnav7jpkhzfdfkbkwm

Chiral Koszul duality [article]

John Francis, Dennis Gaitsgory
2011 arXiv   pre-print
We extend the theory of chiral and factorization algebras, developed for curves by Beilinson and Drinfeld in bd, to higher-dimensional varieties.  ...  This extension entails the development of the homotopy theory of chiral and factorization structures, in a sense analogous to Quillen's homotopy theory of differential graded Lie algebras.  ...  The general theory of monads, 7 implies: Since the functor Bar O , being a left adjoint, commutes with colimits, and since oblv O commutes with colimits and is conservative, we obtain that the functor  ... 
arXiv:1103.5803v4 fatcat:xyaevwh54fb4bgiff7tgtkggce

The cotangent complex and Thom spectra [article]

Nima Rasekh, Bruno Stonek
2020 arXiv   pre-print
In this work we first establish, in the context of ∞-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of E_∞-ring spectra that exist in  ...  Let R be an E_∞-ring spectrum and Pic(R) denote its Picard E_∞-group.  ...  The colimit of G f G G Pic(R) Mod R is the Thom R-module of f , denoted M f . In fact, Pic(R) is an E ∞ -group.  ... 
arXiv:2005.01382v2 fatcat:pcuelhxdwfd27k2e7pp43udtri

Algebraization and Tannaka duality

Bhargav Bhatt
2016 Cambridge Journal of Mathematics  
The goal of this paper is to identify certain naturally occurring colimits of schemes and algebraic spaces.  ...  This corollary answers a question raised in [NS10, §2] and pointed out to us by Nicaise.  ...  Acknowledgements I am very grateful to Vladimir Drinfeld, Johannes Nicaise, and Bjorn Poonen for bringing the algebraization questions treated here to my attention; to Johan de Jong and Ofer Gabber for  ... 
doi:10.4310/cjm.2016.v4.n4.a1 fatcat:44k2yxrnknbqncig5s4iggw3zu

Spectral algebras and non-commutative Hodge-to-de Rham degeneration [article]

D. Kaledin, A. Konovalov, K. Magidson
2019 arXiv   pre-print
We revisit the non-commutative Hodge-to-de Rham Degeneration Theorem of the first author, and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry  ...  Mathew for generously sharing his insights and expertise, and in particular, for helping us with the (sketch of the) proof of Proposition 2.3.  ...  Nikolaus and A. Prihodko for useful discussions, and to MSRI where part of this work was done. We are especially grateful to A.  ... 
arXiv:1906.09518v2 fatcat:xrkqs7enlrcxlarsjmz6rhtuum

The cotangent complex and Thom spectra

Nima Rasekh, Bruno Stonek
2021 Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg  
In this work we first establish, in the context of $$\infty $$ ∞ -categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ E ∞  ...  -ring spectra that exist in the literature are equivalent.  ...  The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.  ... 
doi:10.1007/s12188-020-00226-8 fatcat:tezvbl5xyjb4xowfmohbgxw4cy
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