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Stable Infinity Categories
[article]
2009
arXiv
pre-print
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We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a variety of constructions. ...
this stable infinity category by a universal mapping property. ...
If C is a stable ∞-category, then the opposite ∞-category C op is also stable. Remark 2.14. ...
arXiv:math/0608228v5
fatcat:d76mnnq6wjhozcuc4yh6cjx3se
Stable homotopy categories
1968
Bulletin of the American Mathematical Society
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Accordingly, the "stable categories" of Puppe are referred to below as triangulated categories (the word "stable" is itself used in a quite different way, cf. §1). ...
If F: Gt' ->0t is a stable equivalence of stable additive categories then AHi?-l (A) gives a bijection ïa^ïCfc'. ...
An additive category d is a torsion category if each object A is a torsion object, i.e. one such that for some integer h, h-lA = 0. LEMMA 17.1. JET 2 3 is a torsion category. ...
doi:10.1090/s0002-9904-1968-11871-3
fatcat:fifndjqtnzdodhburigfb2ibim
Realizing stable categories as derived categories
2013
Advances in Mathematics
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Then the stable category mod Z A of the category of Z-graded A-modules is a triangulated category (cf. [1] ). ...
Moreover the stable category mod Z/ Z A of the category of Z/ Z-graded A-modules is a triangulated category (cf. [1] ). ...
doi:10.1016/j.aim.2013.08.017
fatcat:ylbfkq4c3jafxi6qp2557477he
Realizing stable categories as derived categories
[article]
2012
arXiv
pre-print
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First we show that there exists a triangle-equivalence between the stable category of Z-graded A-modules and the derived category of a certain algebra Γ of finite global dimension. ...
Secondly we show that if A has Gorenstein parameter ℓ, then there exists a triangle-equivalence between the stable category of Z/ℓZ-graded A-modules and a derived-orbit category of Γ, which is a triangulated ...
Realizing stable categories as derived-orbit categories In this section, we compare the stable categories of self-injective algebras and derivedorbit categories. ...
arXiv:1201.5487v1
fatcat:ffqd4cyz4baixba2v4sfespfwi
Noncommutative stable homotopy and stable infinity categories
2015
Journal of Topology and Analysis (JTA)
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We show that the triangulated category NSH is topological as defined by Schwede using the formalism of (stable) infinity categories. ...
The noncommutative stable homotopy category NSH is a triangulated category that is the universal receptacle for triangulated homology theories on separable C^*-algebras. ...
stable homotopy category. ...
doi:10.1142/s1793525315500077
fatcat:zxlajtrcyrblrg4duzayvdn7ca
Stable model categories are categories of modules
2003
Topology
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A stable model category is a setting for homotopy theory where the suspension functor is invertible. ...
The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. ...
the original stable model category C. ...
doi:10.1016/s0040-9383(02)00006-x
fatcat:3zf5q5nic5bubdadti3t233cum
Stable categories of Cohen-Macaulay modules and cluster categories
[article]
2015
arXiv
pre-print
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As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay R-modules and the derived category of Λ. ...
By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the 1-cluster category of the path algebra of a Dynkin quiver (i.e ...
In [IO09] , it was shown that the stable categories of modules over d-preprojective algebras of (d − 1)-representationfinite algebras are triangle equivalent to generalized d-cluster categories of stable ...
arXiv:1104.3658v3
fatcat:cfoc26l4sbb4jlydl5ebxf7jna
The N-Stable Category
[article]
2021
arXiv
pre-print
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We identify this "N-stable category" via the monomorphism category and prove Buchweitz's theorem for N-complexes over a Frobenius exact abelian category. ...
A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes ...
); and c) stab(A), the stable category of A. ...
arXiv:2109.07728v2
fatcat:rxqourb5tvht7chtxdlbd25pxa
Balance in Stable Categories
2009
Algebras and Representation Theory
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2006 pre-print arXiv:math/0610122v1 fatcat:xaylykdtynb3le7yobnavtoaou
We study when the stable category of an abelian category modulo a full additive subcategory is balanced and, in case the subcategory is functorially finite, we study a weak version of balance. ...
The results in this second case apply very neatly to (generalizations of) hereditary abelian categories. ...
stable categories. ...
doi:10.1007/s10468-008-9113-6
fatcat:3u6lmvmq55emzppytifs362hnm
Recollements in stable ∞-categories
[article]
2016
arXiv
pre-print
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From this we deduce a generalized associative property for n-fold gluing t_0...t_n, valid in any stable ∞-category. ...
Such a classical result, well-known in the setting of triangulated categories, is recasted in the setting of stable ∞-categories and the properties of the associated (∞-categorical) factorization systems ...
Version 1 of the present paper is sensibly different from the present one; the unexpected (and actually undue) symmetric behavior of stable recollements (Lemma 4.3 of version 1, therein called the Rorschach ...
arXiv:1507.03913v2
fatcat:2nohutatlvgyljezbudd6p4zli
Stable categories and reconstruction
[article]
2010
arXiv
pre-print
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More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A. ...
That set satisfies some obvious properties of Hom-spaces and it generates the stable category of A. Keep now only S and A. Can B be reconstructed ? ...
More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A. ...
arXiv:1008.1976v1
fatcat:wwkjdyhmoveabf3tqnxcjxvqgu
Stable categories and reconstruction
2017
Journal of Algebra
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More precisely, given two self-injective algebras A and B and an equivalence between their stable categories, consider the set S of images of simple B-modules inside the stable category of A. ...
Let M be an A-module with a decomposition M ∼ M 1 ⊕ M 2 in the stable category. ...
Assume now that Hom There is a surjective map g : M → M/M 1 with filtrable kernel such that the composition M can −→ M g −→ M/M 1 is equal to the canonical map M → M/M 1 in the stable category, by Proposition ...
doi:10.1016/j.jalgebra.2016.05.018
fatcat:2cj3q253dbbzrjhzxou2mokkuy
The Stable Monomorphism Category of a Frobenius category
[article]
2009
arXiv
pre-print
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For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable ...
As an application, we give two characterizations to the stable category of Ringel-Schmidmeier (RS3). ...
We show that Mon(A) is a Frobenius exact category and then the stable category Mon(A) modulo projective objects is triangulated; it is called the stable monomorphism category of A. ...
arXiv:0911.1987v2
fatcat:4iql6vwv4bfivkxnx4nuibnjky
Differential Graded Categories are k-linear Stable Infinity Categories
[article]
2016
arXiv
pre-print
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We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. ...
We show the underlying infinity category of this model category is equivalent to the infinity category of k-linear stable infinity categories. ...
stable ∞-categories. ...
arXiv:1308.2587v2
fatcat:z6fgeufggjgiriaow6oruitwb4
The stable monomorphism category of a Frobenius category
2011
Mathematical Research Letters
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For a Frobenius abelian category A, we show that the category Mon(A) of monomorphisms in A is a Frobenius exact category; the associated stable category Mon(A) modulo projective objects is called the stable ...
As an application, we give two characterizations to the stable category of Ringel-Schmidmeier. ...
stable monomorphism category Mon(A). ...
doi:10.4310/mrl.2011.v18.n1.a9
fatcat:xqtvl36nczewlcyewygt57n7nm
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