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Stable Infinity Categories
[article]

2009
*
arXiv
*
pre-print

We prove that the homotopy

arXiv:math/0608228v5
fatcat:d76mnnq6wjhozcuc4yh6cjx3se
*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. ...##
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Stable homotopy categories

1968
*
Bulletin of the American Mathematical Society
*

Accordingly, the "

doi:10.1090/s0002-9904-1968-11871-3
fatcat:fifndjqtnzdodhburigfb2ibim
*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*. ...##
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Realizing stable categories as derived categories

2013
*
Advances in Mathematics
*

Then the

doi:10.1016/j.aim.2013.08.017
fatcat:ylbfkq4c3jafxi6qp2557477he
*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] ). ...##
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Realizing stable categories as derived categories
[article]

2012
*
arXiv
*
pre-print

First we show that there exists a triangle-equivalence between the

arXiv:1201.5487v1
fatcat:ffqd4cyz4baixba2v4sfespfwi
*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*. ...##
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Noncommutative stable homotopy and stable infinity categories

2015
*
Journal of Topology and Analysis (JTA)
*

We show that the triangulated

doi:10.1142/s1793525315500077
fatcat:zxlajtrcyrblrg4duzayvdn7ca
*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*. ...##
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Stable model categories are categories of modules

2003
*
Topology
*

A

doi:10.1016/s0040-9383(02)00006-x
fatcat:3zf5q5nic5bubdadti3t233cum
*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. ...##
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Stable categories of Cohen-Macaulay modules and cluster categories
[article]

2015
*
arXiv
*
pre-print

As a byproduct, we give a triangle equivalence between the

arXiv:1104.3658v3
fatcat:cfoc26l4sbb4jlydl5ebxf7jna
*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*...##
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The N-Stable Category
[article]

2021
*
arXiv
*
pre-print

We identify this "N-

arXiv:2109.07728v2
fatcat:rxqourb5tvht7chtxdlbd25pxa
*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. ...##
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Balance in Stable Categories

2009
*
Algebras and Representation Theory
*

We study when the

doi:10.1007/s10468-008-9113-6
fatcat:3u6lmvmq55emzppytifs362hnm
*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*. ...##
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Recollements in stable ∞-categories
[article]

2016
*
arXiv
*
pre-print

From this we deduce a generalized associative property for n-fold gluing t_0...t_n, valid in any

arXiv:1507.03913v2
fatcat:2nohutatlvgyljezbudd6p4zli
*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 ...##
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Stable categories and reconstruction
[article]

2010
*
arXiv
*
pre-print

More precisely, given two self-injective algebras A and B and an equivalence between their

arXiv:1008.1976v1
fatcat:wwkjdyhmoveabf3tqnxcjxvqgu
*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. ...##
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Stable categories and reconstruction

2017
*
Journal of Algebra
*

More precisely, given two self-injective algebras A and B and an equivalence between their

doi:10.1016/j.jalgebra.2016.05.018
fatcat:2cj3q253dbbzrjhzxou2mokkuy
*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 ...##
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The Stable Monomorphism Category of a Frobenius category
[article]

2009
*
arXiv
*
pre-print

For a Frobenius abelian

arXiv:0911.1987v2
fatcat:4iql6vwv4bfivkxnx4nuibnjky
*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. ...##
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Differential Graded Categories are k-linear Stable Infinity Categories
[article]

2016
*
arXiv
*
pre-print

We describe a comparison between pretriangulated differential graded

arXiv:1308.2587v2
fatcat:z6fgeufggjgiriaow6oruitwb4
*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*. ...##
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The stable monomorphism category of a Frobenius category

2011
*
Mathematical Research Letters
*

For a Frobenius abelian

doi:10.4310/mrl.2011.v18.n1.a9
fatcat:xqtvl36nczewlcyewygt57n7nm
*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). ...
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