Calculus of Fractions and Homotopy Theory

Gabriel, Peter; Zisman, Michel

doi:10.1007/978-3-642-85844-4

Cited by 922 publications

(596 citation statements)

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“…The derived category D(A) is the localization of the category C(A) with respect to the class of quasi-isomorphisms. Thus, its objects are the dg modules and its morphisms are obtained from morphisms of dg modules by formally inverting [53] all quasi-isomorphisms. The projection functor C(A) → D(A) induces a functor H(A) → D(A) and the derived category could equivalently be defined as the localization of H(A) with respect to the class of all quasi-isomorphisms.…”

Section: 3mentioning

confidence: 99%

Differential graded categories

Tabuada¹

2015

University Lecture Series

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Section: 3mentioning

confidence: 99%

Differential graded categories

Tabuada¹

2015

University Lecture Series

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“…This is a variation of the logic presented by G. Roşu: the rules, and also the assumptions for proving the completeness, are slightly different. We use ideas of the classical work [11] of Gabriel and Zisman on the calculus of fractions, as exploited by Hébert, Adámek and Rosický in [18]. We start by recalling that concept.…”

Section: Remarkmentioning

confidence: 99%

A logic of implications in algebra and coalgebra

2009

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Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective w.r.t. that epimorphism. G. Roçu formulated a logic for deriving an implication from other implications. We present two versions of implicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains and codomains). In categories Alg Σ of algebras on a given signature our logic specializes to the implicational logic of R. Quackenbush. In categories Coalg H of coalgebras for a given accessible endofunctor H of sets we derive a logic for implications in the sense of P. Gumm.

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“…where P is a projective resolution of the constant functor O in F(C; O), and where Hom F(C;O) (P, F ) is the cochain complex of O-modules obtained from applying the contravariant functor Hom F(C;O) (−, F ) to the chain complex P. See [6], [7] for more background information on functor cohomology.…”

Section: Typeset By a M S-t E Xmentioning

confidence: 99%

Fusion category algebras

Linckelmann

2004

Journal of Algebra

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This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link: Markus LinckelmannAbstract. The fusion system F on a defect group P of a block b of a finite group G over a suitable p-adic ring O does not in general determine the number l(b) of isomorphism classes of simple modules of the block. We show that conjecturally the missing information should be encoded in a single second cohomology class α of the constant functor with value k × on the orbit categoryF c of F-centric subgroups Q of P of b which "glues together" the second cohomology classes α(Q) of AutF (Q)

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Calculus of Fractions and Homotopy Theory

Cited by 922 publications

References 0 publications

Differential graded categories

Differential graded categories

A logic of implications in algebra and coalgebra

Fusion category algebras

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