Categories for the Working Mathematician

Lane, Saunders Mac

doi:10.1007/978-1-4612-9839-7

Cited by 5,588 publications

(4,972 citation statements)

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“…In the language of monoidal categories, the smash product is a symmetric monoidal product on the category of pointed simplicial sets. We recall the definition of symmetric monoidal product, but for more details see [ML71,VII] or [Bor94,6.1]. Definition 1.1.1.…”

Section: Symmetric Spectramentioning

confidence: 99%

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Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

477

569

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The stable homotopy category, much studied by algebraic topologists, is a closed symmetric monoidal category. For many years, however, there has been no well-behaved closed symmetric monoidal category of spectra whose homotopy category is the stable homotopy category. In this paper, we present such a category of spectra; the category of symmetric spectra. Our method can be used more generally to invert a monoidal functor, up to homotopy, in a way that preserves monoidal structure. Symmetric spectra were discovered at about the same time as the category of S S -modules of Elmendorf, Kriz, Mandell, and May, a completely different symmetric monoidal category of spectra.

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Section: Symmetric Spectramentioning

confidence: 99%

“…Coherence of the natural isomorphisms means that all reasonable diagrams built from the natural isomorphisms also commute [ML71]. When the product is closed, the pairing Hom(X, Y ) : C op × C → C is an internal Hom.…”

Section: Symmetric Spectramentioning

confidence: 99%

Symmetric spectra

Hovey

Shipley

Smith³

1999

J. Amer. Math. Soc.

477

569

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“…If F is both fully faithful and dense, then it is an equivalence [ML98,IV.4], that is, there is a functor G: B −→ A such that both compositions are isomorphic to the respective identities. In this case write A ≃ B. Equivalences preserve and reflect essentially all "categorical" properties and attributes: mono-and epimorphisms, projectives, injectives, etc.…”

Section: A Morita Equivalencementioning

confidence: 99%

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Leuschke

2012

Progress in Commutative Algebra 1

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ABSTRACT. Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.

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“…We assume that the reader is familiar with the elementary concepts of category theory [McL71] and algebraic specifications [GTW78] [EM85].…”

Section: Introductionmentioning

confidence: 99%

Exception handling and term labelling

Bernot

Gall

1993

Lecture Notes in Computer Science

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We propose a new algebraic framework for exception handling which is powerful enough to cope with many exception handling features such that recovery, implicit propagation of exceptions, etc. This formalism is capable of treating all the exceptional cases, including the following ones: "intrinsic" exceptions which are related to the underlying data structure (for instance, popping an emptstack or applying predecessor on zero for natural numbers), exceptions which are relied on "dynamic" properties (as an acces to a non-initialized array cell) or else exceptions which are due to certain limitations (mainly bounded data structures). We show that within the already existing frameworks, the case of bounded data structures with certain recoveries of exceptional values remains unsolved.First, we justify the usefulness of "labelling" some terms in order to easily specify exceptions without inconsistency, and we then define a general framework of label algebras which allows us to "type" terms instead of values. Exception algebras and exception specifications are defined as a direct application of label algebras. Indeed, the usual inconsistency problems raised by exception handling are avoided by the possibility of labelling terms.As a conclusion, we also sketch out how far the application domain of label algebras seems to be much more general than exception handling.

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Categories for the Working Mathematician

Cited by 5,588 publications

References 0 publications

Symmetric spectra

Symmetric spectra

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Exception handling and term labelling

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