Topologies et faisceaux

Verdier, Jean Louis

doi:10.1007/bfb0081553

Cited by 10 publications

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“…Here we denote by C the category of sheaves on C, by i : C → C the inclusion functor and by a : C → C the associated sheaf functor (see [8,Théorème 3.4]). It follows from [9,Proposition 2.3] that Σ * f is left exact and it has a right adjointf * : Let X. be a simplicial object in E, namely a functor op → E. Consider the pull-back p X. : F (X.)…”

Section: F (α F (Y Y)) = F (F Id Yx (F )(Y) ) = (F (λ F ) Y X (Idmentioning

confidence: 99%

Representations of Internal Categories

Yamaguchi

2008

Kyushu J. Math.

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Abstract. Adams gave the notion of a Hopf algebroid generalizing the notion of a Hopf algebra and showed that certain generalized homology theories take values in the category of comodules over the Hopf algebroid associated with each homology theory. A Hopf algebra represents an affine group scheme which is a group in the category of a scheme and the notion of comodules over a Hopf algebra is equivalent to the notion of representations of the affine group scheme represented by a Hopf algebra. On the other hand, a Hopf algebroid represents a groupoid in the category of schemes. Therefore, it is natural to consider the notion of comodules over a Hopf algebroid as representations of the groupoid represented by a Hopf algebroid. This motivates the study of representations of groupoids, and more generally categories, for topologists. The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category, using the notion of a fibered category. IntroductionIn [1], Adams generalized the notion of Hopf algebras in the study of generalized homology theories satisfying certain conditions and showed that such a generalized homology theory, say E * , takes values in the category of comodules over the 'generalized Hopf algebra' associated with E * . The notion introduced by Adams is now called a Hopf algebroid which represents a functor taking values in the category of groupoids. Here 'a groupoid' means a special category whose morphisms are all isomorphisms. A comodule over a Hopf algebroid can be regarded as a representation of the groupoid represented by . The aim of this paper is to set a categorical foundation of representations of an internal category which is a category object in a given category.We begin by reviewing the notion of a fibered category following [7] and an internal category in Section 1, we give a detailed description on the relationship between the notions of a fibered category and a 2-category in Section 2, which is originally observed in [7, Section 8]. There, we show that the 2-category of a fibered category over a given category E is equivalent to the 2-category of 'lax functors' from the opposite category of E to the 2-category of categories. Our construction of fibered categories from lax functors allows us to give the notion of fibered categories represented by internal categories (Example 2.18) and a short definition (Example 2.19) of Grothendieck topoi over a simplicial object in the given site.

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Section: F (α F (Y Y)) = F (F Id Yx (F )(Y) ) = (F (λ F ) Y X (Idmentioning

confidence: 99%

Representations of Internal Categories

Yamaguchi

2008

Kyushu J. Math.

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“…The next step after finding an elementary theory of the category of sets seems by hindsight obvious. Giraud in [26] had given a (nonelementary) characterization of the category of sheaves on a site. Such a category was called a topos by Grothendieck.…”

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confidence: 99%

Book Review: Basic concepts of enriched category theory

Gray¹

1983

Bull. Amer. Math. Soc.

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Topologies et faisceaux

Cited by 10 publications

References 0 publications

Representations of Internal Categories

Representations of Internal Categories

Book Review: Basic concepts of enriched category theory

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