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Higher topos theory
, 2006
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain com ..."
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Cited by 47 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of Gvalued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 32 (20 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
On ∞topoi
, 2003
"... Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalu ..."
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Cited by 12 (0 self)
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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of Gvalued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an EilenbergMacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous realvalued functions on X. If X does not admit many realvalued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields
Factorization Systems For Symmetric CatGroups
 THEORY AND APPLICATIONS OF CATEGORIES, PREPRINT
, 2000
"... This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective fu ..."
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Cited by 6 (0 self)
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This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2categorical sense, in the 2category of symmetric catgroups. An application to exact sequences is given.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Cited by 6 (0 self)
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
On the 2categorical duals of (full and) faithful functors
, 2009
"... We consider the classes of functors F: A → B such that for all C, the precomposition functors (−) ◦ F: C B → C A are faithful respectively full and faithful. Since F is (full and) faithful if and only if all postcomposition functors F ◦(−) : A C → B C are (full and) faithful, these classes of funct ..."
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We consider the classes of functors F: A → B such that for all C, the precomposition functors (−) ◦ F: C B → C A are faithful respectively full and faithful. Since F is (full and) faithful if and only if all postcomposition functors F ◦(−) : A C → B C are (full and) faithful, these classes of functors can be considered 2categorical duals of the classes of (full and) faithful functors. John Baez and Michael Shulman mention these classes of functors briefly in [2, page 48], and give sufficient conditions for functors to be contained. We give necessary and sufficient conditions and some examples. Remark It was pointed out by Mathieu Dupont that the results presented in this note were in fact already proved as soon as 2001 by by Admek, El Bashir, Sobral and Velebil in [1]. I’ll leave the file available for the moment, because it is linked on the nCategory Café. 1
Reprints in Theory and Applications of Categories, No. 23, 2013, pp. 1–165. NONABELIAN COHOMOLOGY IN
"... The following document was accepted as an AMS Memoir but was never published as I will now explain. When it was being typed in the final form for publication (before the day of TEX!) the secretary, who had never used the mathematical electronic text then required lost completely over one half of the ..."
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The following document was accepted as an AMS Memoir but was never published as I will now explain. When it was being typed in the final form for publication (before the day of TEX!) the secretary, who had never used the mathematical electronic text then required lost completely over one half of the manuscript. I had another student’s thesis which needed typing and did not require the electronic text. I had her drop the paper and do the thesis, planning to come back to the paper at a later time. Much later I finally learned to type using TEX and planned to come back to the paper. Unfortunately, a stroke prevented my ever completing it myself. Recently, a former student of mine, Mohammed Alsani, an expert in TEX, offered to type the long manuscript and recently did so. The resulting paper, which I have left unchanged from its original form, except for minor changes made thanks to Mike Barr to make it compatible with TAC, is being presented here in the hope that it may still find some use in the mathematical community. The notion of morphism used here, which Grothendieck liked a lot, and its relation with that of
DOI: 10.1016/j.aim.2004.05.004 Homotopical Algebraic Geometry I:
, 2013
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
Abstract
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of A. Joyal and R. Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a onetoone correspodence between Stopologies on an Scategory T, and certain left exact Bousfield localizations of the model category of prestacks on T. Based on the above results, we study the notion of model topos introduced by C. Rezk, and we relate it to our model categories of stacks over Ssites. In the second part of the paper, we present a parallel theory where Scategories, Stopologies and Ssites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over Ssites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale Ktheory of ring