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Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 25 (9 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
On Sifted Colimits And Generalized Varieties
, 1999
"... Filtered colimits, i.e., colimits over schemes D such that Dcolimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that Dcolimits in Set commute with finite products. An important example: reflexive coequalizers are sif ..."
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Cited by 13 (1 self)
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Filtered colimits, i.e., colimits over schemes D such that Dcolimits in Set commute with finite limits, have a natural generalization to sifted colimits: these are colimits over schemes D such that Dcolimits in Set commute with finite products. An important example: reflexive coequalizers are sifted colimits. Generalized varieties are defined as free completions of small categories under siftedcolimits (analogously to finitely accessible categories which are free filteredcolimit completions of small categories). Among complete categories, generalized varieties are precisely the varieties. Further examples: category of fields, category of linearly ordered sets, category of nonempty sets.
Flatness, preorders and general metric spaces
, 2008
"... This paper studies a general notion of flatness in the enriched context: Pflatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category FlatP(A) ofPflat presheaves over A. This completion is related to the free cocompletion un ..."
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Cited by 2 (0 self)
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This paper studies a general notion of flatness in the enriched context: Pflatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category FlatP(A) ofPflat presheaves over A. This completion is related to the free cocompletion under a class of colimits defined by Kelly. We define a notion of Qaccessible categories for a family Q of indexes. Our FlatP(A) for small A’s are exactly the Qaccessible categories where Q is the class of Pflat indexes. For a category A, for P =P0 the class of all presheaves, FlatP0(A) is the Cauchycompletion of A. Two classes P1 andP2 of interest for general metric spaces are considered. TheP1 andP2 flatness are investigated and the associated completions are characterized for general metric spaces (enrichments over IR+) ¯ and preorders (enrichments over Bool). We get this way two nonsymmetric completions for metric spaces and retrieve the ideal completion for preorders. 1
Categories: A Free Tour
"... Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. ..."
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Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. In the past decades, the subject of mathematics has experienced an explosive increase both in diversity and in the sheer amount of published material. (E.g., the Mathematical Reviews volume of 1950 features 766 pages of reviews, compared to a total of 4550 pages in the six volumes for the rst half of 2000.) It has thus become inevitable that this growth, taking place in numerous and increasingly disconnected branches, be complemented by some form of unifying theory. There have been attempts at such unications in the past, such as Birkhostyle universal algebra or the encyclopedic work of Bourbaki. However, the most successful and universal approach so far is certainly the theory of cat...