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2017
A category \(\mathcal{C}\) is formed by a class of objects \(\mathop{\mathrm{Ob}}\nolimits \mathcal{C}\) and a class of disjoint sets \(\mathop{\mathrm{Hom}}\nolimits (X,Y ) =\mathop{ \mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )\), one set for each ordered pair of objects \(X,Y \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\).
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A category \(\mathcal{C}\) is formed by a class of objects \(\mathop{\mathrm{Ob}}\nolimits \mathcal{C}\) and a class of disjoint sets \(\mathop{\mathrm{Hom}}\nolimits (X,Y ) =\mathop{ \mathrm{Hom}}\nolimits _{\mathcal{C}}(X,Y )\), one set for each ordered pair of objects \(X,Y \in \mathop{\mathrm{Ob}}\nolimits \mathcal{C}\).
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An example of a non-Fourier–Mukai functor between derived categories of coherent sheaves
Inventiones Mathematicae, 2014Orlov’s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier–Mukai functor.
Alice Rizzardo, M. Bergh, A. Neeman
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2022
This thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally promonoidal categories ... The third contribution of this thesis is the study of functors between categories of permutation representations.
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This thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally promonoidal categories ... The third contribution of this thesis is the study of functors between categories of permutation representations.
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2012
In this first chapter of Part 2 we give a general, rapid introduction to the required language from category theory.
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In this first chapter of Part 2 we give a general, rapid introduction to the required language from category theory.
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Derived categories and functors [PDF]
, 1971 For each abelian category A, there is a category D(A), called the derived category of A, whose objects are complexes of objects of A, and whose morphisms are formal fractions of homotopy classes of complex morphisms having as denominators homotopy classes inducing isomorphisms in cohomology.
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, 2014
Note to the reader Introduction 1. Categories, functors and natural transformations 2. Adjoints 3. Interlude on sets 4. Representables 5. Limits 6. Adjoints, representables and limits Appendix: proof of the General Adjoint Functor Theorem Glossary of ...
T. Leinster
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Note to the reader Introduction 1. Categories, functors and natural transformations 2. Adjoints 3. Interlude on sets 4. Representables 5. Limits 6. Adjoints, representables and limits Appendix: proof of the General Adjoint Functor Theorem Glossary of ...
T. Leinster
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Categories of Categories and Categories of Functors
1972The composition of functors in 2.2.6 suggests the study of categories whose objects are categories and whose morphisms are functors. 2.2.7 leads to categories whose objects are functors C→D and whose morphisms are natural transformation. However, familiar antinomies like “the set of all sets.” or “the set of all sets not containing themselves as an ...
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Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
, 2013This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya
Cheol-Hyun Cho +2 more
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2013
From previous paragraphs we have understood what a category is, how to define maps between categories, namely functors, and finally how to define maps between functors, which are natural transformations. It is now possible to group all these definitions in a coherent way and define the category of functors.
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From previous paragraphs we have understood what a category is, how to define maps between categories, namely functors, and finally how to define maps between functors, which are natural transformations. It is now possible to group all these definitions in a coherent way and define the category of functors.
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Monoidal categories and functors
2017The study of monoidal categories originated in the work of Jean Benabou [Ben] and Saunders Mac Lane [ML1]. In this chapter, we review the basics of the theory of monoidal categories.
Vladimir Turaev, Alexis Virelizier
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