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Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Isomorphisms Between Left And Right Adjoints
 Theory Appl. Categ
, 2003
"... There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the le ..."
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Cited by 15 (2 self)
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There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint specified by an appropriate "dualizing object". Typically the left adjoint is well understood while the right adjoint is more mysterious, and the result identifies the right adjoint in familiar terms. We give a categorical discussion of such results. One essential point is to di#erentiate between the classical framework that arises in algebraic geometry and a deceptively similar, but genuinely di#erent, framework that arises in algebraic topology.
Notes on Derived Functors and Grothendieck Duality, Foundations of Grothendieck Duality for Diagrams of Schemes
 Lecture Notes in Math. 1960
, 2009
"... Abstract: This is a polished version of notes begun in the late 1980s, largely available from my home page since then, meant to be accessible to midlevel graduate students. The first three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, o ..."
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Cited by 11 (0 self)
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Abstract: This is a polished version of notes begun in the late 1980s, largely available from my home page since then, meant to be accessible to midlevel graduate students. The first three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded complexes, of the sheaf functors ⊗, Hom, f ∗ and f ∗ (where f is a ringedspace map). Included are some enhancements, for concentrated ( = quasicompact and quasiseparated) schemes, of classical results such as the projection and Künneth isomorphisms. The fourth chapter presents the abstract foundations of Grothendieck Duality—existence and torindependent base change for the right adjoint of the derived functor Rf ∗ when f is a quasiproper map of concentrated schemes, the twisted inverse image pseudofunctor for separated finitetype maps of noetherian schemes, some refinements for maps of finite