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Authors
Nicolas Michel
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In nature, one observes that a K-theory of an object is defined in two steps. First a structured category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a general framework that deals with the first step of this process. The K-theory of an object is defined via a category of locally trivial objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.
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Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old homological algebra (of derived functors between abelian categories) was established. This new homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the old homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the old homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript A Course on Derived Categories (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
Generalized differential cohomology theories, in particular differential K-theory (often called smooth K-theory), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will sta
We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...). One then applies to the category A_C a $K$-theory machine, which provides an infinite loop space that is the $K$-theory K(C) of the object C. We study the first step of this process. What are the kinds of objects to be studied via $K$-theory? Given these types of objects, what structured categories should one associate to an object to obtain $K$-theoretic information about it? And how should the morphisms of these objects interact with this correspondence? We propose a unified, conceptual framework for a number of important examples of objects studied in $K$-theory. The structured categories associated to an object C are typically categories of modules in a monoidal (op-)fibred category. The modules considered are locally trivial with respect to a given class of trivial modules and a given Grothendieck topology on the object Cs category.
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