Modified Traces and the Nakayama Functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

A revised augmented Cuntz semigroup

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

Isogenies, Connectedness, and Lie-Irreducibility

This chapter takes up the proofs of Theorems 8.1 and 8.2. For each prime to p integer n, we have the n'th power homomorphism [n] : G → G. Formation of the direct image is an exact functor from Perv to itself, which maps Neg to itself, in Ƿ to itself, and which (because a homomorphism) is compatible with middle convolution. So for a given object N in Garith, [n]* allows us to view 〈N〉arith as a Tannakian subcategory of 〈[n]* N〉arith, and 〈N〉geom as a Tannakian subcategory of 〈[n]* N〉geom.

The Axiom of Infinity and Transformations j: V → V

We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková–Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor, such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V → V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V → V known to be equivalent to the Axiom of Infinity.

On Representations of Grothendieck Toposes

Results of a representation-theoretic nature have played a major role in topos theory since the beginnings of the subject. For example, Deligne's theorem on coherent toposes, which says that every coherent topos has a continuous embedding into a topos of the form SetI for a discrete set I, is a typical result in the representation theory of toposes. (A continuous functor between toposes is the left adjoint of a geometric morphism. For Grothendieck toposes, it is exactly the same as a continuous functor between them, considered as sites with their canonical topologies. By a continuous functor between sites on left exact categories, we mean a left exact functor taking covers to covers.)A representation-like result for toposes typically asserts that a topos that satisfies some abstract conditions is related to a topos of some concrete kind; the relation between them is usually an embedding of the first topos in the second (concrete) one, for which the embedding satisfies some additional properties (fullness, etc.).

Triples and Localizations(1)

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.

A Remark on Flat and Projective Modules

It is the purpose of this note to give some characterizations of flat and projective modules, partly in ideal theoretical terms, partly in terms of the exterior product of a module (“puissance extérieure“); cf. (1).We shall consider left modules over a ring R with identity element and without proper zero divisors. The left module M is called flat if X ⊗R M is an exact functor on the category of right R-modules X. If M is flat over a commutative domain R, M is necessarily torsion-free. Therefore when looking for flatness of a module M over a commutative domain, one may assume from the start that M is torsion-free.In the following theorem, we shall not restrict ourselves to commutative rings R, but the modules concerned have to be torsion-free, which, of course, should mean that rm = 0 implies r = 0 or m = 0.