Let A be a commutative noetherian ring. We investigate a class of functors from ≪commutative A-algebras≫ to ≪sets≫, which we call coherent. When such a functor F in fact takes its values in ≪abelian groups≫, we show that there are only finitely many prime numbers p such that pF (A) is infinite, and that none of these primes are invertible in A. This (and related statements) yield information ab...

The subcategory of a category and product of categories is defined. The inclusion functor is the injection (inclusion) map E ↪→ which sends each object and each arrow of a Subcategory E of a category C to itself (in C). The inclusion functor is faithful. Full subcategories of C, that is, those subcategories E of C such that HomE(a,b) = HomC(b,b) for any objects a,b of E, are defined. A subcateg...

The theory of product preserving functors and Weil functors is partly extended to infinite dimensional manifolds, using the theory of C∞-algebras.

1 Belian categories 3 1.1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Diagram chase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Snake Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Delta functors . . . . . . . . . . . . . . . . . . . . ....

We study spectral triples on (weighted) groups and consider functors between the categories of weighted groups and spectral triples. We study the properties of weights and the corresponding functor for spectral triples coming from discrete weighted groups.

We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of qua...

We study properties of functors on categories of sets (classes) together with set (class) functions. In particular, we investigate the notion of inclusion preserving functor, and we discuss various monotonicity and continuity properties of set functors. As a consequence of these properties, we show that some classes of set operators do not admit functorial extensions. Then, starting from Aczel’...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.