In this paper, we principally explore flat modules over a commutative ring with identity. We do this in relation to projective and injective modules with the help of derived functors like Tor and Ext. We also consider an extension of the property of flatness and induce analogies with the “special cases” occurring in flat modules. We obtain some results on flatness in the context of a noetherian...

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small t...

Contents 1. Introduction 1 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 13 5. Localization via Brown representatbility 23 6. Well generated triangulated categories 31 7. Localization for well generated categories 38 8. Epilogue: Beyond well-generatedness 46 Appendix A. The abelianization of a triangulated category ...

Coalgebra develops a general theory of transition systems, parametric in a functor T ; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T , a logic for T -coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step trans...

Coalgebra develops a general theory of transition systems, parametric in a functor T ; the functor T specifies the possible one-step behaviours of the system. A fundamental question in this area is how to obtain, for an arbitrary functor T , a logic for T -coalgebras. We compare two existing proposals, Moss’s coalgebraic logic and the logic of all predicate liftings, by providing one-step trans...

Using the skew-Hopf pairing, we obtain ℛ-matrix for two-parameter quantum algebra Uv,t. We further construct a strict monoidal functor

In this paper we present background results in enriched category theory and model necessary for developing categories of functors suitable doing functor calculus.

For a functor F ⊃ Id on the category of metrizable compacta, we introduce a conception of a linear functorial operator T = {TX : Pc(X) → Pc(FX)} extending (for each X) pseudometrics from X onto FX ⊃ X (briefly LFOEP for F ). The main result states that the functor SP G of G-symmetric power admits a LFOEP if and only if the action of G on {1, . . . , n} has a one-point orbit. Since both the hype...

Given a functor T : C → D carrying a class of morphisms S ⊂ C into a class S ⊂ D, we give sufficient conditions in order that T induces an equivalence on the localised categories. These conditions are in the spirit of Quillen’s theorem A. We give some applications in algebaic and birational geometry. Introduction Let T : C → D be a functor and S ⊂ C, S ⊂ D two classes of morphisms containing id...

The recursive construction of a function f : A → Θ consists, paradigmatically, of finding a functor T and maps α : A→ TA and θ : TΘ→ Θ such that f = α ; Tf ; θ. The role of the functor T is to marshall the recursive sub-arguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i...