For each standard affine data satisfying some condition on the coroots, we define a group functor on the category of all rings which is a subgroup of some group functor R 7→ G(R[t, t−1])o T (R), where G is a Chevalley scheme and T is a torus. This functor is then shown to satisfy the conditions of the five axioms required by [Ti4] to be a ‘minimal’ Kac-Moody group.

For $X$ a smooth scheme acted on by linear algebraic group $G$ and $p$ prime, the equivariant Chow ring $CH^*_G(X)\otimes \mathbb{F}_p$ is an unstable algebra over Steenrod algebra. We compute Lannes's $T$-functor applied to \mathbb{F}_p$. As application, we localization of away from $n$-nilpotent modules algebra, affirming conjecture Totaro as special case. The case when point $n = 1$ generali...

The aim of Coalgebraic Logic is to find formalisms that allow reasoning about T-coalgebras uniformly in the functor T. Moss' seminal idea was to consider the set functor T as providing a modality ∇ T , the semantics of which is given in terms of the relation lifting of T. The latter exists whenever T preserves weak pullbacks. In joint work with Marta Bílková, Alexander Kurz and Jiří Velebil, we...

For any set-endofunctor T : Set→ Set there exists a largest subcartesian transformation μ to the filter functor F : Set → Set. Thus we can associate with every T -coalgebra A a certain filter-coalgebra AF. Precisely, when T weakly preserves preimages, μ is natural, and when T weakly preserves intersections, μ factors through the covariant powerset functor P, thus providing for every T -coalgebr...

Author's Note. When this manuscript was submitted in January 1972, the editor asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complete: the only item no...

We study Gentzen-style proof theory of the finitary version of the coalgebraic logic introduced by L. Moss. The logic captures the behaviour of coalgebras for a large class of set functors. The syntax of the logic, defined uniformly with respect to a finitary coalgebraic type functor T , uses a single modal operator ∇T of arity given by the functor T itself, and its semantics is defined in term...

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat c...

In this paper we argue that the category of Stone spaces forms an interesting base category for coalgebras, in particular, if one considers the Vietoris functor as an analogue to the power set functor on the category of sets. We prove that the so-called descriptive general frames, which play a fundamental role in the semantics of modal logics, can be seen as Stone coalgebras in a natural way. T...

Contents 1. Introduction 1 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 14 5. Localization via Brown representability 24 6. Well generated triangulated categories 31 7. Localization for well generated categories 39 8. Epilogue: Beyond well generatedness 47 Appendix A. The abelianization of a triangulated category 4...

In the theory of coalgebras C over a ring R, the rational functor relates the category C∗M of modules over the algebra C∗ (with convolution product) with the category CM of comodules over C. This is based on the pairing of the algebra C∗ with the coalgebra C provided by the evaluation map ev : C∗ ⊗R C → R. The (rationality) condition under consideration ensures that CM becomes a coreflective fu...