Spectral sequences are a powerful bookkeeping tool, used to handle large amounts of information. As such, they have become nearly ubiquitous in algebraic topology and algebraic geometry. In this paper, we take a few results on faith (i.e., without proof, pointing to books in which proof may be found) in order to streamline and simplify the exposition. From the exact couple formulation of spectr...

Let $$(\mathbf{U}, \mathbf{U}^\imath )$$ be a quantum symmetric pair of Kac–Moody type. The $$\imath $$ groups $$\mathbf{U}^\imath and the universal $$\widetilde{\mathbf{U}}^\imath can viewed as generalization Drinfeld doubles $$\widetilde{\mathbf{U}}$$ . In this paper we formulate establish Serre–Lusztig relations for in terms divided powers, which are an -analog Lusztig’s higher order Serre g...

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopěnka’s principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. Th...

It is shown that the category of non-Archimedean metric spaces with l-Lipschitz maps can be embedded as a coreflectlve non-bireflective subcategory in the category of fuzzy uniform spaces. Consequential characterizations of topological and unif’orm properties are derived.

A protolocalisation of a regular category is a full reflective regular subcategory, whose reflection preserves pullbacks of regular epimorphisms along arbitrary morphisms. We devote special attention to the epireflective protolocalisations of an exact Mal’cev category; we characterise them in terms of a corresponding closure operator on equivalence relations. We give some examples in algebra an...

We give some new examples for which the Serre problem is solvable by using invariant pseudodistances.