Singular curves and quasi-hereditary algebras

In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of $X$ and the derived category of finite length modules over a certain artinian quasi-hereditary ring $Q$ depending purely on the local singularity types of $X$. Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on $X$. Moreover, in the case $X$ is rational and projective we construct a finite dimensional quasi-hereditary algebra $\Lambda$ such that the triangulated category of perfect complexes on $X$ embeds into $D^b(\Lambda-\mathsf{mod})$ as a full subcategory.Comment: minor changes; to appear in IMR

NIP omega-categorical structures: the rank 1 case

We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections 3 and

Approximation in quantale-enriched categories

Our work is a fundamental study of the notion of approximation in V-categories and in (U,V)-categories, for a quantale V and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of V- and (U,V)-categories. We fully characterize continuous V-categories (resp. (U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale V and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory.Comment: 17 page

A fresh perspective on canonical extensions for bounded lattices

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of thetheory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploscica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphismsthe Ploscica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploscica's paper. This leads to a construction of canonical extension valid for all bounded lattices,which is shown to be functorial, with the property that the canonical extension functor decomposes asthe composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors

Parabolic subgroups of Garside groups II: ribbons

We introduce and investigate the ribbon groupoid associated with a Garside group. Under a technical hypothesis, we prove that this category is a Garside groupoid. We decompose this groupoid into a semi-direct product of two of its parabolic subgroupoids and provide a groupoid presentation. In order to established the latter result, we describe quasi-centralizers in Garside groups. All results hold in the particular case of Artin-Tits groups of spherical type

Non-commutative crepant resolutions: scenes from categorical geometry

Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.Comment: 57 pages; final version, to appear in "Progress in Commutative Algebra: Ring Theory, Homology, and Decompositions" (Sean Sather-Wagstaff, Christopher Francisco, Lee Klingler, and Janet Vassilev, eds.), De Gruyter. Incorporates many small bugfixes and adjustments addressing comments from the referee and other

Bourn-normal monomorphisms in regular Mal'tsev categories

Normal monomorphisms in the sense of Bourn describe the equivalence classes of an internal equivalence relation. Although the definition is given in the fairly general setting of a category with finite limits, later investigations on this subject often focus on protomodular settings, where normality becomes a property. This paper clarifies the connections between internal equivalence relations and Bourn-normal monomorphisms in regular Mal'tesv categories with pushouts of split monomorphisms along arbitrary morphisms, whereas a full description is achieved for quasi-pointed regular Mal'tsev categories with pushouts of split monomorphisms along arbitrary morphisms.Comment: This vesion fixes one error present in the last section of the previous versio