Maria Cristina Pedicchio

Some remarks on Maltsev and Goursat categories

Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.

Arithmetical categories and commutator theory

We characterize Maltsev categories in terms of internal groupoids (internal pregroupoids) and their associated commutators. As a consequence we get a description of arithmetical categories.

Categorical foundations : special topics in order, topology, algebra, and Sheaf theory

Introduction Walter Tholen 1. Ordered sets via adjunction R. J. Wood 2. Locales Jorge Picado, Ales Pultr and Anna Tozzi 3. A functional approach to general topology Maria Manuel Clementino, Eraldo Giuli and Walter Tholen 4. Regular, protomodular and abelian categories Dominique Bourn and Marino Gran 5. Aspects of monads John MacDonald and Manuela Sobral 6. Algebraic categories Maria Cristina Pedicchio and Fabrizio Rovatti 7. Sheaf theory Claudia Centazzo and Enrico M. Vitale 8. Beyond barr exactness: effective descent morphisms George Janelidze, Manuela Sobral and Walter Tholen.

A characterization of quasi-toposes

A locally presentable category 6 is the category of models of some theory F which can be described using a-limits, for a regular cardinal a. Our aim is to find conditions which ensure good properties of the category d of models: when is 6 Cartesian closed? When is d a quasi-topos? When is d a topos? and so on. In this paper, we focus our attention on some semantic conditions. A locally a-presentable category 8 can thus be presented as the category of cc-left exact presheaves on a small category

Groupoidal completely distributive lattices

9 with a-colimits. Clearly, each subpresheaf S of an E-left exact presheaf F has an a-left exact closure SC F. This closure operation plays a key role in the theory of a-left exact functors and we study, first, conditions on dz~Lex(%?) which ensure the universality of that closure operation. We prove this universality to be equivalent to the universality of strongly epimorphic families in &, but also to the fact that a-Lex(%‘) is an epireflective subcategory of a Grothendieck topos. This Grothendieck topos will play an important role in the rest of the paper; its construction depends heavily on the fact that, under our assumptions, a-left exact subpresheaves are stable under arbitrary intersections, from the point of view of the internal logic of the topos 4 of presheaves. Among the epireflective subcategories of a Grothendieck topos, we find the categories of separated objects for a given topology in this topos. These

Some remarks on internal pregroupoids in varieties

Abstract The Adamek and Pedicchio proof that top op is a quasi-variety is adapted to show that the opposite of the category of pre-ordered sets is also a quasi-variety. The constructive proof given requires a description of power objects in terms of (constructively) completely distributive lattices and such a description is provided by the Carboni and Walters notion of “groupoidal” object in a cartesian bicategory.

Left exact presheaves on a small pretopos

We investigate internal pregroupoid structures in congruence modular and congruence distributive varieties.

Frames and Grids

Given a small exact category E with finite colimits, we prove that the category Lex(E) of left exact presheaves on E is exact precisely when in E, the equivalence relation generated by a reflexive symmetric relation R is a finite iterate of R. This is in particular the case when E is Noetherian, that is, every ascending chain of subobjects is stationary. When this condition is satisfied and moreover E is a pretopos, Lex(E) becomes a topos. Various examples are given, distinguishing the possible situations

Summary of Contents

M. Barr and M.-C. Pedicchio introduced the category Grids of grids in order to show that the opposite of the category Top of topological spaces is a quasivariety. J. Adámek and M.-C. Pedicchio proved that there exists a duality D between the category TopSys of topological systems (defined by S. Vickers) and the category Grids. In both papers a description of the full subcategory D(Top) of the category Grids is given. In this paper we describe internally all grids isomorphic to the objects of the full coreflective subcategory D(Loc) of the category Grids, i.e. we characterize internally all grids of the form D(C), where C is a localic topological system (here Loc is the category of locales regarded as a full subcategory of TopSys). Since, obviously, the category Frm of frames is equivalent to D(Loc), we can say that in this paper those grids which could be called frames are characterized internally. An internal characterization of all grids which correspond (in the above sense) to the frames having T1 spectra and a generalization of the well-known fact that the spectrum of a locale is a sober space are obtained as well.

Diagram chasing in Mal'cev categories

Volume 1 Chapter 1. [Reserved] Chapter 2. Pre-Litigation Management and Avoidance Chapter 3. [Reserved] Chapter 4. Selection of Outside Counsel Chapter 5. Requests for Proposals, Bidding and Presentations Chapter 6. Marketing to Potential Corporate Clients Chapter 7. Optimizing the Number of Outside Counsel through Convergence and Partnering Strategies Chapter 8. Fee Arrangements Chapter 9. Engagement Letters (Including Written Corporate Policies and Procedures) Chapter 10. The Planning Process Chapter 11. Budgeting and Controlling Costs Chapter 12. Evaluating Legal Risks and Costs with Decision Tree Analysis Chapter 13. Communication Methods and Skills Chapter 14. Billing Chapter 15. Expenses and Disbursements Chapter 16. The Relationship Between the Legal Department and the Corporation Chapter 17. Law Department Management Chapter 18. Law Firm Staffing Chapter 19. Legal Research Management Chapter 20. Local and Specialized Outside Counsel