A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category

REVIEWS-Sketches of an elephant: A topos theory compendium

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Revetements Etales Et Groupe Fondamental Sga 1

Basic concepts of enriched category theory

Was sind und was sollen die Zahlen

Algebraic theories, introduced as a concept in the 1960s, have been a fundamental step towards a categorical view of general algebra. Moreover, they have proved very useful in various areas of mathematics and computer science. This carefully developed book gives a systematic introduction to algebra based on algebraic theories that is accessible to both graduate students and researchers. It will facilitate interactions of general algebra, category theory and computer science. A central concept is that of sifted colimits - that is, those commuting with finite products in sets. The authors prove the duality between algebraic categories and algebraic theories and discuss Morita equivalence between algebraic theories. They also pay special attention to one-sorted algebraic theories and the corresponding concrete algebraic categories over sets, and to S-sorted algebraic theories, which are important in program semantics. The final chapter is devoted to finitary localizations of algebraic categories, a recent research area.

Algebraic Theories: A Categorical Introduction to General Algebra

We introduce a notion of a subtractive category. It generalizes the notion of a pointed subtractive variety of universal algebras in the sense of A. Ursini. Subtractive categories are closely related to Mal’tsev and additive categories: (i) a category C with finite limits is a Mal’tsev category if and only if for every object X in C the category Pt(X)=((X,1X)↓(C↓X)) of “points over X” is subtractive; (ii) a pointed category C with finite limits is additive if and only if C is subtractive and half-additive.

Subtractive Categories

We study closedness properties of internal relations in finitely complete categories, which leads to developing a unified approach to: Mal'tsev categories, in the sense of A. Carboni, J. Lambek and M. C. Pedicchio, that generalize Mal'tsev varieties of universal algebras; unital categories, in the sense of D. Bourn, that generalize pointed Jonsson-Tarski varieties; and subtractive categories, introduced by the author, that generalize pointed subtractive varieties in the sense of A. Ursini.

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Closedness properties of internal relations i: a unified approach to mal'tsev, unital and subtractive categories

We introduce a notion of an extended operation which should serve as a new tool for the study of categories like Mal’tsev, unital, strongly unital and subtractive categories. However, in the present paper we are only concerned with subtractive categories, and accordingly, most of the time we will deal with extended subtractions, which are particular instances of extended operations. We show that these extended subtractions provide new conceptual characterizations of subtractive categories and moreover, they give an enlarged “algebraic tool” for working in a subtractive category—we demonstrate this by using them to describe the construction of associated abelian objects in regular subtractive categories with finite colimits. Also, the definition and some basic properties of abelian objects in a general subtractive category is given for the first time in the present paper.

Subtractive Categories and Extended Subtractions

A good theory of ideals in regular multi-pointed categories

As J. W. Snow showed, every linear Mal’tsev condition on a variety $${\mathcal{V}}$$
 of universal algebras, is equivalent to a relational condition on $${\mathcal{V}}$$
 . Using slightly different relational reformulations of linear Mal’tsev conditions, we develop a purely categorical approach to these conditions.

Zurab Janelidze

Papers

Subtractive Categories

Closedness properties of internal relations i: a unified approach to mal'tsev, unital and subtractive categories

Subtractive Categories and Extended Subtractions

A good theory of ideals in regular multi-pointed categories

Closedness properties of internal relations V: Linear Mal'tsev conditions