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For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models. As another example, one may take the subcategory of Cat consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms.
Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group.
Algebra (from Arabic: الجبر, romanized: al-jabr, lit. 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols. The following category includes articles about algebra.
Given a group G and a commutative ring R, one can construct RG, known as the group algebra; it is an R-module equipped with a multiplication. A group is the same as a category with a single object in which all morphisms are isomorphisms (where the elements of the group correspond to the morphisms of the category), so the following construction generalizes the definition of the group algebra ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain ...