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Even if the shapes, sizes, and objects are radically different, they will appear as a group if they are close. Refers to the way smaller elements are "assembled" in a composition. Also called "grouping", the principle concerns the effect generated when the collective presence of the set of elements becomes more meaningful than their presence as ...
For example: suppose T is the theory of groups. Then Mod( T ) is the category of groups, and the compact objects in Mod( T ) are the finitely presented groups. The compact objects in the derived category D ( R − Mod ) {\displaystyle D(R-{\text{Mod}})} of R -modules are precisely the perfect complexes .
An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. A localic group is a group object in the category of locales. The group objects in the category of groups (or monoids) are the abelian groups.
A container, usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem. A set of objects, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that ...
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. [7] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group.
Each group is named by Small Groups library as G o i, where o is the order of the group, and i is the index used to label the group within that order.. Common group names: Z n: the cyclic group of order n (the notation C n is also used; it is isomorphic to the additive group of Z/nZ)
In natural language and physical science, a physical object or material object (or simply an object or body) is a contiguous collection of matter, within a defined boundary (or surface), that exists in space and time. Usually contrasted with abstract objects and mental objects. [1] [2]
A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at ...