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In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets.A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous.
Given a groupoid object (R, U), the equalizer of , if any, is a group object called the inertia group of the groupoid. The coequalizer of the same diagram, if any, is the quotient of the groupoid. Each groupoid object in a category C (if any) may be thought of as a contravariant functor from C to the category of groupoids.
A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible. [1] More explicitly, a groupoid is a set of objects with . for each pair of objects x and y, a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y; we write f : x → y to indicate that f is an element of G(x,y);
A group is said to act on another mathematical object if every group element can be associated to some operation on and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. [ 50 ]
On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts (see Maschke's theorem). These parts, in turn, are much more easily manageable than the whole V (via Schur's lemma).
As a monoidal category, any 2-group G has a unit object I G. The automorphism group of I G is an abelian group by the Eckmann–Hilton argument, written Aut(I G) or π 2 G. The fundamental group of G acts on either side of π 2 G, and the associator of G defines an element of the cohomology group H 3 (π 1 G, π 2 G).
A simple example is the category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.