TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an isomorphism class is a collection of
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

to each other. Isomorphism classes are often defined if the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are ordinals and
graphs Graph may refer to: Mathematics *Graph (discrete mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes an ...
. However, there are circumstances in which the isomorphism class of an object conceals vital internal information about it; consider these examples: * The
associative algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s consisting of
coquaternion In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra In mathematics Mathematics (from Ancient Greek, Greek: ) ...
s and 2 × 2
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
are isomorphic as
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
. Yet they appear in different contexts for application (plane mapping and kinematics) so the isomorphism is insufficient to merge the concepts. *In
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps come with homotopy, homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Be ...
, the
fundamental group In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

of a
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...
$X$ at a point $p$, though technically denoted $\pi_1\left(X,p\right)$ to emphasize the dependence on the base point, is often written lazily as simply $\pi_1\left(X\right)$ if $X$ is
path connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
. The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other; however, unless $\pi_1\left(X,p\right)$ is abelian this isomorphism is non-unique. Furthermore, the classification of
covering space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s makes strict reference to particular
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
s of $\pi_1\left(X,p\right)$, specifically distinguishing between isomorphic but
conjugate Conjugation or conjugate may refer to: Linguistics * Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.

References

{{DEFAULTSORT:Isomorphism Class Abstract algebra