Editing Equality (mathematics) (section)

=== Isomorphism ===
{{Main|Isomorphism}}

In mathematics, especially in [[abstract algebra]] and [[category theory]], it is common to deal with objects that already have some internal [[Mathematical structure|structure]].  An [[isomorphism]] describes a kind of structure-preserving correspondence between two objects, establishing them as essentially identical in their structure or properties.<ref name="Isomorphism2024" /><ref>{{Citation |last=Leinster |first=Tom |title=Basic Category Theory |date=2016-12-30 |page=12 |arxiv=1612.09375}}</ref>

More formally, an isomorphism is a bijective [[Map (mathematics)|mapping]] (or [[morphism]]) <math>f</math> between two [[Set (mathematics)|sets]] or structures <math>A</math> and <math>B</math> such that <math>f</math> and its inverse <math>f^{-1}</math> preserve the [[Operation (mathematics)|operations]], [[Relation (mathematics)|relations]], or [[Function (mathematics)|functions]] defined on those structures.<ref name="Isomorphism2024">{{Cite encyclopedia |date=2024-11-25 |title=Isomorphism |encyclopedia=Encyclopædia Britannica |url=https://www.britannica.com/science/isomorphism-mathematics |access-date=2025-01-12}}</ref> This means that any operation or relation valid in <math>A</math> corresponds precisely to the operation or relation in <math>B</math> under the mapping. For example, in [[group theory]], a [[group isomorphism]] <math>f: G \mapsto H </math> satisfies <math>f(a * b) = f(a) * f(b)</math> for all elements <math>a, b,</math> where <math>*</math> denotes the group operation.<ref>{{harvnb|Pinter|2010|p=94}}.</ref>

When two objects or systems are isomorphic, they are considered indistinguishable in terms of their internal structure, even though their elements or representations may differ. For instance, all [[cyclic groups]] of order <math>\infty</math> are isomorphic to the integers, <math>\Z,</math> with addition.<ref>{{harvnb|Pinter|2010|p=114}}.</ref> Similarly, in [[linear algebra]], two [[vector spaces]] are isomorphic if they have the same [[Dimension (vector space)|dimension]], as there exists a [[Linear isomorphism|linear bijection]] between their elements.<ref>{{Cite book |last=Axler |first=Sheldon |url=https://linear.axler.net/LADR4e.pdf |title=Linear Algebra Done Right |publisher=[[Springer (publisher)|Springer]] |page=86}}</ref>

The concept of isomorphism extends to numerous branches of mathematics, including [[graph theory]] ([[graph isomorphism]]), [[topology]] ([[homeomorphism]]), and algebra (group and [[Ring isomorphism|ring isomorpisms]]), among others. Isomorphisms facilitate the classification of mathematical entities and enable the transfer of results and techniques between similar systems. Bridging the gap between isomorphism and equality was one motivation for the development of [[category theory]], as well as for [[homotopy type theory]] and [[univalent foundations]].<ref>{{cite journal |last1=Eilenberg |first1=S. |last2=Mac Lane |first2=S. |date=1942 |title=Group Extensions and Homology |journal=Annals of Mathematics |volume=43 |issue=4 |pages=757–831 |issn=0003-486X |jstor=1968966}}</ref><ref>{{cite encyclopedia |last=Marquis |first=Jean-Pierre |date=2019 |title=Category Theory |url=https://plato.stanford.edu/entries/category-theory/ |access-date=26 September 2022 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |publisher=Department of Philosophy, [[Stanford University]]}}</ref><ref>{{cite book |last1=Hofmann |first1=Martin |title=Twenty Five Years of Constructive Type Theory |last2=Streicher |first2=Thomas |author2-link=Thomas Streicher |date=1998 |publisher=Clarendon |isbn=978-0-19-158903-4 |editor1-last=Sambin |editor1-first=Giovanni |series=Oxford Logic Guides |volume=36 |pages=83–111 |chapter=The groupoid interpretation of type theory |mr=1686862 |editor2-last=Smith |editor2-first=Jan M. |chapter-url=https://books.google.com/books?id=pLnKggT_In4C&pg=PA83}}</ref>