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In mathematics, (in general) what we mean by identification between two things? Shoud we find a bijection between the sets that we take this thing? or it is something else, if so please explain.

Andrés E. Caicedo
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Mary Maths
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    Usually we want to have an isomorphism, e.g., of abstract groups. See also this question. Depending on the "category" we would require more, e.g., for topological groups an isomorphism of abstract groups which is a homeomorphism for the underlying topological spaces. – Dietrich Burde Jan 15 '20 at 20:14
  • See also the notion of an equivalence relation, i.e. a relation possessing properties of symmetry, reflexivity, and transitivity. The term "identification" may be used with such a relation in connection to partitioning a set into equivalent items. – hardmath Jan 18 '20 at 02:28

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In general the term "isomorphism" is used to denote "things" which are identical in respect of the particular mathematical properties of interest in the particular case.

So isomorphic groups have the same cardinality (bijection) but also behave the same way under the group operation.

There are other terms used eg in topology "homeomorphic" - this refers not just to the "things" but the fact also that the bijection is continuous both ways - so sometimes the properties of the maps/functions between things are important as well as the things themselves.

In each field of (pure) mathematics the question "when are two objects to be treated as essentially the same" is one of the key questions, and is answered in relation to the particular properties of interest in that field.

Mark Bennet
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