Editing Equivalence of categories (section)

{{Use American English|date = January 2019}}
{{Short description|Abstract mathematics relationship}}
{{More footnotes|date=June 2015}}
In [[category theory]], a branch of abstract [[mathematics]], an '''equivalence of categories''' is a relation between two [[Category (mathematics)|categories]] that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

If a category is equivalent to the [[dual (category theory)|opposite (or dual)]] of another category then one speaks of
a '''duality of categories''', and says that the two categories are '''dually equivalent'''.

An equivalence of categories consists of a [[functor]] between the involved categories, which is required to have an "inverse" functor. However, in contrast to the situation common for [[isomorphism]]s in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping. Instead it is sufficient that each object be ''[[natural transformation|naturally isomorphic]]'' to its image under this composition. Thus one may describe the functors as being "inverse up to isomorphism". There is indeed a concept of [[isomorphism of categories]] where a strict form of inverse functor is required, but this is of much less practical use than equivalence.