In fundamental physics, notably in quantum field theory and string theory one often says that a non-trivial equivalence between two models (in theoretical physics) is a duality.
While most of these dualities refer to equivalences between quantum field theories, they find their conceptual explanation in string theory. See at
Relation to mathematical duality
In some cases such as Montonen-Olive duality/S-duality the equivalence involves some actual duality in the mathematical sense, as in replacing the gauge group by its Langlands dual group. In T-duality only simple cases exhibit such obviously “dual” behaviour and in general cases such as U-duality really only the notion of equivalence remains.
One way to pseudo-formalize accurately most of what is usually meant by “duality” in physics might be the following.
Write for a moduli stack of prequantum field theory data consisting of species of fields and of Lagrangians/action functionals defined on these.
For instance for well-understood mirror symmetry this would be the usual moduli space of Calabi-Yau manifolds regarded as the Lagrangian data for the 2d (2,0)-superconformal QFT.
One imagines that quantization gives a map from such prequantum data to a moduli stack of actual quantum field theories
For instance, in the case of mirror symmetry this would be the TCFT-construction that takes a Calabi-Yau manifold to the corresponding Calabi-Yau category.
The 1-image of this map would be the moduli space of Lagrangian quantum field theories
By assumption this is now a 1-epimorphism and hence an atlas of moduli stacks.
The physical concept of duality, such as in mirror symmetry, says that two points in the space of Lagrangian data are “dual” to each other, if they become equivalent as quantum field theories after quantization.
Mathematically this means that the space of such “dualities” is the homotopy fiber product
By definition, an element of is two Lagrangians and a choice of equivalence of their associated quantum field theories:
This construction is the first step in associating the groupoid object in an (infinity,1)-category which is induced by the atlas “quantization” via Giraud's theorem of Higher Topos Theory.
This groupoid has as moduli stack of objects and as moduli stack of 1-morphisms . Its corresponding stack realization is
There is also a duality in the description of physics:
duality between algebra and geometry in physics: