Let be a magma, that is a set equipped with a binary operation written as multiplication or juxtaposition. Then the same set may be equipped with another binary operation which we will write as . Specifically,
x * y \coloneqq y x .
This defines a new magma, the opposite of , denoted (also sometimes or ).
If is a ring or a -algebra, the same definition applies, and we see that is again a ring or a -algebra (including such special cases of algebra as an associative algebra, Lie algebra, etc). However, one can also interpret this situation as internal to Ab or Mod; see below.
The notion of magma makes sense in any monoidal category . The notion of opposite does not make sense in this general context, because we must switch the order of the variables and in (1). It does make sense in a braided monoidal category, although now there are two ways to write it, depending on whether we use the braiding or its inverse to switch the variables. In a symmetric monoidal category, the definition not only makes sense but gives the same result either way.
In particular, a monoidal category has two kinds of opposites: one as a mere category (an oidified monoid) and one as a monoidal object (a categorified monoid). We denote the first as and the second as .
If we categorify and oidify, then we get the concept of 2-category. Again, a -category has kinds of opposites, again denoted and . So reverses the 1-morphisms while reverses the 2-morphisms. See opposite 2-category.