Opposite magmas (monoids / groups / rings / algebras)
Every magma has an opposite in which the operation goes the other direction. This is especially applied when is a monoid, group, ring, or algebra (nonassociative or associative).
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,
This defines a new magma, the opposite of , denoted (also sometimes or ).
If is a monoid or a group (or semigroup, quasigroup, loop, etc), the same definition applies, and we see that is again a monoid or a group (etc).
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.
In other categories
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 magma object in Mod is a nonassociative algebra over , a monoid object in is an associative algebra over , and a monoid object in Ab is a ring. So all of these have opposites.
If is commutative?, then . In fact, this isomorphism lives over (or over the underlying monoidal category ), so we may write to denote this.
The concept of monoid may be oidified to that of category; the concept of opposite monoid is then oidified to that of opposite category.
The concept of monoid may also be categorified to that of monoidal category; the concept of opposite monoid is then categorified to that of opposite monoidal category?.
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.
An -category has kinds of opposites. See (or write) opposite n-category?. A monoidal n-category? has kinds of opposites.