nLab
opposite magma

Idea

Every magma $A$ has an opposite $A^{\mathrm{op}}$ in which the operation goes the other direction. This is especially applied when $A$ is a monoid, group, ring, or algebra (nonassociative or associative).

Definitions

In $\mathrm{Set}$

Let $A$ be a magma, that is a set $\mid A\mid$ equipped with a binary operation $\mid A\mid \times \mid A\mid \to \mid A\mid$ written as multiplication or juxtaposition. Then the same set $\mid A\mid$ may be equipped with another binary operation which we will write as $*$. Specifically,

(1)$$x*y≔yx.$$x * y \coloneqq y x .

This defines a new magma, the opposite of $A$, denoted $A^{\mathrm{op}}$ (also sometimes $A^{*}$ or $A^{\perp }$).

If $A$ is a monoid or a group (or semigroup, quasigroup, loop, etc), the same definition applies, and we see that $A^{\mathrm{op}}$ is again a monoid or a group (etc).

If $A$ is a ring or a $K$-algebra, the same definition applies, and we see that $A^{\mathrm{op}}$ is again a ring or a $K$-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 $K$Mod; see below.

In other categories

The notion of magma makes sense in any monoidal category $C$. The notion of opposite does not make sense in this general context, because we must switch the order of the variables $x$ and $y$ 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 $K$Mod is a nonassociative algebra over $K$, a monoid object in $K\mathrm{Mod}$ is an associative algebra over $K$, and a monoid object in Ab is a ring. So all of these have opposites.

Commutative magmas

If $A$ is commutative?, then $A^{\mathrm{op}}\cong A$. In fact, this isomorphism lives over $\mathrm{Set}$ (or over the underlying monoidal category $C$), so we may write $A^{\mathrm{op}}=A$ to denote this.

Categorifications

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 $A$ 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 $A^{\mathrm{op}}$ and the second as $A^{\mathrm{co}}$.

If we categorify and oidify, then we get the concept of 2-category. Again, a $2$-category $A$ has $2$ kinds of opposites, again denoted $A^{\mathrm{op}}$ and $A^{\mathrm{co}}$. So $A^{\mathrm{op}}$ reverses the 1-morphisms while $A^{\mathrm{co}}$ reverses the 2-morphisms. See opposite 2-category.

An $n$-category has $n$ kinds of opposites. See (or write) opposite n-category?. A monoidal n-category? has $n+1$ kinds of opposites.