nLab opposite magma

Opposite magmas

Opposite magmas


The opposite of a magma – hence of a set with a binary operation (x,y)xy(x,y) \mapsto x y – has the same underlying set of elements, but binary operation changed by reversing the order of the factors: (x,y)yx(x,y) \mapsto y x.

This is often considered for the case that AA a monoid, group, ring, or algebra (nonassociative or associative), in which case one speaks of the opposite group, opposite ring, etc.


In SetSet

Let AA be a magma, that is a set |A|{|A|} equipped with a binary operation

|A|×|A| |A| (x,y) xyxy \array{ {\left\vert A \right\vert} \times {\left\vert A \right\vert} &\longrightarrow& {\left\vert A \right\vert} \\ (x,y) &\mapsto& x \cdot y \mathrlap{ \coloneqq {\color{blue}x} {\color{red}y} } }

written as multiplication or juxtaposition.

Then the opposite magna A opA^{op} (also denoted A *A^* or A A^\perp) has the same underlying set

|A op||A| \left\vert A^{op} \right\vert \;\coloneqq\; \left\vert A \right\vert

but binary operation that of AA but with the ordering in the pair of arguments reversed:

(1)|A op|×|A op| |A op| (x,y) x*yyx. \array{ {\left\vert A^{op} \right\vert} \times {\left\vert A^{op} \right\vert} &\longrightarrow& {\left\vert A^{op} \right\vert} \\ (x,y) &\mapsto& x * y \mathrlap{ \coloneqq {\color{red}y}{\color{blue}x} } \,. }

If AA is a monoid or a group (or semigroup, quasigroup, loop, etc), the same definition applies, and we see that A opA^op is again a monoid or a group (etc).

If AA is a ring or a KK-algebra, the same definition applies, and we see that A opA^op is again a ring or a KK-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 KK\,Mod; see below.

In other categories

The notion of magma makes sense in any monoidal category CC. The notion of opposite does not make sense in this general context, because we must switch the order of the variables xx and yy 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 KK\,Mod is a nonassociative algebra over KK, a monoid object in KModK Mod is an associative algebra over KK, and a monoid object in Ab is a ring. So all of these have opposites.

Commutative magmas

If AA is commutative, then A opAA^op \cong A. In fact, this isomorphism lives over SetSet (or over the underlying monoidal category CC), so we may write A op=AA^op = A 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 AA 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 opA^op and the second as A coA^co.

If we categorify and oidify, then we get the concept of 2-category. Again, a 22-category AA has 22 kinds of opposites, again denoted A opA^op and A coA^co. So A opA^op reverses the 1-morphisms while A coA^co reverses the 2-morphisms. See opposite 2-category.

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

Last revised on June 5, 2021 at 14:23:21. See the history of this page for a list of all contributions to it.