nLab
opposite magma

Opposite magmas (monoids / groups / rings / algebras)

Idea

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

Definitions

In SetSet

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

(1)x*yyx. x * y \coloneqq y x .

This defines a new magma, the opposite of AA, denoted A opA^op (also sometimes A *A^* or A A^\perp).

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.

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 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.

Revised on September 20, 2010 19:10:49 by Toby Bartels (64.89.59.162)