symmetric monoidal (∞,1)-category of spectra
The opposite of a magma – hence of a set with a binary operation $(x,y) \mapsto x y$ – has the same underlying set of elements, but binary operation changed by reversing the order of the factors: $(x,y) \mapsto y x$.
This is often considered for the case that $A$ a monoid, group, ring, or algebra (nonassociative or associative), in which case one speaks of the opposite group, opposite ring, etc.
Let $A$ be a magma, that is a set ${|A|}$ equipped with a binary operation
written as multiplication or juxtaposition.
Then the opposite magna $A^{op}$ (also denoted $A^*$ or $A^\perp$) has the same underlying set
but binary operation that of $A$ but with the ordering in the pair of arguments reversed:
If $A$ is a monoid or a group (or semigroup, quasigroup, loop, etc), the same definition applies, and we see that $A^op$ 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 $A^op$ 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.
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 $K Mod$ is an associative algebra over $K$, and a monoid object in Ab is a ring. So all of these have opposites.
If $A$ is commutative, then $A^op \cong A$. In fact, this isomorphism lives over $Set$ (or over the underlying monoidal category $C$), so we may write $A^op = A$ 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 $A$ 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 $A^op$ and the second as $A^co$.
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 $A^op$ and $A^co$. So $A^op$ reverses the 1-morphisms while $A^co$ 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.
Last revised on January 26, 2021 at 06:24:41. See the history of this page for a list of all contributions to it.