symmetric monoidal (∞,1)-category of spectra
The opposite of a magma – hence of a set with a binary operation – has the same underlying set of elements, but binary operation changed by reversing the order of the factors: .
This is often considered for the case that a monoid, group, ring, or algebra (nonassociative or associative), in which case one speaks of the opposite group, opposite ring, etc.
Let be a magma, that is a set equipped with a binary operation
written as multiplication or juxtaposition.
Then the opposite magma (also denoted or ) has the same underlying set
but binary operation that of but with the ordering in the pair of arguments reversed:
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.
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.
under delooping a monoid to a pointed one-object category, passing to the opposite monoid corresponds to passing to the opposite category.
the analogue for monoidal categories is the reverse monoidal category
Last revised on May 27, 2025 at 11:59:58. See the history of this page for a list of all contributions to it.