abstract duality: opposite category,
Categories generalize (are a horizontal categorification of) monoids, groups and algebras, and forming the opposite category corresponds to forming the opposite of a group, of a monoid, of an algebra.
source and target:
then is the category with
the same (isomorphic) collections of objects and morphisms
the same identity-assigning map
switched source and target maps
the same composition operation,
or more precisely, the composition operation of is
where the isomorphism in the middle is the unique one induced from the universality of the pullback.
Notice that hence the composition law does not change when passing to the opposite category. Only the interpretation of in which direction the arrows point does change. So forming the opposite category is a completely formal process. Nevertheless, due to the switch of source and target, the opposite category is usually far from being equivalent to . See the examples below.
The definition has a direct generalization to enriched category theory.
and composition given by
The unit maps are those of under the identification .
Note that the braiding of is used in defining composition for . So, we cannot define the opposite of a -enriched category if is merely a monoidal category, though -enriched categories still make perfect sense in this case. If is a braided monoidal category there are (at least) two ways to define “”, resulting in two different “opposite categories”: we can use either the braiding or the inverse braiding. If is symmetric these two definitions coincide.
The opposite category can be regarded as a dual object of in the monoidal bicategory of -categories and -profunctors. (Note that this does not characterize up to equivalence, but only up to Morita equivalence, i.e. up to Cauchy completion.) When is symmetric, then is also symmetric monoidal, so there is only one notion of dual object. When is braided, then is not symmetric and has two notions of dual: a left dual and a right dual. These are exactly the two different opposite categories referred to above (the “left opposite” and “right opposite”).
The nerve of is the simplicial set that is degreewise the same as , but in each degree with the order of the face and the order of the degeneracy maps reversed. See opposite quasi-category for more details.
the opposite group is the group whose underlying set (underlying object, underlying vector space, etc.) is the same as that of
but whose product operation is that of but combined with a switch of the order of the arguments:
So for two elements we have that their product in the opposite group is
Under this identification of groups with one-object categories, passing to the opposite category corresponds precisely to passing to the opposite group
The opposite of an opposite category is the original category:
This is also true for -enriched categories when is symmetric monoidal, but not when is merely braided. However, in the latter case we can say , i.e. the two different notions of “opposite category” are inverse to each other (as is always the case for left and right dualization operations in a non-symmetric monoidal (bi)category).
Every algebraic structure in a category, for instance the notion of monoid in a monoidal category , has a co-version, where in the original definition the direction of all morphisms is reversed – for instance the co-version of a monoid is a comonoid. Of an algebra its a coalgebra, etc.
Passing to the opposite category is a realization of abstract duality.
This goes as far as defining some entities as objects in an opposite category–in particular, all generalizations of geometry which characterize spaces in terms of algebras. The idea of noncommutative geometry is essentially to define a category of spaces as the opposite category of a category of algebras. Similarly, a locale is opposite to a frame.
Are there examples where algebras are defined as dual to spaces?
constitutes an equivalence of categories from the opposite category of Set to that of complete atomic Boolean algebras. See at Set – Properties – Opposite category and Boolean algebras
For the definition in enriched category theory see page 12 of