Example: An equivalence of the form $ \mathscr{A}^\op \simeq \mathscr{B} $ is sometimes called a duality between $ \mathscr{A} $ and $ \mathscr{B} $ . One says that $ \mathscr{A} $ is dual to $ \mathscr{B} $ . There are many famous dualities in which $ \mathscr{A} $ is a category of algebras and $ \mathscr{B} $ is a category of spaces; recall the slogan `algebra is dual to geometry’ from Example X.