We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first characterize, in terms of exact squares, when pseudomonads on a bicategory extend to its bicategory of two-sided discrete fibrations. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories satisfying our criterion and thus extending to profunctors. Among these, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.

翻译：本文对Barr于1970年的里程碑式论文进行了双范畴推广。在该论文中，Barr描述了如何将Set-幺半延拓至关系范畴，并利用此方法将拓扑空间刻画为超滤子幺半的关系代数。通过将双边离散纤维化在双范畴中扮演关系的角色，我们首先以精确平方为工具，刻画了双范畴上的伪幺半何时能延拓至其双边离散纤维化的双范畴。作为一个广泛的示例类，我们证明了每个Set-幺半均可在满足我们准则的范畴2-范畴上诱导一个伪幺半，从而延拓至profunctor。在此基础上，我们聚焦于超完备化伪幺半，其伪代数为超范畴：我们将其profunctorial延拓的归一化lax代数刻画为超收敛空间——这是拓扑空间的一种近期引入的范畴化形式。