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Logic and Computation Seminar

Monday, November 29, 2021 - 3:30pm

Dusko Pavlovic

University of Hawaii at Manoa


University of Pennsylvania

online - link available upon request

Dedekind defined the real numbers as the elements of a tight bicompletion of the linear order of the rational numbers. A bicompletion of a poset is tight when it preserves any infima and suprema that already exist. In the 1930s, MacNeille generalized Dedekind's approach and constructed tight bicompletions of arbitrary posets. In his Lectures on Completions of Categories (published in 1966 as vol. 24 of  Springer Lecture Note in Mathematics), Lambek spelled out the categorical generalizations of all of the main poset completions except of the one due to Dedekind and MacNeille, which he enunciated as an open problem. In 1972, Isbell showed that the group Z_4 cannot have a tight bicompletion under limits and colimits.  

Isbell's result, however, did not close Lambek's Problem, but broadened it. While the tight bicompletions of categories may not exist for all limits and colimits, for general reasons they must exist for suitable tight families of limits and colimits, where every limit is also a colimit and every colimit is also a limit. Lambek's Problem thus boils down to characterizing the tight limits and colimits for which tight bicompletions exist. I present a solution of this problem. 

Remarkably, the general reasons why tight bicompletions under must exist, and the techniques for their construction, emerged from a particular application in data analysis, where the categorical approach captures source dependencies. A concrete practical application gave rise to a solution of a category theoretical problem.