Miika Tuominen will defend the thesis on Monday, March 24. The title is
“Completions and DK-equivalences of Θn-spaces”.
Abstract: Higher categories are categories with an additional structure of higher-dimensional morphisms between those of lower dimension. Examples of higher categories include the Morita category in algebra and extended cobordism categories in geometry and mathematical physics. Usually in a higher category, including in these examples, the composition operation for each dimension of morphisms is associative and unital only in a weak sense, which is most conveniently described in the language of homotopy theory. An (∞, n)-category is a homotopy-theoretic, weak, n-dimensional category and can be defined in terms of a Θn-space, which is a space-valued contravariant functor on a certain generating subcategory Θn of strict n-categories. In order to give the functor the algebraic structure of weak composition operations, so-called Segal conditions have to be imposed, and to ensure that the present homotopy theory recognizes when a morphism is invertible, additional completeness conditions are required. In this thesis work, we study the completeness conditions by constructing a resolution that takes a Θn-space satisfying the Segal conditions to one that also satisfies the completeness conditions while preserving its key properties. This completion construction generalizes classical work of Rezk in the one-dimensional case and allows us to understand completeness as the property of being local with respect to the class of DK-equivalences, which are maps that behave like essentially surjective and fully faithful n-functors. Our completion also sheds light on the closely related notion of univalence in theoretical computer science.
Everyone is invited to attend.