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Fibration categories and models for higher categories

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If you have a question about this talk, please contact Tamara von Glehn.

This talk reports on two related pieces of work in progress. Homotopy type theory is a language for certain categories of spaces, but depends only on the structure of a (type-theoretic) fibration category rather than a model category. Fibration categories are easier to construct than model categories, and although they do not support the full range of constructions known from model categories, often it is easier to see a higher-dimensional entity as an object of a fibration category. For example, simplicial sets form a model category, but semisimplicial sets only form a type-theoretic fibration category.

The first part concerns models of higher categories in (marked) presheaves on a category C. A standard example is Verity’s model structure on weak complicial sets, where C = Delta. When C is direct, one typically only obtains a fibration category, as is the case for opetopic sets. We propose a route for relating these two models by constructing an intermediate model in weak semicomplicial sets.

In the second part, we look at models of higher categories based on space-valued presheaves where spaces are now assumed to be modelled by just a (type-theoretic) fibration category. Recently, Paolo Capriotti has suggested complete semi-Segal spaces as a replacement for the technology of complete Segal spaces not available in this context, yielding a notion of univalent (omega,1)-categories (assuming omega-Reedy limits). This only works in the presence of completeness: due to the lack of degeneracies, semi-Segal spaces do not correspond to non-univalent (i.e., pre) (omega,1)-categories. We define a direct category D such that D-Segal spaces, space-valued presheaves on D with an analogue of the Segal condition, fill this role. There is an analogue of the completeness condition, and complete D-Segal spaces correspond to semi-Segal spaces.

This talk is part of the Category Theory Seminar series.

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