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Presenting higher categories and weak functors via multi-opetopic nerves and terminal coalgebras

University of British Columbia

In this thesis, we construct a convenient presentation of weak n-categories for 0 <= n <= omega whose corresponding weak n-functors in particular do not strictly preserve units. The approach constructs the finite-dimensional higher categories inductively using multi-opetopic sets, analogous to the construction of Segal categories, and multi-simplicial nerves of Tamsamani. We prove that our construction is correct in dimension two by providing a fully faithful and weakly essentially surjective functor from the category of bicategories and pseudofunctors to our weak 2-categories. For the infinite-dimensional case, we realise the weak omega-categories as formal limits of their finite-dimensional truncations, using coalgebras over an appropriate endofunctor. We also prove that these weak omega-categories admit an equivalent characterisation as infinitary opetopic sets subject to constraints analogous to the finite-dimensional case. Finally, we specialise our construction to infinity-groupoids and prove that the coalgebraic structure induces a canonical functor from nice topological spaces that defines the Poincaré infinity-groupoid construction. We show that the Poincaré infinity-groupoid has the correct higher morphisms in all dimensions and retains the information about all homotopy groups of the space. Moreover, we show that this construction preserves and reflects weak equivalences. We conclude by proposing a construction that likely recovers a space from its Poincaré infinity-groupoid which conjecturally establishes a version of the Homotopy Hypothesis.

Text

2021-04-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77998

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/