#
Weak $\infty$-categories via terminal coalgebras

##
Eugenia Cheng and Tom Leinster

Higher categorical structures are often defined by induction on dimension,
which *a priori* produces only finite-dimensional structures. In
this paper we show how to extend such definitions to infinite dimensions
using the theory of terminal coalgebras, and we apply this method to
Trimble's notion of weak n-category. Trimble's definition makes explicit
the relationship between n-categories and topological spaces; our extended
theory produces a definition of Trimble $\infty$-category and a
fundamental $\infty$-groupoid construction.

Furthermore, terminal coalgebras are often constructed as limits of a
certain type. We prove that the theory of Batanin - Leinster weak
$\infty$-categories arises as just such a limit, justifying our
approach to Trimble $\infty$-categories. In fact we work at the
level of monads for $\infty$-categories, rather than $\infty$-categories
themselves; this requires more sophisticated technology but also provides
a more complete theory of the structures in question.

Keywords:
$\infty$-category, $\omega$-category, n-category, higher category,
terminal coalgebra, final coalgebra

2010 MSC:
18D05 (primary), 18C15

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 34, pp 1073-1133.

Published 2019-10-22.

http://www.tac.mta.ca/tac/volumes/34/34/34-34.pdf

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