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Gray-categories model algebraic tricategories

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Giovanni Ferrer

Lack described a Quillen model structure on the category GrayCat of Gray-categories and Gray-functors, for which the weak equivalences are the weak 3-equivalences. Restricted to Gray-groupoids, the resulting homotopy category is equivalent to the homotopy category of 3-types.
In this note, we adapt the technique of Gurski, Johnson, and Osorno to show the localization of GrayCat at the weak equivalences is equivalent to the category of algebraic tricategories and pseudo-natural equivalence classes of weak 3-functors. This finishes establishing the homotopy hypothesis for algebraic trigroupoids.

Keywords:
Gray-categories, algebraic tricategories, homotopy hypothesis

2020 MSC:
18N20 and 18N40

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 29, pp 1136-1155.

Published 2022-09-15.

http://www.tac.mta.ca/tac/volumes/38/29/38-29.pdf

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