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A localization of bicategories via homotopies

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Descotte M.E., Dubuc E.J., and Szyld M.

Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C,W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C.
We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows.
This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C_{fc},W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.

Keywords:
localization, bicategory, homotopy

2020 MSC:
18N10, 18N40, 18N55

*Theory and Applications of Categories,*
Vol. 35, 2020,
No. 23, pp 845-874.

Published 2020-06-12.

http://www.tac.mta.ca/tac/volumes/35/23/35-23.pdf

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