Homotopical models for metric spaces and completeness

arXiv2212.00147MaRDI QIDQ6419156

Isaiah Dailey, Semir Mujevic, Chloe Shupe, Clara Huggins

Publication date: 30 November 2022

Abstract: We develop model structures in which homotopy theory can be used on Lawvere metric spaces, with a focus on extended, Cauchy complete Lawvere, and Cauchy complete extended metric spaces. The motivating example for one of these model structures is the proof of the Karoubian model structure on

m a t h b f C a t

which has been described in the literature, although no formal proof of its existence was given. We then construct model structures on the categories

m a t h b b R_{+} e x t - m a t h b f C a t

, of Lawvere metric spaces, and

{m a t h b b R_{+} e x t - m a t h b f C a t}^{m a t h r m s y m}

, of symmetric Lawvere metric spaces. The fibrant-cofibrant objects in these three model structures are the extended metric spaces, the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. In particular, we show that the two of these model structures which model extended metric spaces are suitably ``unique while the other bears a striking resemblance to the Karoubian model structure on $m a t h b f C a t$ .

Mathematics Subject Classification ID

Enriched categories (over closed or monoidal categories) (18D20) Homotopical algebra, Quillen model categories, derivators (18N40)

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