Geodesic
Homotopy theory of normed sets~II. Model categories
Algebra i analiz, Tome 30 (2018) no. 1, pp. 32-95.

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This paper is a continuation of the paper by the same author published in no. 2017:6 of this journal, where the foundations of the theory of normed and graded sets, and other algebraic structures were laid out. Here these foundations are used to present a homotopy theory of normed and graded sets, and other algebraic structures, by introducing combinatorial model structures on categories of relevant simplicial objects. We also construct a homotopy theory of metric spaces, which turns out to be deeply related to that of normed sets.
Keywords: normed sets, normed groups, norms, normed algebraic structures, graded algebraic structures, filtered algebraic structures, fuzzy sets, linear logic, presheaf categories, finitary monads, generalized rings, metric spaces, model categories, homotopy categories, higher categories. 
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N. V. Durov. Homotopy theory of normed sets~II. Model categories. Algebra i analiz, Tome 30 (2018) no. 1, pp. 32-95. http://geodesic.mathdoc.fr/item/AA_2018_30_1_a2/

[1] Adámek J., Rosický J., Locally presentable and accessible categories, London Math. Soc. Lecture Note Ser., 189, Cambridge Univ. Press, Cambridge, 1994 | MR

[2] Day B.,, “On closed categories of functors”, Lecture Notes in Math., 137, Springer, Berlin, 1970, 1–38 | DOI | MR

[3] Durov N., New approach to Arakelov geometry, Ph. D. thesis, Bonn Univ.; 2007, arXiv: 0704.2030

[4] Durov N., Classifying vectoids and generalisations of operads, 2011, arXiv: 1105.3114

[5] Durov N. V., “Homotopy theory of normed sets I. Basic constructions”, Algebra i analiz, 29:6 (2017), 35–98 | MR

[6] Durov N. V., Homotopy theory of normed sets III. Graded and normed monads, in preparation, 2017

[7] Gabriel P., Zisman M., Calculus of fractions and homotopy theory, Ergeb. Math. Grenzgeb., 35, Springer, Berlin, 1967 | MR

[8] Hovey M., Model categories, Math. Surveys Monogr., 99, Amer. Math. Soc., Providence, RI, 1999 | MR

[9] “Functorial semantics of algebraic theories”, Lawvere F. W., 50 (1963), 869–872 | MR

[10] Lurie J., Higher topos theory, http://www.math.harvard.edu/~lurie/papers/higertopoi.pdf

[11] Paugam F., Overconvergent global analytic geometry, 2015, arXiv: 1410.7971v2 | MR

[12] Quillen D., Homotopical algebra, Lecture Notes in Math., 43, Springer-Verlag, Berlin, 1967 | DOI | MR