Geodesic
Homotopy theory of normed sets~I. Basic constructions
Algebra i analiz, Tome 29 (2017) no. 6, pp. 35-98.

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We would like to present an extension of the theory of $\mathbb R_{\ge0}$-graded (or "$\mathbb R_{\ge0}$-normed") sets and monads over them as defined in recent paper by Frederic Paugam. We extend the theory of graded sets in three directions. First of all, we show that $\mathbb R_{\ge0}$ can be replaced with more or less arbitrary (partially) ordered commutative monoid $\Delta$, yielding a symmetric monoidal category $\mathcal N_\Delta$ of $\Delta$-normed sets. However, this category fails to be closed under some important categorical constructions. We deal with this problems by embedding $\mathcal N_\Delta$ into a larger category $\mathrm{Sets}^\Delta$ of $\Delta$-graded sets. Next, we show that most constructions make sense with $\Delta$ replaced by a small symmetric monoidal category $\mathcal I$. In particular, we have a symmetric monoidal category $\mathrm{Sets}^\mathcal I$ of $\mathcal I$-graded sets. We use these foundations for two further developments: a homotopy theory for normed and graded sets, essentially consisting of a well-behaved combinatorial model structure on simplicial $\mathcal I$-graded sets and a theory of $\Delta$-graded monads. This material will be exposed elsewhere.
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~I. Basic constructions. Algebra i analiz, Tome 29 (2017) no. 6, pp. 35-98. http://geodesic.mathdoc.fr/item/AA_2017_29_6_a1/

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