Homotopy theory of normed sets I. Basic constructions

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.