Homotopy theory of normed sets I. Basic constructions

Abstract:

We would like to present an extension of the theory of $\mathbb {R}_{\geq 0}$-graded (or “$\mathbb {R}_{\geq 0}$-normed”) sets and monads over them as defined in recent paper by Frederic Paugam.

The theory of graded sets is extended in three directions. First of all, it is shown that $\mathbb {R}_{\geq 0}$ 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. This problem is dealt with by embedding $\mathcal {N}_\Delta$ into a larger category $\mathit {Sets}^\Delta$ of $\Delta$-graded sets.

Next, it is shown 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 $\mathit {Sets}^\mathcal {I}$ of $\mathcal {I}$-graded sets.

These foundations are used 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.