On the Universal Property of Derived Manifolds

Abstract

It is well known that any model for derived manifolds must form a higher category. In this paper, we propose a universal property for this higher category, classifying it up to equivalence. Namely, the $\infty$-category $\mathbf{DMfd}$ of derived manifolds has finite limits, is idempotent complete, and receives a functor from the category of manifolds which preserves transverse pullbacks and the terminal object, and moreover is universal with respect to these properties. We then show this universal property is equivalent to another one, intimately linking the $\infty$-category of derived manifolds to the theory of $C^\infty$-rings. More precisely, $\mathbb{R}$ is a $C^\infty$-ring object in $\mathbf{DMfd}$, and the pair $\left(\mathbf{DMfd},\mathbb{R}\right)$ is universal among idempotent complete $\infty$-categories with finite limits and a $C^\infty$-ring object. We then show that (a slight extension beyond the quasi-smooth setting of) Spivak's original model satisfies our universal property.