Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory

Abstract

Informally, an orbifold is a smooth space whose points may have finitely many internal symmetries. Formally, however, the notion of orbifold has been presented in a number of different guises -from Satake's V -manifolds to Moerdijk and Pronk's proper étale groupoids -which do not on their face resemble the informal definition. The reason for this divergence between formalism and intuition is that the points of spaces cannot have internal symmetries in traditional, set-level foundations. The extra data of these symmetries must be carried around and accounted for throughout the theory. More drastically, maps between orbifolds presented in the usual ways cannot be defined pointwise, and instead must use intermediary spaces.In this paper, we will put forward a definition of orbifold in synthetic differential cohesive homotopy type theory: an orbifold is a microlinear type for which the type of identifications between any two points is properly finite. In homotopy type theory, a point of a type may have internal symmetries, and we will be able to construct examples of orbifolds by defining their type of points directly. All maps between these orbifolds may be defined pointwise; the mapping space between two orbifolds is merely the type of functions.We will justify this synthetic definition by proving, internally, that every proper étale groupoid is an orbifold. In this way, we will show that the synthetic theory faithfully extends the usual theory of orbifolds. Along the way, we will investigate the microlinearity of higher types such as étale groupoids, showing that the methods of synthetic differential geometry generalize gracefully to higher analogues of smooth spaces. We will also investigate the relationship between the Dubuc-Penon and open-cover definitions of compactness in synthetic differential geometry, and show that any discrete, Dubuc-Penon compact subset of a second-countable manifold is subfinitely enumerable.

Commuting Cohesions

“…In addition to allowing us to formalize the theorem about Čech nerves of open covers as Theorem 6.1.5, our type theory will be able to handle the equivariant differential cohesion used by Sati and Schreiber in their Proper orbifold cohomology [43], as well as the nested focuses of Schreiber's supergeometric solid cohesion [44]. This extends the work of Cherubini [17] and the first author [34,35,36] of giving synthetic accounts of the constructions of Schreiber [44] and Sati-Schreiber [43].…”

“…See, e.g. Section 4.1 of [36] for a list of these axioms. In any case, if R S is a type of smooth reals, then we will take differential cohesion to mean that R S detects connectivity.…”