Some differential equations in synthetic differential geometry

Synthetic differential geometry within homotopy type theory I

This paper gives a first step towards developing synthetic differential geometry within homotopy type theory. Its model theory will be discussed in a subsequent paper.

Synthetic geometry of differential equations: I. Jets and comonad structure

We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as super-geometry, higher (stacky) differential geometry, or even a combination of both. A motivation for such a level of generality is the eventual goal of solving the open problem of covariant geometric pre-quantization of locally variational field theories, which may include fermions and (higher) gauge fields. (abridged)

合成微分喷射束是简化的

Synthetic Differential Jet Bundles are Reduced

本文证明了从弗雷歇光滑流形到Cahiers拓扑的转换保持无限喷射束的项目极限，从而为偏微分方程解理论提供了更强大的余单子形式化。

Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of $A$-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart.

Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory

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 d…

“激进综合”微分几何的双向模型

Bi-directional models of `radically synthetic' differential geometry

本文构建了首个具有双向环R的预黏合拓扑斯模型，并证明其预序与原始模型一致，为激进综合微分几何提供了新基础。