# A Sheaf-Theoretic Construction of Shape Space

arXiv: https://arxiv.org/abs/2204.09020 · HTML: https://mlsanity.com/paper/2204.09020/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2022.

## Built on by (most cited first, 3 of 3)

- [Probabilistic U-Net with Kendall Shape Spaces for Geometry-Aware Segmentations of Images](https://mlsanity.com/paper/2410.14017/) (2024)
- [On the Sheafification of Higher-Order Message Passing](https://mlsanity.com/paper/2509.23020/) (2025)
- [A Sheaf-Theoretic and Topological Perspective on Complex Network Modeling and Attention Mechanisms in Graph Neural Models](https://mlsanity.com/paper/2601.21207/) (2026)

## Abstract (arXiv)

We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.