Lingyuan Ye

Fredrik Bakke, Jonathan Sterling, Mark Damuni Williams et al.

In domains, categories, and toposes, the Sierpiński cone construction glues onto a space a universal closed point lying below all the other points. Although this is a lax colimit, it also enjoys a well-known right-handed universal property: the Sierpiński cone classifies partial maps defined on an open subspace. The situation proves more subtle in synthetic models of space based on extending homotopy type theory with an interval, as in several recent approaches to synthetic higher categories and domains: although globally it may well be the case that the Sierpiński cone classifies partial maps, this property cannot hold of all parameterised types without degenerating the theory. On the other hand, there are reflective subuniverses within which the classifying property nonetheless holds. We show that the largest subuniverse in which the Sierpiński cone classifies partial maps is the accessible localisation at a family of embeddings parameterised in the interval, and this subuniverse is contained within the Segal types; this containment is moreover strict in the sense that when the interval is non-trivial, it is not possible for all Segal types to lie in the subuniverse. We finally extend these results from Sierpiński cones to mapping cylinders, providing a new right-handed universal property for the latter.

Lingyuan Ye

This paper shows that the sheaf representation of finitely presented Heyting algebras constructed by Ghilardi and Zawadowski is, from an algebraic perspective, equivalent to the construction of profinite completion. We show that the dual category of profinite Heyting algebras is an infinitary extensive regular category, and its ex/reg-completion is exactly the aforementioned sheaf topos, which we refer to as the K-topos. We show how certain properties of uniform interpolation can be generalised to the context of arbitrary profinite Heyting algebras, and are consequences of the internal logic of the K-topos. Along the way we also establish various topos-theoretic properties of the K-topos.

Rui Peng, Ziru Liu, Lingyuan Ye et al.

Accurately modeling the relationship between perturbations, transcriptional responses, and phenotypic changes is essential for building an AI Virtual Cell (AIVC). However, existing methods typically constrained to modeling direct associations, such as Perturbation $\rightarrow$ RNA or Perturbation $\rightarrow$ Morphology, overlook the crucial causal link from RNA to morphology. To bridge this gap, we propose TRIDENT, a cascade generative framework that synthesizes realistic cellular morphology by conditioning on both the perturbation and the corresponding gene expression profile. To train and evaluate this task, we construct MorphoGene, a new dataset pairing L1000 gene expression with Cell Painting images for 98 compounds. TRIDENT significantly outperforms state-of-the-art approaches, achieving up to 7-fold improvement with strong generalization to unseen compounds. In a case study on docetaxel, we validate that RNA-guided synthesis accurately produces the corresponding phenotype. An ablation study further confirms that this RNA conditioning is essential for the model's high fidelity. By explicitly modeling transcriptome-phenome mapping, TRIDENT provides a powerful in silico tool and moves us closer to a predictive virtual cell.