Directed univalence in simplicial homotopy type theory
<|reference_start|>Directed univalence in simplicial homotopy type theory: Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic (higher) category theory -- and programming languages -- where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathcal{S}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathcal{S}$ is directed univalent. The construction of $\mathcal{S}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathcal{S}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.<|reference_end|>
@article{gratzer2024directed,
    title={Directed univalence in simplicial homotopy type theory},
    author={Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz},
    journal={arXiv preprint arXiv:2407.09146},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.09146},
    primaryClass={cs.LO math.AT math.CT}
}