Displayed Type Theory and Semi-Simplicial Types

Abstract: We introduce Displayed Type Theory (dTT), a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$-topos, while the simplicial mode is interpreted by Reedy fibrant augmented semi-simplicial diagrams in that model. This simplicial structure is represented inside the theory by a primitive notion of display or dependency, guarded by modalities, yielding a partially-internal form of unary parametricity. Using the display primitive, we then give a coinductive definition, at the simplicial mode, of a type $\mathsf{SST}$ of semi-simplicial types. Roughly speaking, a semi-simplicial type $X$ consists of a type $X_0$ together with, for each $x:X_0$, a displayed semi-simplicial type over $X$. This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of $\mathsf{SST}$ then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality.

• The paper introduces dTT as a novel framework that systematically encodes semi-simplicial types within Homotopy Type Theory.
• It details the use of telescopes and partial substitutions to model complex homotopy-coherent structures with multi-modal modalities.
• The approach paves the way for computational implementations in algebraic topology and type systems, enabling deeper explorations of higher-dimensional coherence.

Displayed Type Theory and Semi-Simplicial Types

The paper entitled "Displayed Type Theory and Semi-Simplicial Types" presents a novel framework called Displayed Type Theory (dTT). This framework is aimed at addressing complexities involved in defining and working with semi-simplicial types within Homotopy Type Theory (HoTT). Overcoming the limitations of existing approaches to codifications of generalized homotopy-coherent structures, dTT provides a systematic way to encode semi-simplicial types in a syntactically coherent and semantically generalized manner.

The authors introduce dTT as a multi-modal homotopy type theory encompassing discrete and simplicial modalities. These modalities interact through display, a primitive operation representing a form of unary parametricity. Display attempts to align dTT with the semantic interpretation of categories and presheaves, enabling it to internalize semi-simplicial types effectively.

Structural Overview and Semantics

Central to dTT's design is its ability to model telescopes (finite sequences of types dependent on preceding types), along with substitution mechanisms known as partial substitutions. This functionality extends to meta-abstractions, classifying types in their own right, thereby harmonizing with a cubical, pseudo-categorical model where each type admits a computed display.

The semantics as construed in the paper are built upon existing type-theoretic models augmented with the ability to represent Reedy fibrant presheaves over finite direct categories (or inverse diagrams). This provides the compositional flexibility necessary for scalable and coherent semi-simplicial structure manipulation within HoTT.

Computational Mechanisms: Display and D́ecalage

Display operates akin to logical relations in parametricity frameworks, specifically adapted to contexts whose governance necessitates partially internalized terms. The accompanying d́ecalage operator repositions these contexts, preserving or adjusting modular dependencies according to specified rules, and {these parameters set context-modifying rules}.

Impact and Applications

The canonical implementation of semi-simplicial types as presented enables the systematization of algebraic hierarchy within homotopy types. The framings in dTT induce a constructive framework that scaffolds definitions and operations on semi-simplicial types, fostering migrations from abstract homotopical structures to concrete type constructions.

Future Directions

The paper provides a backbone for further exploration into how Displayed Type Theory can generalize the comprehension of complex categorical constructs like fibrations and higher-dimensional type coherence. By addressing inherent difficulties in definitions requiring infinite coherence, dTT sets a foundation for further exploration and potential computational implementations of homotopy-theoretic semantics in other type-theoretic contexts.

The implications of this work are theoretically and practically significant, with dTT not only extending the point-set homotopy logic to higher-type categorical frameworks, but also enhancing the synthesis of algebraic topology and type systems for a more profound comprehension of mathematical structures.

In sum, this framework establishes a reusable, robust structure for type theorists, facilitating further inquiries and applications of multi-modal type systems in higher categorical equivalences and their representations.