[Paper Review] Homotopy type theory: the logic of space

Michael Shulman|arXiv (Cornell University)|Mar 8, 2017

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper proposes homotopy type theory (HoTT) as a foundational framework where spaces—rather than sets—are the fundamental mathematical objects, enabling synthetic reasoning about topology and homotopy theory. By interpreting types as ∞-groupoids (homotopy spaces), HoTT unifies algebraic structures with spatial and homotopical properties, allowing direct development of synthetic homotopy theory without relying on topological spaces or simplicial sets, thus revealing deeper structural orthogonality between spatial, smooth, and homotopical structures.

ABSTRACT

This is an introduction to type theory, synthetic topology, and homotopy type theory from a category-theoretic and topological point of view, written as a chapter for the book "New Spaces for Mathematics and Physics" (ed. Gabriel Catren and Mathieu Anel).

Motivation & Objective

• To advocate for a foundational shift from set theory to type theory where spaces are fundamental objects.
• To demonstrate how dependent type theory naturally supports synthetic treatments of topology and homotopy.
• To unify diverse notions of space—topological, smooth, algebraic, homotopical—under a single formal system.
• To show that homotopy type theory enables direct, computationally meaningful reasoning about spaces without combinatorial models.
• To argue that multiple foundational systems (toposes) are equally valid, with choice depending on context, not absolutism.

Proposed method

• Uses Martin-Löf dependent type theory as the foundational formal system, with types interpreted as spaces.
• Applies the univalence axiom to equate isomorphic types, enabling structural reasoning about spaces.
• Interprets types as ∞-groupoids (homotopy spaces), allowing synthetic homotopy theory.
• Constructs free toposes from type theory syntax to model different kinds of spaces.
• Uses higher topos theory to interpret types in homotopical and higher-categorical contexts.
• Introduces synthetic reasoning by treating spatial and homotopical properties as intrinsic to types, not derived from sets.

Experimental results

Research questions

• RQ1 How can type theory serve as a foundation for mathematics that inherently includes spatial structure?
• RQ2 What is the role of homotopy types (∞-groupoids) in unifying different notions of space?
• RQ3 How does synthetic homotopy theory differ from classical homotopy theory in its foundational approach?
• RQ4 Can foundational systems other than ZFC be equally valid, and how do they relate to one another?
• RQ5 How do different spatial structures—topological, smooth, homotopical—coexist and interact in a unified framework?

Key findings

• Homotopy type theory provides a synthetic framework where all mathematical objects inherently carry spatial and homotopical structure.
• The univalence axiom allows isomorphic types to be treated as equal, enabling structural reasoning that aligns with geometric intuition.
• Types in HoTT naturally interpret as ∞-groupoids, allowing direct development of homotopy theory without simplicial or topological models.
• Different toposes—such as those modeling continuous, smooth, or discrete spaces—can be constructed from type theory, each with its own logical rules.
• The framework supports mixed structures: objects can be both topological and homotopical, or smooth and homotopical, without conflict.
• The pluralistic view of foundations allows switching between toposes as needed, with consistent translation rules, analogous to reference frames in relativity.