© 2019

Temporal Type Theory

A Topos-Theoretic Approach to Systems and Behavior


Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 29)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Patrick Schultz, David I. Spivak
    Pages 1-15
  3. Patrick Schultz, David I. Spivak
    Pages 17-37
  4. Patrick Schultz, David I. Spivak
    Pages 39-45
  5. Patrick Schultz, David I. Spivak
    Pages 47-86
  6. Patrick Schultz, David I. Spivak
    Pages 87-114
  7. Patrick Schultz, David I. Spivak
    Pages 115-132
  8. Patrick Schultz, David I. Spivak
    Pages 133-156
  9. Patrick Schultz, David I. Spivak
    Pages 157-177
  10. Back Matter
    Pages 179-235

About this book


This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. 

The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action.  

Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource.  It can also serve as a supplemental text for a specialized graduate topics course. 


temporal type theory category theory topoi systems engineering sheave theory hybrid dynamical systems

Authors and affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Bibliographic information