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Translation Invariance

  • Patrick Schultz
  • David I. Spivak
Chapter
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 29)

Abstract

As discussed in Sect. 1.2.2, the topos \(\mathsf {Shv}(S_{\mathbb {I\hspace {1.1pt} R}})\) of sheaves on the interval domain is slightly unsatisfactory as a model of behaviors. For example, to serve as a compositional model of dynamical systems, we do not want the set of possible behaviors in some behavior type to depend on any global time. In this chapter, we define a topos \(\mathcal {B}\) of “translation-invariant behavior types” by defining a translation-invariant version of the interval domain and a corresponding site, denoted \(\mathbb {I\hspace {1.1pt} R}_{/\rhd }\) and \(S_{\mathbb {I\hspace {1.1pt} R}/\rhd }\), respectively, and letting \(\mathcal {B}\mathrel{\mathop:}= \mathsf {Shv}(S_{\mathbb {I\hspace {1.1pt} R}/\rhd })\).

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Patrick Schultz
    • 1
  • David I. Spivak
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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