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Hochschild cohomology in toposes

Cameron Michie, Ivan Tomasic

Abstract

We develop a theory of internal Hochschild cohomology in a ringed topos. We construct it via the internal Hochschild cochain complex, as well as through derived functor/topos cohomology theory, and discuss its relationship to the absolute Hochschild cohomology. By specialising to the topos of difference sets, we obtain a theory of internal difference Hochschild cohomology, and compare it to the absolute Hochschild cohomology through the Grothendieck and hypercohomology spectral sequences. We provide a systematic and detailed treatment of tensor products in suitable toposes in hope to complete the existing literature.

Hochschild cohomology in toposes

Abstract

We develop a theory of internal Hochschild cohomology in a ringed topos. We construct it via the internal Hochschild cochain complex, as well as through derived functor/topos cohomology theory, and discuss its relationship to the absolute Hochschild cohomology. By specialising to the topos of difference sets, we obtain a theory of internal difference Hochschild cohomology, and compare it to the absolute Hochschild cohomology through the Grothendieck and hypercohomology spectral sequences. We provide a systematic and detailed treatment of tensor products in suitable toposes in hope to complete the existing literature.
Paper Structure (41 sections, 80 theorems, 211 equations)

This paper contains 41 sections, 80 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Homological algebra internal to a topos
  3. Symmetric monoidal closed structure of k-Mod
  4. Internal logic of a topos
  5. Internal Hochschild cohomology in a topos
  6. Internal Hochschild cochain complex
  7. Derived functor definition of Hochschild cohomology
  8. Internal Hochschild cohomology in difference sets
  9. The topos of difference sets
  10. Functors on sigma-Set
  11. Tensored structure of k-Mod over sigma-Set
  12. Low dimensional internal difference Hochschild cohomology
  13. Degree 0
  14. Degree 1
  15. Example
  16. ...and 26 more sections

Key Result

Lemma 3.1

If $i>j$, then $\underline{\partial}^i\circ\underline{\partial}^j=\underline{\partial}^j\circ\underline{\partial}^{i-1}$.

Theorems & Definitions (164)

  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4: Internally projective/injective object
  • Definition 2.1.5: Enriched projective/injective object
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 3.1.1: Interal Hochschild cohomology
  • ...and 154 more