---
date: 2022-04-24 17:02
modification date: Sunday 24th April 2022 17:02:48
title: "THH"
aliases: ["THH", "Topological Hochschild homology", "topological Hochschild cohomology", "topological Hochschild cohomology", "cyclic homology", "negative  cyclic homology", ]
---

Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
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# THH

Cyclic cohomology (and homology) is a well developed theory which was first designed to  
handle the [[leaf spaces]] of [[foliations]] as well as group rings of discrete groups

# Defining THH

- Take $A \in \cat{C}$, where $\mathcal C$ is a "nice" [monoidal category](monoidal%20category.md), and $A$ is an algebra object in it. We'll call the monoidal operation $\tensor$.
- We'll make a simplicial object $\complex{\THoH}(A)$:
  - $\THoH_n(A) = A^{\tensor n+1}$. If it's to be simplicial, need to specify the face/degeneracy maps:
  - Face maps: collapse by cyclic multiplication
  - Degeneracy maps: use the unit of $A$, can replace any tensor symbol with it. Have a unit map that goes from the unit to $A$, so somehow this gets you "up" one level (?)
- Now take its geometric realization $\abs{\complex\THoH(A)}$
  - In general, take $\mathrm{hocolim}_\Delta \complex\THoH(A)$


![](attachments/Pasted%20image%2020220505160537.png)
![](attachments/Pasted%20image%2020220505160609.png)
![](attachments/Pasted%20image%2020220505160626.png)
![](attachments/Pasted%20image%2020220505160647.png)
![](attachments/Pasted%20image%2020220505160709.png)

# Results

![](attachments/Pasted%20image%2020220505161041.png)
![](attachments/Pasted%20image%2020220505161206.png)

See [[cyclotomic spectra]].
![](attachments/Pasted%20image%2020220515014305.png)
# Loday construction

![](attachments/Pasted%20image%2020220505160745.png)

# Spectral sequences

![](attachments/Pasted%20image%2020220505160812.png)

# As derived centers

![](attachments/Pasted%20image%2020220505161021.png)

# Examples


See [examples of THH computations](Unsorted/examples%20of%20THH%20computations.md).