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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 \(A \in \mathsf{C}\), where \(\mathcal C\) is a “nice” monoidal category, and \(A\) is an algebra object in it. We’ll call the monoidal operation \(\otimes\).
-
We’ll make a simplicial object \({ {{\operatorname{THH}}}_{\scriptscriptstyle \bullet}} (A)\):
- \({\operatorname{THH}}_n(A) = A^{\otimes n+1}\). 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 \({\left\lvert {{ {{\operatorname{THH}}}_{\scriptscriptstyle \bullet}} (A)} \right\rvert}\)
- In general, take \(\mathrm{hocolim}_\Delta { {{\operatorname{THH}}}_{\scriptscriptstyle \bullet}} (A)\)
Results
See cyclotomic spectra. 
# Loday construction
Spectral sequences
As derived centers
