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I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC).

I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. Loday. This is very fantastic written book. Can someone suggest me some good reference like this for THH/TC?

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As a first introduction I like these notes by Achim Krause and Thomas Nikolaus. They do require some familiarity with spectra and stable homotopy theory though.

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Ib Madsen's survey

MR1474979 (98g:19004) Madsen, Ib Algebraic K-theory and traces. Current developments in mathematics, 1995 (Cambridge, MA), 191–321, Int. Press, Cambridge, MA, 1994.

was written to be such an introduction, but is 25 years old. Its technical foundations are given in terms of FSPs (functors with smash product). A more recent source is the book

MR3013261 Dundas, Bjørn Ian; Goodwillie, Thomas G; McCarthy, Randy The local structure of algebraic K-theory. Algebra and Applications, 18. Springer-Verlag London, Ltd., London, 2013. xvi+435 pp. ISBN: 978-1-4471-4392-5; 978-1-4471-4393-2

which works with $\Gamma$-rings (= monoids in $\Gamma$-spaces). Both of these predate the connection to crystalline and syntomic cohomology.

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