Day convolution and the Hodge filtration on THH

This thesis is divided into two chapters. In the first, given symmetric monoidal occategories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary properties. Thesis Supervisor: Clark Barwick Title: Cecil and Ida B. Green Career Development Associate Professor

[1]  P. Scholze,et al.  Integral p-adic Hodge theory , 2016 .

[2]  D. Kaledin.  Spectral Mackey functors and equivariant algebraic K-Theory ( I ) , 2016 .

[3]  C. Barwick,et al.  Dualizing cartesian and cocartesian fibrations , 2014, 1409.2165.

[4]  David Ayala,et al.  Factorization homology of stratified spaces , 2014, 1409.0848.

[5]  Michael A. Mandell,et al.  Relative cyclotomic spectra and topological cyclic homology via the norm , 2014 .

[6]  T. Tradler,et al.  Higher Hochschild Homology, Topological Chiral Homology and Factorization Algebras , 2010, 1011.6483.

[7]  J. Lurie.  Higher Topos Theory , 2006, math/0608040.

[8]  R. Street,et al.  MACKEY FUNCTORS ON COMPACT CLOSED CATEGORIES , 2007, 0706.2922.

[9]  Daniel R. Grayson.  The Motivic Spectral Sequence* , 2004 .

[10]  R. McCarthy,et al.  HKR Theorem for Smooth S-algebras , 2003, math/0306243.

[11]  V. Minasian.  André–Quillen spectral sequence for THH , 2003 .

[12]  N. Kuhn.  The McCord Model for the Tensor Product of a Space and a Commutative Ring Spectrum , 2002, math/0202042.

[13]  C. Breuil.  Integral $p$-adic Hodge Theory , 2002 .

[14]  Birgit Richter,et al.  Hochschild and cyclic homology via functor homology , 2002 .

[15]  T. Pirashvili.  Hodge decomposition for higher order Hochschild homology , 2000 .

[16]  J.-L. Loday,et al.  Opérations sur l'homologie cyclique des algèbres commutatives , 1989 .

[17]  W. Fulton,et al.  Riemann-Roch Algebra , 1985 .

[18]  Brian Day,et al.  Construction of biclosed categories , 1971, Bulletin of the Australian Mathematical Society.

[19]  P. Deligne.  Theorie de Hodge I , 1970 .

[20]  B. Kostant,et al.  Differential Forms on Regular Affine Algebras , 1962 .

[21]  L. Hesselholt,et al.  Higher Algebra , 1937, Nature.