Morel and Voevodsky constructed the $\mathbb{A}^1$-motivic homotopy category $SH(k)$. One purpose of motivic homotopy theory is to incorporate various cohomology theories into a single framework so that one can discuss relations between cohomology theories more efficiently. However, there are still lots of non $\mathbb{A}^1$-invariant cohomology theories, e.g. Hodge cohomology and topological cyclic homology. There is no way to represent these in $SH(k)$. In this talk, I will explain the construction of the logarithmic motivic homotopy category $logSH(k)$ and how we can represent $TC$ in $logSH(k)$.
This work is joint with Federico Binda and Paul Arne Østvær.