**Title: **What are these lax limits and actions doing in my traces?

**Abstract: **Trace methods play a key role in both computations of and celebrated theorems on algebraic K-theory. One might wonder if there is a geometric picture that can be sketched relating K, THH, and TC. This has a positive answer thanks to a series of papers by Ayala, Mazel-Gee, and Rozenblyum, where THH of a "space" X may be thought of as functions on loops on X and TC as the ones whose values are traces of the monodromy of vector bundles on X, in an appropriate sense. The goal of this talk is to draw some pictures to help (me) intuit why lax behavior comes up in making such heuristics precise and orient us towards AMGR's stratified noncommutative geometry. Willingness to blackbox a few facts about (∞,2)-categories will be assumed, but minimally.