# Twisting cochains and twisted complexes

###### 26th July, 2019

This week was the Young Topologists Meeting at EPFL in Lausanne, and had courses by Julie Bergner (on 1- and 2-Segal spaces) and Vidit Nanda (on the topology of data), as well as a bunch of great talks by various postdocs and PhD students. I was lucky enough to get the chance to talk about twisting cochains and twisted complexes for half an hour, and you can find the slides here.

One talk that I really liked was Rachel Hardeman’s introduction to A-homotopy theory, which is a really interesting way of describing the homotopy of graphs — whereas, with classical homotopy theory, all cyclic graphs are equivalent, in A-homotopy theory, you can distinguish between them. In particular, C_n is only contractible for n\leqslant4. You can read more about this in her paper on the arXiv.

Also really great one was Maru Sarazola’s talk on constructing model and Waldhausen category structures via cotorsion pairs: a method that lets you get model structures on abelian categories that play nicely with all the abelian structure, but where you have some freedom to choose the morphisms that you want to be weak equivalences. I don’t think she currently has notes for this stuff typed up anywhere, but (from what I gather) it’s in the works!