For primes $l≠p$, $l$-adic cohomology provides a good cohomology theory for varieties over a field of positive characteristic $p$. For $l=p$, there is a whole zoo of $p$-adic cohomology theories. While rigid cohomology provides a well-behaved $p$-adic cohomology theory for all varieties of characteristic $p$, it has the drawback of having only rational coefficients. On the other hand, crystalline cohomology provides a good integral $p$-adic cohomology theory for smooth and proper varieties. In this talk, we explain a joint work with Veronika Ertl and Atsushi Shiho towards the construction of a good integral $p$-adic cohomology theory beyond the smooth and proper case.

For a scheme $X$, we will introduce the so called Galois category $\operatorname{Gal}(X)$ of $X$ due to Barwick, Glasman and Haine. Its objects are geometric points of $X$ and its morphisms are \'etale specializations of such. It is naturally equipped with a pro-finite topology and plays the role of an \'etale version of an exit-path category: Continuous representations of $\operatorname{Gal}(X)$ with values in finite sets are equivalent to constructible \'etale sheaves on $X$. After recalling the necessary ingredients, we will discuss how one can use the language of condensed/ pyknotic mathematics to generalize the exodromy equivalence to a much larger class of sheaves on $X$.