The K-adic filtration | Higher Topos Theory notes

Construction 5.85.

Let $K \in \Cong(T)$ be a congruence. By Theorem 5.84, congruences form a subalgebra of $\Acyc(T)$ . We can therefore form powers $$K^n$$ , giving congruences for all $n \geq 0$ sitting in a chain

\dots \subseteq K^3 \subseteq K^2 \subseteq K \subseteq \All.

We call this the $$K$$ -adic filtration. If $$K$$ is of small generation, so are its powers, and the filtration gives a tower of quotient logoi

T \to \dots \to T/K^3 \to T/K^2 \to T/K \to T/\All \iso *.

Let $(K^{n+1})^\perp$ denote the right class of the modality determined by $K^{n+1}$ . We define the $$(n+1)$$ -st layer of the filtration by

K^n/K^{n+1}:=K^n\cap(K^{n+1})^\perp.

For $X\in T$ , let $(K^n/K^{n+1})_X\subseteq T_{/X}$ be the full subcategory spanned by the maps $Y\to X$ in this class. At the terminal object, it can equivalently be described as the fiber

Thus an object of $(K^n/K^{n+1})_1$ is $K^{n+1}$ -local and becomes terminal after localization at $$K^n$$ . The same description in the slice $T_{/X}$ gives the layer at an arbitrary object $$X$$ .