Normalization operator

Contents

1. Definition§

Let $\mathcal C$ be a category with Local state classifier $\Xi$ and cocone maps $\xi_X:X\to\Xi$. Its normalization operator is the component at the classifier itself:

$$\nu=\xi_\Xi:\Xi\to\Xi.$$

See Definition 4.1.

2. Properties§

Proposition 2.1 (Normalization lemma)§★

If $\mathcal C$ is cartesian closed, then $\operatorname{id}_\Xi\leq\nu$ in the canonical internal semilattice order. Equivalently, internally, $x\wedge\nu(x)=x$ for every local state $x$.

See Proposition 4.9.

3. Examples§

For right actions of a group $G$, the operator sends a subgroup to its normalizer (Example 4.2).
For a localic Grothendieck topos, it is the identity on the terminal classifier (Example 4.4).
In general it need not be idempotent or order-preserving; the group-action and species examples exhibit this failure (Example 4.8).
For regular languages, the Nerode congruence and syntactic congruence belong to exactly the same internal filters of the classifier of Sigma-sets. The proof uses the normalization lemma. See Corollary 6.5.

The classifier of a hyperconnected quotient is the internal filter defining that quotient. This is Theorem 3.1.

5. References§

Ryuya Hora, Normalization of a subgroup, in a topos, and of a word-congruence, arXiv:2511.05012v1.