Local state classifier
Contents
A local state classifier collects the information about an object that is preserved by passing to subobjects.
1. Definition§
Let $\mathcal C$ be a category and let $\mathcal C_{\mathrm{mono}}$ have the same objects and only the monomorphisms of $\mathcal C$. A local state classifier is a colimit of the inclusion $\mathcal C_{\mathrm{mono}}\to\mathcal C$, if it exists. Write $\Xi$ for the colimit object and $\xi_X:X\to\Xi$ for its cocone maps.
Thus $\xi_Ym=\xi_X$ for every monomorphism $m:X\to Y$, and every family satisfying these equations factors uniquely through $\Xi$. Such a family is called a locally determined cocone.
See Definitions 3.4–3.5.
2. Examples§
- In sets, $\Xi$ is a singleton (Example 3.6).
- In directed graphs, $\Xi$ has one vertex and two loop edges, recording whether an edge of the original graph is a loop (Example 3.9).
- For right actions of a group $G$, $\Xi$ is the set of subgroups with conjugation action; $\xi_X$ sends an element to its stabilizer (Example 3.10).
- In a presheaf topos, its value at an object consists of the quotients of the corresponding representable presheaf (Example 3.22).
3. Properties§
Proposition 3.1 (Existence and semilattice structure)§★
Every Grothendieck topos has a local state classifier. In a cartesian closed category, a local state classifier carries a canonical internal meet-semilattice structure: the top is $\xi_1$, where $1$ is terminal, and the meet is induced by products.
Theorem 3.2 (Hyperconnected quotients)§★
For a topos with local state classifier $\Xi$ and subobject classifier $\Omega$, the following correspond bijectively:
- hyperconnected quotients;
- internal filters of $\Xi$;
- internal meet-semilattice homomorphisms $\Xi\to\Omega$.
Here filters are internal filters in the meet-semilattice $\Xi$. See Theorem 4.1 and Appendix A.
Corollary 3.3 (Localic and Boolean cases)§★
A Grothendieck topos is localic exactly when its local state classifier is terminal. For a Boolean topos with a local state classifier, every quotient is hyperconnected, so the theorem classifies all its quotients.
Theorem 3.4 (Classifier of a hyperconnected quotient)§★
Let $\mathcal E$ be an elementary topos with local state classifier $\Xi$, and let $F\hookrightarrow\Xi$ be an internal filter. The induced hyperconnected quotient consists of the objects $X$ for which $\xi_X$ factors through $F$. The object $F$ belongs to that quotient and, with these factorizations as cocone maps, is its local state classifier.
See Theorem 3.1. Its proof uses the Normalization operator.
4. Related concepts§
5. References§
- Ryuya Hora, Internal Parameterization of Hyperconnected Quotients, Theory and Applications of Categories 42 (2024), no. 11, 263–313.
- Ryuya Hora, Normalization of a subgroup, in a topos, and of a word-congruence, arXiv:2511.05012v1, Theorem 3.1.