nLab rigid topology

Idea

A Grothendieck topology $J$ on a small Cauchy complete category $\mathcal{C}$ is rigid when the corresponding sheaf topos $Sh(\mathcal{C},J)$ is of the form $Set^{\mathcal{D}^{op}}$ for some full subcategory $\mathcal{D}\hookrightarrow \mathcal{C}$. In particular, $Sh(\mathcal{C},J)$ is an essential subtopos of $Set^{\mathcal{C}^{op}}$.

Definition

Let $\mathcal{C}$ be a small Cauchy complete category and $J$ a Grothendieck topology on $\mathcal{C}$, an object $U\in\mathcal{C}$ is called $J$-irreducible if the only covering sieve is the maximal sieve.

$J$ is called rigid when for every object $X\in \mathcal{C}$ there exists a $J$-covering sieve generated by the family of all morphisms from $J$-irreducible objects to $X$.

Properties

Proposition

Let $J$ be a rigid topology on the small Cauchy complete category $\mathcal{C}$ and $J|_{\mathcal{D}}$ the induced Grothendieck topology on the full subcategory $\mathcal{D}$ of $J$-irreducible objects. Then

$$Sh(\mathcal{C},J)\simeq Sh(\mathcal{D},J|_{\mathcal{D}})\simeq Set^{\mathcal{D}^{op}}\qquad .$$

The first equivalence follows from the comparison lemma and the second equivalence from the fact that $J|_{\mathcal{D}}$ is the minimal topology on $\mathcal{D}$ whence every presheaf is a sheaf (cf. Johnstone, C2.2.18). Since $Set^{\mathcal{D}^{op}}\hookrightarrow Set^{\mathcal{C}^{op}}$ arises by Kan extension of the (full) subcategory inclusion $\mathcal{D}\hookrightarrow \mathcal{C}$ the subtopos inclusion is in fact essential.

Examples

• Trivially, for any $\mathcal{C}$ all objects $X\in\mathcal{C}$ are $J_{min}$-irreducible for the minimal topology $J_{min}$ consisting of only the maximal sieves whence $J_{min}$ is rigid.

• Similarly, $J_{max}$ the collection of all sieves is rigid because then no object is $J_{max}$-irreducible which in turn says by the definition of rigidity that $\empty\in J_{max}(X)$ for any $X$ which in turn implies that every sieve $\empty\subseteq S\in J_{max}(X)$.

• On a finite Cauchy complete category $\mathcal{C}$ every Grothendieck topology is rigid (cf. Johnstone, C2.2.21); By a remark there on p.562 the same holds if merely all slices $\mathcal{C}/X$ are finite as happens e.g. in the case of the semi-simplex category $\Delta_+$ where any object $[n]$ receives only maps from $[n']$ with $n'\leq n$.

Related entries

• level of a topos

• Aufhebung

References

• Peter Johnstone, Sketches of an Elephant II, Oxford UP 2002. (C2.2.18, pp.560-563)

• Jérémie Marquès, A Criterion for Categories on which every Grothendieck Topology is Rigid, arXiv:2407.1847 (2024). (abstract)