nLab canonical topology

Definition

The canonical topology on a category $C$ is the Grothendieck topology on $C$ which is the largest subcanonical topology. More explicitly, a sieve $R$ is a covering for the canonical topology iff every representable functor is a sheaf for every pullback of $R$. Such sieves are called universally effective-epimorphic.

Examples

On a Grothendieck topos

If $C$ is a Grothendieck topos, then the canonical covering sieves are those that are jointly epimorphic. Moreover, in this case the canonical topology is generated by small jointly epimorphic families, since $C$ has a small generating set.

The canonical topology of a Grothendieck topos is also special in that every sheaf is representable; that is, $C \simeq Sh_{canonical}(C)$.

Notice that if $(D,J)$ is a site of definition for the topos $C$, then this says that

Proposition

$$Sh_J(D) \simeq Sh_{canonical}(Sh_J(D)) \,.$$

(e.g. Johnstone, prop C 2.2.7, Makkai-Reyes, lemma 1.3.14)

Proof

For simplicity, assume $(\mathbb{C}, J)$ is a small subcanonical site. The quasi-inverse of the Yoneda embedding $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$ has a very simple description: it is the functor that sends a sheaf $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ to its restriction along the Yoneda embedding $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$.

Indeed, suppose $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ is a sheaf. We claim that $F$ is determined up to unique isomorphism by its restriction along the embedding $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$. Indeed, let $X : \mathbb{C}^\mathrm{op} \to \mathbf{Set}$ be a $J$-sheaf. Then $X$ is the colimit of a canonical small diagram of representable sheaves on $(\mathbb{C}, J)$ in a canonical way. Consider the colimiting cocone on $X$: it is a universal effective epimorphic family and is therefore a covering family in the canonical topology on $\mathbf{Sh}(\mathbb{C}, J)$. Thus, $F (X)$ is indeed determined up to unique isomorphism by the restriction of $F$ to $\mathbb{C}$. We must also show that the restriction is actually a sheaf on $(\mathbb{C}, J)$; but this is true because $J$-covering sieves in $\mathbb{C}$ become universal effective epimorphic families in $\mathbf{Sh}(\mathbb{C}, J)$.

Thus we obtain a functor $\mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J)) \to \mathbf{Sh}(\mathbb{C}, J)$ that is left quasi-inverse to the embedding $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$, and the argument above shows that it is also a right quasi-inverse.

Related concepts

• subcanonical topology

• trivial topology

References

A textbook account in topos theory is in

• Peter Johnstone, Sketches of an Elephant

• Makkai, Gonzalo Reyes, First Order Categorical Logic

Discussion of examples:

• Cynthia Lester, Covers in the Canonical Grothendieck Topology [arXiv:1909.03384]

• Cynthia Lester, The canonical Grothendieck topology and a homotopical analog, PhD thesis (2019) [uoregon:1794/24924 pdf]

• Cynthia Lester, The canonical Grothendieck topology and a homotopical analog [arXiv:1909.03188]

Discussion in the refined context of higher topos theory is in

• Jacob Lurie, section 6.2.4 and around prop. 5.5.2.2., remark 6.3.5.17 of Higher Topos Theory

• David Carchedi, MO discussion Canonical topology for infinity topoi revisited.