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.
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
(e.g. Johnstone, prop C 2.2.7, Makkai-Reyes, lemma 1.3.14)
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.
A textbook account in topos theory is in
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.
Last revised on April 16, 2023 at 11:02:07. See the history of this page for a list of all contributions to it.