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


On a Grothendieck topos

If CC 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 CC has a small generating set.

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

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

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

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


For simplicity, assume (,J)(\mathbb{C}, J) is a small subcanonical site. The quasi-inverse of the Yoneda embedding Sh(,J)Sh(Sh(,J))\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:Sh(,J) opSetF : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set} to its restriction along the Yoneda embedding Sh(,J)\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J).

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

Thus we obtain a functor Sh(Sh(,J))Sh(,J)\mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J)) \to \mathbf{Sh}(\mathbb{C}, J) that is left quasi-inverse to the embedding Sh(,J)Sh(Sh(,J))\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

Discussion of examples:

Discussion in the refined context of higher topos theory is in

Last revised on April 16, 2023 at 11:02:07. See the history of this page for a list of all contributions to it.