nLab sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes



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Let (𝒞,τ)(\mathcal{C}, \tau) be a small site. Then the full subcategory inclusion i:Sh(𝒞,τ)PSh(𝒞)i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C}) of its category of sheaves (Grothendieck topos) into its category of presheaves is a reflective subcategory inclusion

Sh(𝒞,τ)iLPSh(𝒞) Sh(\mathcal{C},\tau) \underoverset {\underset{i}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} PSh(\mathcal{C})

such that the reflector L:PSh(𝒞)Sh(𝒞)L \colon PSh(\mathcal{C}) \to Sh(\mathcal{C}) preserves finite limits (the reflector being sheafification).

Moreover, up to equivalence, every Grothendieck topos arises this way: Given a small category 𝒞\mathcal{C} there is a bijection between

  1. the equivalence classes of left exact reflective subcategories PSh(𝒞)\mathcal{E} \hookrightarrow PSh(\mathcal{C}) of the category of presheaves

  2. Grothendieck topologiesτ\tau on 𝒞\mathcal{C},

which is such that Sh(𝒞,τ)\mathcal{E} \simeq Sh(\mathcal{C}, \tau).

(e.g. Borceux 94, prop. 3.5.4, cor. 3.5.5, Johnstone, C.2.1.11, or Kelly).


(accessible embedding is implied)

In the situation of prop. it follows that the inclusion i:Sh(𝒞,τ)PSh(𝒞)i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C}) is an accessible functor, hence an accessible reflective subcategory inclusion.


Every Grothendieck topos like Sh(𝒞,τ)Sh(\mathcal{C}, \tau) and PSh(𝒞)PSh(\mathcal{C}) is a locally presentable category (this prop., Borceux 94, vol 3, prop. 3.4.16). Therefore with prop. the statement follows by the adjoint functor theorem for locally presentable categories (this prop.).


(generalization to (∞,1)-toposes)

For (∞,1)-toposes the accessibility of the reflection is no longer implied in general, contrary to prop. above, but needs to be required. It is however still implied for topological localizations (Lurie, Cor.


Last revised on February 9, 2024 at 03:18:44. See the history of this page for a list of all contributions to it.