# Universal property of the category of $\mathcal{S}$-sheaves and the definition of Topos

Let $$\mathcal{C}$$ be a small category; let $$\mathcal{S}$$ be any family of maps in $$\text{Psh}\left(\mathcal{C}\right)$$.

Call $$X\in \text{Psh}\left(\mathcal{C}\right)$$ an $$\mathcal{S}$$-sheaf when $$\text{Hom}_{\text{Psh}\left(\mathcal{C}\right)}(-,X)=h_X$$ is such that $$h_X(s)$$ is an isomorphism for every $$s\in \mathcal{S}$$.

1 The full subcategory of $$\mathcal{S}$$-sheaves, call it $$P_{\mathcal S}$$, comes equipped with a localization functor $$L_{\mathcal S}$$ which is left adjoint to the inclusion, so that $$P_{\mathcal S}\subset \text{Psh}\left(\mathcal{C}\right)$$ is a reflective subcategory.

2 For every co-complete category $$\mathcal{E}$$ and for every co-continuous functor $$F:\mathcal{C}\to \mathcal{E}$$ with the property that $$\text{Lan}_yF(s)$$ is invertible for every $$s \in \mathcal S$$, there exists a unique co-continous functor $$\overline F:P_{\mathcal S}\to \mathcal {E}$$ such that $$\overline F L_{\mathcal S}y=F$$ (where $$y$$ is the Yoneda embedding).

3 Suppose that $$\mathcal S$$ is a family of subfunctors of representable functors and $$\mathcal S$$ is a Grothendieck topology. Then in this case $$P_{\mathcal S}$$ is a topos, and every (Grothendieck) topos arises in this way for some choice of $$\mathcal S$$

I would like to know a reference for 1,2,3. Is there a book where the notion of topos is defined in this way?

If you don't know a reference but can sketch me an idea of the proofs I would still be happy.

• You can find similar information about Q1 in Borceux Handbook, first tome. It's a slightly more general setting than LPAC (remember that your toposes are left exact localizations! This means that S has to be pullback stable.), but the ideas are the same. See in particular the section on the "orthogonal subcategory problem", Thm 5.4.7 – fosco Dec 5 '19 at 19:39
• Thanks I will check Borceux – giuseppe Dec 5 '19 at 20:00
• These conditions aren't enough to specify a category of sheaves. These categories are called 'locally presentable', and the objects are called 'S-local objects'. When the set S of maps is closerd under certain kinds of fibre products, such that the localization functor is exact, then they are an honest category of sheaves. It follows almost tautologically that the universal property is the univ. property of the localization in Pr^L of locally presentable categories with left adjoint functors between them. – Harry Gindi Dec 5 '19 at 21:24
• @HarryGindi I just called $S$-sheaves what you call $S$-local objects.. I never said they were already an honest topos, unless $S$ is a topology. So are you just arguing about my terminology? – giuseppe Dec 5 '19 at 22:43
• @giuseppe I was telling you the right terminology so you can look up the answer yourself… There are standard references that answer your question. – Harry Gindi Dec 5 '19 at 23:18

3. Choose your favourite generator $$S$$ of your topos $$i: S \subset\mathcal{E}$$, and use the nerve of $$i$$, $$N(i): \mathcal{E} \to \mathsf{Set}^{S^\circ}$$. Since in a topos every generator is dense, the nerve is fully faithful, moreover, it preserves limits and by the general AFT (here any argument would work, you can also use the universal property of the presheaf construction), it has a left adjoint. You can check (it's easier if $$S$$ is closed under finite limits) that the left adjoint must preserve finite limits (because $$\mathcal{E}$$ is a topos) . This kind of representation argument will work for every cocomplete category with a dense generator and sits at the very core of Thm. 1.46 in LPAC.