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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.

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    $\begingroup$ 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 $\endgroup$ – fosco Dec 5 '19 at 19:39
  • $\begingroup$ Thanks I will check Borceux $\endgroup$ – giuseppe Dec 5 '19 at 20:00
  • $\begingroup$ 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. $\endgroup$ – Harry Gindi Dec 5 '19 at 21:24
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    $\begingroup$ @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? $\endgroup$ – giuseppe Dec 5 '19 at 22:43
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    $\begingroup$ @giuseppe I was telling you the right terminology so you can look up the answer yourself… There are standard references that answer your question. $\endgroup$ – Harry Gindi Dec 5 '19 at 23:18
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  1. (&3?) Can follow from the theory of orthogonality classes. A reference is Thm 1.38 and 1.39 in Locally presentable and accessible categories by Adamek and Rosicky.

  2. I do not have a reference, but this is really a standard technique. In my mind, this follows from the theory of Kan extensions. A place where this kind of arguments is carried on is in my preprint with Julia Ramos Gonzales, Gabriel-Ulmer duality for topoi and its relation with site presentations, Prop 4.3. The latter is far from being a natural reference, and probably someone can suggest better. In the spirit of this question, I should probably mention Cor VII.7.4 in Sheaves in Geometry and Logic by Mac Lane and Moerdijk.

  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.

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  • $\begingroup$ thanks! I will check the references and wait in case someone wants to add something about 2 and 3 and then I can accept. $\endgroup$ – giuseppe Dec 5 '19 at 20:00
  • $\begingroup$ Take also your time to decide if these references are satisfying for you! $\endgroup$ – Ivan Di Liberti Dec 5 '19 at 20:49
  • $\begingroup$ Another exposition of (2) can be found at nlab:reflective localization. Some of the ideas date back to Gabriel-Zisman, Calculus of fractions and homotopy theory, although I don't think they dealt with the cocomplete case. $\endgroup$ – Mike Shulman Dec 8 '19 at 19:48

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