Products of topoi as tensor products

The initial object of \(\Logos\), equivalently the terminal object of \(\Topos\), is \(\An\) by Example 4.12. It remains to compute binary coproducts in \(\Logos\).Write \(T = \PSh(C)[\Sigma^{-1}]\) and \(S = \PSh(D)[\Gamma^{-1}]\) with \(C,D \in \Cat^{\lex}\). We claim that a coproduct of \(C\) and \(D\) in \(\Cat^{\lex}\) is given by the product category \(C \times D\). We have left exact inclusions \(C \simeq C \times \{*\} \hookrightarrow C \times D\) and \(D \simeq \{*\} \times D \hookrightarrow C \times D\), and restriction along them produces a functor We claim that this functor is an equivalence, with inverse sending a pair \((f,g)\) to the functor \(h\colon C \times D \to E\) defined by Indeed it is easy to see that \(h(x,*) = f(x)\) and \(h(*,y) = g(y)\), and conversely we always have \(h(x,y) = h(x,*) \times h(*,y)\) for any \(h\) since we have the relation \((x,y) = (x,*) \times (*,y)\) in \(C \times D\). Using Proposition 2.43, we conclude that for every topos \(T'\) we have an equivalence and hence the logos \(\PSh(C \times D) \simeq \PSh(C) \otimes \PSh(D)\) is a coproduct of \(\PSh(C)\) and \(\PSh(D)\) in \(\Logos\), i.e. it is a product of topoi.Now consider the following commutative diagram of tensor products in \(\PrL\):

Using the universal properties of localizations and tensor products, we observe that this is a pushout square in \(\PrL\). Passing to right adjoints then produces a pullback square in \(\PrR\), and we see that we may identify \(T \otimes S\) with the intersection as a full subcategory of \(\PSh(C) \otimes \PSh(D) = \PSh(C \times D)\).To conclude, we use that for every small category \(E\) and presentable category \(U\) there is a natural equivalence Hence if \(L\colon U' \to U\) is a left exact localization with fully faithful right adjoint \(R\), then the induced functor identifies with postcomposition by \(L\) on \(\Fun(E\catop,-)\). Note that this is again a left exact localization: it is left exact since finite limits in functor categories are computed pointwise, and its right adjoint is pointwise postcomposition by \(R\), hence fully faithful. Applying this to \(\PSh(D)\to S\) (with \(E=C\)) shows that \(\PSh(C)\otimes S \subseteq \PSh(C)\otimes \PSh(D)\) is a left exact localization; by symmetry, the same holds for \(T\otimes \PSh(D) \subseteq \PSh(C)\otimes \PSh(D)\). Intersections of accessible left exact localizations remain accessible left exact localizations by [Lurie 2009, Lemma 6.3.3.4].Thus the intersection is the left exact localization of \(\PSh(C\times D)\) at the union of the two induced localizing classes (equivalently, at the strongly saturated class they generate). In particular, \(T\otimes S\) is a topos. By comparing universal properties, we now conclude that \(T\otimes S\) is the coproduct of \(T\) and \(S\) in \(\Logos\), i.e. the product of \(T\) and \(S\) in \(\Topos\).