Limits and colimits of topoi

Given a diagram \(I \to \Logos, i \mapsto T_i\) of logoi, define \(T := \lim_i T_i\), which is again a presentable category. Given an indexing diagram \(K\), a functor \(K^{\triangleright} \to T\) is a colimit diagram if and only if it becomes a colimit diagram in each \(T_i\). The analogous statement holds for finite limit diagrams. In particular, colimits and finite limits in \(T\) are computed pointwise in the categories \(T_i\). Note that we have It then follows immediately from the characterization of logoi given in Proposition 2.16 that \(T\) is again a logos. Furthermore, we see that a functor \(S \to T\) is a logos morphism if and only if each composite \(S \to T_i\) is a logos morphism. It follows that the limit in \(\Cat\) is in fact a limit in \(\Logos\), as desired.

Let \(T_{\bullet}\colon I\to\Logos\) be a small diagram. By uniformization for accessible categories and functors, we may choose a regular cardinal \(\kappa\) such that each \(T_i\) is \(\kappa\)-accessible, every transition morphism is strongly \(\kappa\)-accessible, and the full subcategory of \(\kappa\)-compact objects is closed under finite limits. The transition morphisms therefore restrict to a diagram \(C_{\bullet}\colon I\to\Cat\) of small categories with finite limits. For the simultaneous choice of \(\kappa\), see the uniformization argument in [Lurie 2009, Remark 5.4.2.13]; closure of the compact objects under finite limits follows after a further enlargement from [Lurie 2009, Proposition 5.4.7.4].For each \(i\), the colimit extension \(\PSh(C_i)\to T_i\) of the inclusion \(C_i\hookrightarrow T_i\) is left exact by Proposition 2.43. Its right adjoint is the restricted Yoneda functor, which is fully faithful because \(C_i\) is a collection of generators. Composing this localization with the canonical localization \(\An[C_i]\to\PSh(C_i)\) from the proof of Proposition 4.26, we obtain a left exact Bousfield localization Choose a small class of morphisms \(\Sigma_i\subseteq\An[C_i]\) whose generated congruence is \(\ker(q_i)\). Thus \(q_i\) induces an equivalence \(\lra{C_i\mid\Sigma_i}\simeq T_i\).These presentations are compatible with the diagram. Indeed, for an arrow \(a\colon i\to j\) in \(I\), let \(a^*\colon T_i\to T_j\) denote the transition morphism and also its restriction \(C_i\to C_j\). The two logos morphisms given by \(a^*q_i\) and by \(q_j\An[a^*]\) agree on the generators \(C_i\), hence are equivalent by the universal property of the free logos. It follows that \(\An[a^*]\) sends \(\ker(q_i)\) into \(\ker(q_j)\) and therefore induces the original transition morphism between the presented logoi.Now form the small category \(C:=\colim_iC_i\), and let \(\Sigma\) be the union of the images of the small classes \(\Sigma_i\) in \(\An[C]\). Consider the logos The functors \(C_i\to C\) induce a cocone \(T_i\to T\). To verify its universal property, let \(S\) be a logos. Then Under this equivalence, a logos morphism \(\An[C]\to S\) inverts \(\Sigma\) if and only if each of its restrictions to \(\An[C_i]\) inverts \(\Sigma_i\). By Lemma 4.25, such data are precisely a compatible family of logos morphisms \(T_i\to S\). Hence so \(T\) is the colimit of \(T_{\bullet}\) in \(\Logos\).

We sketch the construction, following [Lurie 2009, Section 6.3.3]. Given a functor \(I \to \Topos\), consider the associated cartesian fibration \(p\colon E \to I\catop\). The category of sections \(\Gamma(p)\) is a topos by [Lurie 2009, Lemma 6.3.3.2]; its colimits and finite limits are computed fiberwise, as also follows from Chapter B. Assume that \(I\) is filtered. The full subcategory \(\Gamma_{\cart}(p)\) of cartesian sections is an accessible left exact localization of \(\Gamma(p)\) by [Lurie 2009, Proposition 6.3.3.3]. Consequently, \(\Gamma_{\cart}(p)\) is again a topos. Moreover, the evaluation functors are morphisms of topoi by [Lurie 2009, Proposition 6.3.3.5].Let us recall why the left exact localization is the central point. After replacing the indexing category by a cofinal filtered partially ordered set, let \(B\subseteq I\) be a cofinal subset. Write \(\Gamma_B(p)\subseteq\Gamma(p)\) for the full subcategory of sections which are right Kan extensions of their restrictions to \(B\). Restriction to \(B\) admits right Kan extension as a fully faithful right adjoint, so \(\Gamma_B(p)\) is an accessible localization of \(\Gamma(p)\). The reflector is left exact because both restriction and right Kan extension preserve finite limits.The cartesian sections are precisely the intersection of the subcategories \(\Gamma_B(p)\) as \(B\) ranges over all cofinal subsets. Indeed, a cartesian section is a right Kan extension from every cofinal subset. Conversely, given an arrow \(i\to j\), take the cofinal subset of objects lying above \(j\). If a section is a right Kan extension from this subset, its comparison morphism along \(i\to j\) is cartesian. The intersection of a small collection of accessible left exact localizations is again an accessible left exact localization by [Lurie 2009, Lemma 6.3.3.4]. This proves the asserted description of \(\Gamma_{\cart}(p)\); see [Lurie 2009, Proposition 6.3.3.3] for the complete argument.The category \(\Gamma_{\cart}(p)\) computes the limit of the underlying categories \(E_i\). It remains to verify the universal property in \(\Topos\). A functor from a topos \(S\) to \(\Gamma_{\cart}(p)\) is a morphism of topoi if and only if all its composites with the evaluation functors are morphisms of topoi. The nontrivial direction is [Lurie 2009, Proposition 6.3.3.8]. Hence the limiting cone in \(\Cat\) is also limiting in \(\Topos\).

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 \(\Topos\).

Combine Proposition 4.30, Proposition 4.31, Proposition 4.32 and Proposition 4.33.