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Higher Topos Theory · Chapter 5 · Section 5.4

Open and closed immersions

5.4. Open and closed immersions

An open subset of a topological space determines both an open subtopos and a complementary closed subtopos, and the ambient topos can be reconstructed by gluing these two pieces. We now develop the corresponding picture for arbitrary topoi. We first characterize open immersions as slices over \((-1)\)-truncated objects, then construct their closed complements using the theory of congruences, and finally describe the resulting recollement.

Definition 5.44.

A morphism \(\Uu \to \Xx\) is an open immersion if it is both étale and a monomorphism. A morphism in \(\Logos\) is an open localization if its corresponding morphism in \(\Topos\) is an open immersion.

Lemma 5.45.

Let \(j_*\colon \Xx_{/U} \to \Xx\) be an étale morphism of topoi, \(U \in \Xx\). Then \(j_*\) is an open immersion if and only if \(U\) is \((-1)\)-truncated.

Proof
We saw in Proposition 4.38 that the base change of \(j_*\) along itself is given by the functor \((\Xx_{/U})_{/U \times U} \to \Xx_{/U}\). This is an equivalence if and only if the object \((U \times U) \in \Xx_{/U}\) is terminal, i.e. if the projection map \(\pr_1\colon U \times U \to U\) is an isomorphism. This in turn is equivalent to the diagonal \(\Delta\colon U \to U \times U\) being an isomorphism, which by definition means that \(U\) is \((-1)\)-truncated.

If \(U \hookrightarrow X\) is an open embedding of topological spaces, then by definition its complement \(Z := X \setminus U\) is a closed subspace. Its toposic analogue is as follows:

Definition 5.46.

Let \(j_*\colon \Uu \hookrightarrow \Xx\) be an open immersion of topoi, and let \(j^*\colon \Xx \to \Uu\) be its left adjoint. We define the closed subtopos complementary to \(\Uu\) as

\[\Xx / \Uu \quad := \quad \{\,X \in \Xx \mid j^*(X) = *\,\} \quad \subseteq \quad \Xx.\]

More generally, we say that a morphism of topoi \(i\colon \Zz \to \Xx\) is a closed immersion if there exists a \((-1)\)-truncated object \(U \in \Xx\) such that \(i_*\colon \Zz \to \Xx\) induces an equivalence \(\Zz \iso \Xx / \Uu\).

Lemma 5.47.

Given an open immersion \(j\colon \Uu \hookrightarrow \Xx\), the closed complement \(\Zz := \Xx / \Uu\) is a topos, and the inclusion \(i\colon \Zz \hookrightarrow \Xx\) is a morphism of topoi.

Proof
By Lemma 5.45, we may identify \(\Uu\) with the slice \(\Xx_{/U}\) for some \((-1)\)-truncated object \(U\) of \(\Xx\). Given \(X \in \Xx\), we then have \(X \in \Zz\) if and only if the projection map \(X \times U \to U\) is an isomorphism. Equivalently, given an object \(Y \in \Xx\), every morphism \(Y \to U\) admits a unique lift \(Y \to U \times X\). We conclude that the subcategory \(\Zz = \Xx / \Uu\) consists precisely of those objects that are local with respect to the following class of morphisms:
\[S_U := \{\,\emptyset \to V \mid \text{there exists a map } V \to U\,\}.\]
Choose a small set of generators \(\mathcal G\) of \(\Xx\). The objects \(g\colon G\to U\), with \(G\in\mathcal G\) and \(g\) ranging through all morphisms to \(U\), form a small set of generators of the slice \(\Xx_{/U}\). It therefore suffices to impose locality with respect to the corresponding set
\[S_U^0:=\{\,\emptyset\to G\mid G\in\mathcal G, g\colon G\to U\,\},\]
Indeed, every object of \(\Xx_{/U}\) is a colimit of these generators, and mapping out of this colimit identifies locality with respect to \(S_U^0\) with locality with respect to the entire class \(S_U\). Consequently, Proposition A.15 gives a left adjoint \(\Xx\to\Zz\) to the inclusion.It remains to show that this left adjoint preserves finite limits, or equivalently that the strong saturation of \(S_U^0\), which agrees with the strong saturation of \(S_U\), is a congruence. The class \(S_U\) is closed under base change, so \((S_U)^s=(S_U)^m\) by Proposition 5.32. Moreover, its morphisms are monomorphisms, whence \((S_U)^m=(S_U)^c\) by Corollary 5.43. We deduce that the localization \(\Xx\to\Zz\) is left exact, exhibiting \(\Zz\) as a topos and its inclusion into \(\Xx\) as a morphism of topoi.

If \(j\colon \Uu \hookrightarrow \Xx\) is an open immersion with closed complement \(i\colon \Zz \hookrightarrow \Xx\), we may recover the topos \(\Xx\) from the topoi \(\Uu\) and \(\Zz\) together with the composite \(i^*j_*\colon \Uu \to \Zz\).

Construction 5.48.

Let \(\Uu\) and \(\Zz\) be categories with finite limits, and let \(F\colon \Uu \to \Zz\) preserve terminal objects. We define the category \(\Xx\) as the comma category of \(F\):

\[\Xx \quad := \quad \Uu \times_{\Zz,\,F} \Ar(\Zz),\]

where the functor \(\Ar(\Zz)\to\Zz\) is evaluation at the target. In other words, the objects of \(\Xx\) are triples \((U,Z,\varphi)\) where \(U \in \Uu\), \(Z \in \Zz\) and \(\varphi\colon Z \to F(U)\).

Note that the two forgetful functors \(j^*\colon \Xx \to \Uu\) and \(i^*\colon \Xx \to \Zz\) admit canonical right adjoints

\[j_*\colon \Uu \hookrightarrow \Xx, U \mapsto (U,F(U),\id_{F(U)}), \qquadtext{ and } i_*\colon \Zz \hookrightarrow \Xx, Z \mapsto (*,Z,Z\to F(*)\iso *).\]

Lemma 5.49.

Consider topoi \(\Uu\) and \(\Zz\) and an accessible left exact functor \(F\colon \Uu \to \Zz\).

  1. The associated recollement \(\Xx\) is a topos,

  2. The functor \(j_*\colon \Uu \to \Xx\) is an open immersion of topoi,

  3. The functor \(i_*\colon \Zz \to \Xx\) is a closed immersion of topoi.

Proof
The comma category \(\Xx\) is accessible, and the results of Proposition B.1 show that it admits all small colimits and finite limits. Hence \(\Xx\) is presentable. Moreover, the forgetful functor
\[(j^*,i^*)\colon \Xx\longrightarrow \Uu\times\Zz\]
is cocontinuous, left exact, and conservative. The Giraud conditions for \(\Xx\) may therefore be checked after applying this functor, where they follow from the corresponding conditions in \(\Uu\) and \(\Zz\). Thus \(\Xx\) is a topos.Let
\[V:=(*,\emptyset,\emptyset\to F(*))\in\Xx.\]
This object is \((-1)\)-truncated. Base change along \(V\to *\) identifies \(\Xx_{/V}\) with \(\Uu\): a morphism \((U,Z,\varphi)\to V\) forces \(Z\iso\emptyset\), and is then determined by \(U\). Under this equivalence, the inverse-image functor of the étale morphism \(\Xx_{/V}\to\Xx\) identifies with \(j^*\), and its right adjoint identifies with \(j_*\). Thus \(j_*\) is an open immersion by Lemma 5.45.Finally, an object \((U,Z,\varphi)\) belongs to the closed complement of this open immersion precisely when
\[j^*(U,Z,\varphi)=U\iso *.\]
Since \(F(*)\iso *\), such an object is uniquely of the form \((*,Z,Z\to *)=i_*(Z)\). Hence \(i_*\) identifies \(\Zz\) with the closed complement of \(\Uu\), and is therefore a closed immersion.