
## 7.15 Topoi

Here is a definition of a topos which is suitable for our purposes. Namely, a topos is the category of sheaves on a site. In order to specify a topos you just specify the site. The real difference between a topos and a site lies in the definition of morphisms. Namely, it turns out that there are lots of morphisms of topoi which do not come from morphisms of the underlying sites.

Definition 7.15.1 (Topoi). A topos is the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ of sheaves on a site $\mathcal{C}$.

1. Let $\mathcal{C}$, $\mathcal{D}$ be sites. A morphism of topoi $f$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is given by a pair of functors $f_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $f^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ such that

1. we have

$\mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(f^{-1}\mathcal{G}, \mathcal{F}) = \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, f_*\mathcal{F})$

bifunctorially, and

2. the functor $f^{-1}$ commutes with finite limits, i.e., is left exact.

2. Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be sites. Given morphisms of topoi $f :\mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and $g :\mathop{\mathit{Sh}}\nolimits (\mathcal{E}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ the composition $f\circ g$ is the morphism of topoi defined by the functors $(f \circ g)_* = f_* \circ g_*$ and $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$.

Suppose that $\alpha : \mathcal{S}_1 \to \mathcal{S}_2$ is an equivalence of (possibly “big”) categories. If $\mathcal{S}_1$, $\mathcal{S}_2$ are topoi, then setting $f_* = \alpha$ and $f^{-1}$ equal to a quasi-inverse of $\alpha$ gives a morphism $f : \mathcal{S}_1 \to \mathcal{S}_2$ of topoi. Moreover this morphism is an equivalence in the $2$-category of topoi (see Section 7.36). Thus it makes sense to say “$\mathcal{S}$ is a topos” if $\mathcal{S}$ is equivalent to the category of sheaves on a site (and not necessarily equal to the category of sheaves on a site). We will occasionally use this abuse of notation.

Two examples of topoi. The empty topos is topos of sheaves on the site $\mathcal{C}$, where $\mathcal{C}$ has a single object $\emptyset$ and a single morphism $\text{id}_\emptyset$ and a single covering, namely the empty covering of $\emptyset$. We will sometimes write $\emptyset$ for this site. This is a site and every sheaf on $\mathcal{C}$ assigns a singleton to $\emptyset$. Thus $\mathop{\mathit{Sh}}\nolimits (\emptyset )$ is equivalent to the category having a single object and a single morphism. The punctual topos is the topos of sheaves on the site $\mathcal{C}$ which has a single object $pt$ and one morphism $\text{id}_{pt}$ and whose only covering is the covering $\{ \text{id}_{pt}\}$. We will simply write $pt$ for this site. It is clear that the category of sheaves $=$ the category of presheaves $=$ the category of sets. In a formula $\mathop{\mathit{Sh}}\nolimits (pt) = \textit{Sets}$.

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a morphism of topoi. Note that $f_*$ commutes with all limits and that $f^{-1}$ commutes with all colimits, see Categories, Lemma 4.24.5. In particular, the condition on $f^{-1}$ in the definition above guarantees that $f^{-1}$ is exact. Morphisms of topoi are often constructed using either Lemma 7.21.1 or the following lemma.

Lemma 7.15.2. Given a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ corresponding to the functor $u : \mathcal{C} \to \mathcal{D}$ the pair of functors $(f^{-1} = u_ s, f_* = u^ s)$ is a morphism of topoi.

Proof. This is obvious from Definition 7.14.1. $\square$

Remark 7.15.3. There are many sites that give rise to the topos $\mathop{\mathit{Sh}}\nolimits (pt)$. A useful example is the following. Suppose that $S$ is a set (of sets) which contains at least one nonempty element. Let $\mathcal{S}$ be the category whose objects are elements of $S$ and whose morphisms are arbitrary set maps. Assume that $\mathcal{S}$ has fibre products. For example this will be the case if $S = \mathcal{P}(\text{infinite set})$ is the power set of any infinite set (exercise in set theory). Make $\mathcal{S}$ into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section 3.11). We claim that $\mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ is equivalent to the category of sets.

We first prove this in case $S$ contains $e \in S$ which is a singleton. In this case, there is an equivalence of topoi $i : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{S})$ given by the functors

7.15.3.1
$$\label{sites-equation-sheaves-pt-sets} i^{-1}\mathcal{F} = \mathcal{F}(e), \quad i_*E = (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E))$$

Namely, suppose that $\mathcal{F}$ is a sheaf on $\mathcal{S}$. For any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) = S$ we can find a covering $\{ \varphi _ u : e \to U\} _{u \in U}$, where $\varphi _ u$ maps the unique element of $e$ to $u \in U$. The sheaf condition implies in this case that $\mathcal{F}(U) = \prod _{u \in U} \mathcal{F}(e)$. In other words $\mathcal{F}(U) = \mathop{Mor}\nolimits _{\textit{Sets}}(U, \mathcal{F}(e))$. Moreover, this rule is compatible with restriction mappings. Hence the functor

$i_* : \textit{Sets} = \mathop{\mathit{Sh}}\nolimits (pt) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{S}), \quad E \longmapsto (U \mapsto \mathop{Mor}\nolimits _{\textit{Sets}}(U, E))$

is an equivalence of categories, and its inverse is the functor $i^{-1}$ given above.

If $\mathcal{S}$ does not contain a singleton, then the functor $i_*$ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty $\tilde e \in S$ and a map $\varphi : \tilde e \to \tilde e$ whose image is a singleton. For any sheaf $\mathcal{F}$ set

$\mathcal{F}(e) := \mathop{\mathrm{Im}}( \mathcal{F}(\varphi ) : \mathcal{F}(\tilde e) \longrightarrow \mathcal{F}(\tilde e) )$

and show that this is a quasi-inverse to $i_*$. Details omitted.

Remark 7.15.4. (Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Although we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose $\mathcal{C}$ and $\mathcal{D}$ are given sites. Consider a functor $\Phi : \mathcal{C} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$. Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on $\Phi$.

1. Is it true, given a sheaf $\mathcal{F}$ on $\mathcal{D}$, that the rule $U \mapsto \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\Phi (U), \mathcal{F})$ defines a sheaf on $\mathcal{C}$? If so, this defines a functor $\Phi _* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

2. Is it true that $\Phi _*$ has a left adjoint? If so, write $\Phi ^{-1}$ for this left adjoint.

3. Is it true that $\Phi ^{-1}$ is exact?

If the last question still has the answer “yes”, then we obtain a morphism of topoi $(\Phi _*, \Phi ^{-1})$. Moreover, given any morphism of topoi $(f_*, f^{-1})$ we may set $\Phi (U) = f^{-1}(h_ U^\# )$ and obtain a functor $\Phi$ as above with $f_* \cong \Phi _*$ and $f^{-1} \cong \Phi ^{-1}$ (compatible with adjoint property). The upshot is that by working with the collection of $\Phi$ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of $\Phi$ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of conditions (2) and (3) above; we do this in the case of points of topoi in Section 7.32.

Remark 7.15.5. (Skip on first reading.) Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A quasi-morphism of sites $f : \mathcal{D} \to \mathcal{C}$ (see Remark 7.14.9) gives rise to a morphism of topoi $f$ from $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ to $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ exactly as in Lemma 7.15.2.

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