## 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 $\mathcal{C}$.

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

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

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

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.
$\square$

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