Grothendieck this, Grothendieck that

I wanted to title this post with something more directly relevant to its topic, but so many things here are named after Grothendieck that I couldn’t help it. I swear, the only reason more people haven’t heard of him before is because he’s best known for maths that takes years to get to, but this man was an incredible mathematician (maybe check this out).

The plan today is to dissect sheaves a bit more. Precisely, I mean we’re going to be slicing (and dicing) categories of sheaves. Recall that sheaves are defined relative to some site $(\mathcal C,J)$ , which is a category $\mathcal{C}$ equipped with a Grothendieck topology $J$ . Any category equivalent to the category $\mathbf{Sh}(\mathcal C,J)$ of sheaves over some site is called a Grothendieck topos. So, the goal is to show that slices of Grothendieck topoi are again Grothendieck topoi. The key construction here is the aptly-named Grothendieck construction (well, actually we only need it in a special case where it is also called the category of elements, but no matter). This is an exercise in Mac Lane and Moerdijk’s book, so it should be easy, right?

Fix a Grothendieck topos $\mathbf{Sh}(\mathcal C,J)$ , and let $X$ be a sheaf over $J$ . The slice $(\mathbf{Sh}(\mathcal C,J)\downarrow X)$ is the category of sheaves that lie over $X$ , and we want to realise this as a category of sheaves also. The underlying category of our desired site $(\mathcal C',J')$ should then satisfy that presheaves $(\mathcal C')^{\mathrm{op}}\to\mathbf{Set}$ correspond to natural transformations to $X$ . Let’s try to deduce $\mathcal C'$ from this characterisation.

First, recall by the Yoneda Lemma that for $U\in\mathcal C$ , natural transformations $h_U\Rightarrow X$ correspond to elements of $X(U)$ . This means every element $x\in X(U)$ has to correspond to some presheaf $(\mathcal C')^{\mathrm{op}}\to\mathbf{Set}$ . As elements $x\in X(U)$ are probably the “simplest” examples of natural transformations to $X$ , they should probably correspond to the “simplest” presheaves; that is, the representable ones. Therefore, it’s probably not unreasonable to guess that the objects of $\mathcal C'$ are given by the (generalised) elements of $X$ . As for the morphisms of $\mathcal C'$ , the Yoneda Lemma also gives us that these should be precisely the natural transformations between the corresponding representable presheaves. This means they correspond to natural transformations $h_U\Rightarrow h_V$ that lie over $X$ , and are thus (by Yoneda again) the morphisms $U\to V$ which send the chosen element of $X(V)$ to the chosen element of $X(U)$ . This is exactly the Grothendieck construction $\int X$ (and probably explains why it’s called the “category of elements” of $X$ in this case).

Definition 1. Given a (pseudo)functor $X:\mathcal{C}^{\mathrm{op}}\to\mathbf{Cat}$ from an ordinary category $\mathcal{C}$ to the 2-category of (small) categories, the Grothendieck construction produces the category $\int X$ , where:

the objects of $\int X$ are pairs $(U, x)$ where $U\in\mathcal C_0$ and $x\in X(U)_0$
the morphisms $(f,\alpha):(U,x)\to(V,y)$ are given by morphisms $f:U\to V$ in $\mathcal{C}$ and $\alpha: x \to X(f)(y)$ in $X(V)$ .

In particular, if $X:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$ (where we view sets as discrete categories), then the objects of $\int X$ are pairs $(U, x)$ with $x\in X(U)$ , and the morphisms $f:(U,x)\to(V,y)$ are those $f:U\to V$ for which $X(f)(y)=x$ .

This turns out to be exactly what we need. We get a canonical functor $\beta:\mathbf{PSh}(\int X)\to (\mathbf{PSh}(\mathcal C)\downarrow X)$ by sending a presheaf $A$ to the presheaf $\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}, U\mapsto\coprod_{x\in X(U)}A(U, x)$ , which has a natural transformation $\beta A\Rightarrow X$ induced by sending $A(U, x)\to\{x\}\subseteq X(U)$ . In particular, note that the representable presheaf $h_{V,y}$ has $h_{V,y}(U, x) = \{f:U\to V \mid X(f)(y)=x\}$ , and so $\coprod_{x\in X(U)}h_{V,y}(U,x) \cong h_V(U)$ (because $X(f)(y)$ has to be some unique $x\in X$ ). The induced projection $h_V\Rightarrow X$ sends a morphism $f:U\to V$ to the element $X(f)(y)\in X(U)$ , which (by the proof of the Yoneda Lemma) corresponds to the element $y\in X(V)$ . Therefore, $\beta$ sends $h_{V,y}\mapsto (y\in X(V))$ , as expected.

Intuitively, this functor $\beta:\mathbf{PSh}(\int X)\to (\mathbf{PSh}(\mathcal C)\downarrow X)$ bundles up the action of $A:(\int X)^{\mathrm{op}}\to\mathbf{Set}$ on all of the elements of $X$ according to its shape $U$ . This suggests a natural inverse: if we view a presheaf $Y/X$ (i.e., $Y$ lying over $X$ ) as a “bundle” over each element of $X$ , then we should be able to extract each fibre to produce a presheaf over $\int X$ . Indeed, this intuition defines the functor $\phi:(\mathbf{PSh}(\mathcal C)\downarrow X)\to\mathbf{PSh}(\int X)$ that sends a presheaf $\pi:Y\Rightarrow X$ over $X$ to the presheaf $(\int X)^{\mathrm{op}}\to\mathbf{Set}, (U, x)\mapsto \pi_U^{-1}\{x\}$ . The functors $\phi$ and $\beta$ define an equivalence of categories, showing that the slices of a presheaf topos is again a presheaf topos.

Now we need only specialise this equivalence to sheaves if $\mathcal C$ carries a topology $J$ . Recall that if $X$ is a sheaf, then covering sieves $S\subseteq h_V$ induce bijections $X(V)\cong \mathop{\mathrm{Nat}}(S,X)$ . We want a topology on $\int X$ such that $A:(\int X)^{\mathrm{op}}\to\mathbf{Set}$ is a sheaf iff $\beta A$ produces a sheaf $\mathcal C^{\mathrm{op}}\to\mathbf{Set}$ over $X$ . This means for every covering sieve $S\subseteq h_V$ that natural transformations $S\Rightarrow \beta A$ correspond to elements of $\beta A(V) = \coprod_{y\in X(V)}A(V, y)$ . This suggests that natural transformations $S\Rightarrow \beta A$ should admit a canonical decomposition based on the elements of $X(V)$ .

This is why we need $X$ to be a sheaf. If $S\subseteq h_V$ is a covering sieve, then any presheaf $Y/X$ induces for every natural transformation $S\Rightarrow Y$ a natural transformation $S\Rightarrow X$ which corresponds to a unique $y\in X(V)$ . This makes $S$ lie over $X$ , so write $S_y := \phi(S/X)$ for its corresponding presheaf over $\int X$ . Then, we have just shown that every natural transformation $S\Rightarrow Y$ corresponds to a unique natural transformation $S_y\Rightarrow \phi(Y/X)$ for some uniquely determined $y\in X(V)$ . This defines a canonical bijection $\mathop{\mathrm{Nat}}(S,Y) \cong \coprod_{y\in X(V)}\mathop{\mathrm{Nat}}(S_y,\phi(Y/X))$ . Therefore, define the Grothendieck topology on $\int X$ by defining the covering sieves of $h_{V,y}$ to be precisely these $S_y$ for $S\subseteq h_V$ a covering sieve in $\mathcal{C}$ , then $Y/X$ is a sheaf on $\mathcal{C}$ if and only if $\phi(Y/X)$ is a sheaf on $\int X$ .

Explicitly, $S_y(U, x)$ is the subset of $S(U)$ on those $f:U\to V$ such that $X(f)(y)=x$ , so this choice of covering makes intuitive sense as well. Indeed, we are saying that a sieve covers $(U, x)$ if it is the fibre at $x$ of a sieve in $\mathcal{C}$ that covers $U$ . To summarise, we have proven the following:

Proposition 2. Let $(\mathcal C, J)$ be a site, and $X:\mathcal{C}^{\mathrm{op}}\to\mathbf{Set}$ a sheaf. Then, the fibres of covering sieves in $J$ defines a Grothendieck topology $J'$ on $\int X$ such that we have an equivalence of categories $(\mathbf{Sh}(\mathcal C,J)\downarrow X) \simeq \mathbf{Sh}(\int X, J')$ . In particular, the slice of a Grothendieck topos is another Grothendieck topos.