### Sheaves Do Not Belong to Algebraic Geometry

#### Posted by Tom Leinster

…and here’s a proof.

They are, of course, very *useful* in algebraic geometry (as is the
equals sign). Also, human beings discovered them while developing algebraic
geometry, which is why many of them still make the association.

But as we’ll see, sheaves are an inevitable consequence of general ideas that have nothing to do with algebraic geometry. In fact, sheaves (and various related notions) arise automatically from two completely general categorical constructions, together with one almost imperceptibly small topological observation.

Before I give you the proof, let me make clear that it isn’t due
to me. I don’t know who it *is* due to — I’ve never
seen it in print — but I suspect it was known before I
was even born. (*Update*: see Joachim Kock’s comment for a reference.) People who I’ve told this argument to seem to like it,
so I wrote it up in a little note
a few years ago; then a recent conversation reminded me of it, so I
thought I’d air it here.

**First categorical construction** Let $\mathbf{A}$ be a
small category, $\mathbf{E}$ a category with small colimits, and
$J: \mathbf{A} \to \mathbf{E}$ any functor. Then there is an induced
adjunction
$\mathbf{Set}^{\mathbf{A}^{op}}
\begin{aligned}
\stackrel{\displaystyle\stackrel{\displaystyle - \otimes J}{\longrightarrow}}{\stackrel{\longleftarrow}{Hom(J, -)}}
\end{aligned}
\mathbf{E}.$
The right adjoint $Hom(J, -)$ is defined by
$(Hom(J, E))(A)
=
Hom (J(A), E)$
($E \in \mathbf{E}$, $A \in \mathbf{A}$). The left adjoint $- \otimes J$ is defined by the adjointness, and can be described as a certain coend or colimit.

Example: if $J: \Delta \to \mathbf{Top}$ is the standard simplex functor then $Hom(J, -)$ is the singular simplicial set functor and $- \otimes J$ is geometric realization.

**Second categorical construction** Any adjunction restricts
canonically to an equivalence between full subcategories.

Precisely, let $\mathbf{C} \begin{aligned} \stackrel{\displaystyle \stackrel{F}{\displaystyle\longrightarrow}}{ \stackrel{\longleftarrow}{G}} \end{aligned} \mathbf{D}$ be an adjunction ($F$ left adjoint to $G$), with unit $\eta: 1 \to G F$ and counit $\varepsilon: F G \to 1$. Let $\bar{\mathbf{C}}$ be the full subcategory of $\mathbf{C}$ consisting of those objects $C$ for which $\eta_C: C \to G F(C)$ is an isomorphism, and dually $\bar{\mathbf{D}}$. Then the adjunction $(F, G, \eta, \varepsilon)$ restricts to an equivalence between $\bar{\mathbf{C}}$ and $\bar{\mathbf{D}}$.

**Almost imperceptibly small topological observation** Any
open subset of a topological space can be regarded as a space in
its own right, and when one open set is contained in another, there is
an induced inclusion of spaces.

Precisely, let $S$ be a topological space. Write $\mathbf{O}(S)$ for the poset of open subsets of $S$, regarded as a category (in which each hom-set has at most one element). Write $\mathbf{Top}/S$ for the category of spaces over $S$: objects are continuous maps into $S$, and maps are commutative triangles. Then there is a canonical functor $J: \mathbf{O}(S) \to \mathbf{Top}/S,$ sending an open set $U$ to the inclusion $U \hookrightarrow S$.

**Punchline** Fix a topological space $S$. The category
$\mathbf{Top}/S$ has small colimits, since $\mathbf{Top}$ does.

Applying the first categorical construction to the functor $J$ just defined produces an adjunction $(presheaves on S) = \mathbf{Set}^{\mathbf{O}(S)^{op}} \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Top}/S = (spaces over S).$ The two functors here are the ones you’d guess.

Applying the second construction now gives an equivalence of categories $(sheaves on S) = \mathbf{Sh}(S) \begin{aligned} \stackrel{\displaystyle\longrightarrow}{\longleftarrow} \end{aligned} \mathbf{Et}(S) = (étale spaces over S).$ This can be interpreted as the definition of sheaf, étale space, etc., or as a theorem, according to taste.

Going right and then left in the adjunction gives the associated sheaf, or sheafification, of a presheaf. Going left and then right gives the ‘étalification’ of a space over $S$.

## Re: Sheaves Do Not Belong to Algebraic Geometry

Beautiful!

Small grammar note: ‘étale map’, but ‘étalé space’.

It is probably a sign of something that I never thought that sheaves inherently belonged to algebraic geometry. (Unlike schemes.)