Lawvere–Tierney topologies (topologies 6/6)

Posted on 12 April 2022 by Remy

In the post of two weeks ago, we showed how Grothendieck topologies form a neat framework for the categorical aspects of the more familiar (at least to algebraic geometers) Grothendieck pretopologies. In this final post of the series, we take this one step further, to the notion of a Lawevere–Tierney topology on an arbitrary elementary topos $\mathscr E$ .

Definition. An elementary topos is a category $\mathscr E$ that has finite limits, is Cartesian closed, and has a subobject classifier (see last week’s post).

The only example we’re interested in today is the category $\mathscr E = [\mathscr C^{\operatorname{op}},\mathbf{Set}]$ of presheafs (of sets) on a small category $\mathscr C$ , which we saw last week has subobject classifiers. But in fact, any Grothendieck topos (sheaves of sets on a (small) site) is an example of an elementary topos, so one could even introduce further topologies on those.

Example. The category $\mathbf{Set}_{\text{fin}}$ of finite sets is an elementary topos, but not a Grothendieck topos since it is neither complete nor cocomplete.

Before giving the definition, we need to define one more structure on $\Omega$ : the meet. Recall that the intersection (or meet) of two monomorphisms $U \hookrightarrow X \hookleftarrow V$ is the fibre product

$U \wedge V := U \underset X\times V \hookrightarrow X.$

The intersection of $\mathbf 1 \times \Omega \hookrightarrow \Omega \times \Omega$ and $\Omega \times \mathbf 1 \hookrightarrow \Omega \times \Omega$ is the monomorphism $\mathbf 1 \hookrightarrow \Omega \times \Omega$ given by $(t,t)$ , which is classified by a map $\wedge \colon \Omega \times \Omega \to \Omega$ . Since $\mathbf 1 \to \Omega$ is the universal monomorphism, we see that $\mathbf 1 \hookrightarrow \Omega \times \Omega$ is the universal intersection of two subobjects, i.e. if $U \to X$ and $V \to X$ are classified by $f \colon X \to \Omega$ and $g \colon X \to \Omega$ respectively, then $U \wedge V$ is classified by the composition

$X \overset{(f,g)}\longrightarrow \Omega \times \Omega \overset\wedge\longrightarrow \Omega.$

(If we denote this simply by $f \wedge g \colon X \to \Omega$ , then $\wedge \colon \Omega \times \Omega \to \Omega$ is $\operatorname{pr}_1 \wedge \operatorname{pr}_2$ .)

Definition. Let $\mathscr E$ be an elementary topos with subobject classifier $t \colon \mathbf 1 \to \Omega$ . Then a Lawvere–Tierney topology on $\mathscr E$ is a morphism $j \colon \Omega \to \Omega$ such that the following diagrams commute:

$\begin{array}{ccc}\mathbf 1\!\! & \stackrel t\to\!\! & \!\!\!\Omega \\ & \!\!\underset{t\!\!\!}{}\searrow\!\! & \downarrow j\!\! \\[-.2em] & & \!\!\!\Omega,\!\!\end{array}\qquad\qquad\begin{array}{ccc}\Omega\!\! & \stackrel j\to\!\! & \!\!\!\Omega \\ & \!\!\underset{j\!\!\!}{}\searrow\!\! & \downarrow j\!\! \\[-.3em] & & \!\!\!\Omega,\!\!\end{array}\qquad\qquad\quad\begin{array}{ccc}\Omega \times \Omega\!\!\! & \stackrel\wedge\to\!\! & \!\!\!\Omega \\ \!\!\!\!\!\!\!\!\!\!\!\!j\times j \downarrow & & \downarrow j\!\! \\ \Omega \times \Omega\!\!\! &\stackrel\wedge\to\!\! & \!\!\!\Omega.\!\!\end{array}$

We saw two weeks ago that a Grothendieck topology is a certain subpresheaf $J \subseteq \mathbf{Siv}$ , and last week that $\mathbf{Siv}$ is a subobject classifier $\Omega$ on $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ . Thus a subpresheaf $J \subseteq \Omega$ is classified by a morphism $j \colon \Omega \to \Omega$ , which we saw last week is given by $S \mapsto (S \in J)$ .

Lemma. The subpresheaf $J \subseteq \Omega$ is a Grothendieck topology on $\mathscr C$ if and only if $j \colon \Omega \to \Omega$ is a Lawvere–Tierney topology on $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ . In particular, Grothendieck topologies on $\mathscr C$ are in bijective correspondence with Lawvere–Tierney topologies on $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ .

Thus Lawvere–Tierney topologies are an internalisation of the notion of Grothendieck topology to an arbitrary elementary topos $\mathscr E$ .

Proof of Lemma. By definition of the morphism $j$ , we have a pullback square

$\begin{array}{ccc}J & \to & \mathbf 1 \\ \downarrow & & \downarrow \\ \Omega & \stackrel j\to & \Omega.\!\end{array}$

The first commutative diagram in the definition above means that the top arrow has a section $\mathbf 1 \to J$ such that the composition $\mathbf 1 \to J \hookrightarrow \Omega$ is $t$ , i.e. $\mathbf 1 \subseteq J$ as subobjects of $\Omega$ . Since $t \colon \mathbf 1 \to \Omega$ is the map taking $1 \in \mathbf 1(X)$ to the maximal sieve $h_X \subseteq h_X$ for any $X \in \mathscr C$ , this means exactly that $h_X \in J(X)$ for all $X \in \mathscr C$ , which is condition 1 of a Grothendieck topology. For the second, consider the pullback

$\begin{array}{ccccc}J' & \to & J & \to & \mathbf 1 \\ \downarrow & & \downarrow & & \downarrow \\ \Omega & \stackrel j\to & \Omega & \stackrel j\to & \Omega.\!\end{array}$

The condition $jj=j$ means that $J' \cong J$ as subobjects of $\Omega$ . We already saw that $\mathbf 1 \subseteq J$ for a Grothendieck or Lawvere–Tierney topology, so pulling back along $j$ gives $J \subseteq J'$ . Thus the second diagram in the definition of a Lawvere–Tierney topology commutes if and only if $J' \subseteq J$ , i.e. if $S \in \Omega(X) = \mathbf{Siv}(X)$ with $j_X(S) \in J(X)$ , then $S \in J(X)$ . But $j_X \colon \Omega(X) \to \Omega(X)$ is given by $S \mapsto (S \in J)$ , so this is exactly axiom 3 of a Grothendieck topology.

For the third diagram, we first claim that $j_X \colon \Omega(X) \to \Omega(X)$ is monotone for all $X \in \mathscr C$ if and only if $J$ satisfies axiom 2 of a Grothendieck topology. Indeed, if $j$ is monotone and $S, S' \in \Omega(X)$ satisfy $S \subseteq S'$ and $S \in J(X)$ , then the inclusion $h_X = (S \in J) \subseteq (S' \in J)$ shows $(S' \in J) = h_X$ , so $S' \in J(X)$ by axiom 3. Conversely, if $J$ satisfies axiom 2 and $S,S' \in \Omega(X)$ satisfy $S \subseteq S'$ , then for any $f \colon Y \to X$ we have $f^*S \subseteq f^*S'$ , so $f^*S \in J(Y) \Rightarrow f^*S' \in J(Y)$ , i.e. $(S \in J) \subseteq (S' \in J)$ .

The third diagram in the definition above says that the map $j_X \colon \Omega(X) \to \Omega(X)$ given by $S \mapsto (S \in J)$ is a morphism of meet semilattices. This implies in particular that $j_X$ is monotone, as $S \subseteq S'$ if and only if $S \wedge S' = S$ , so the third diagram above implies axiom 2 of a Grothendieck topology.

Conversely, if $J$ is a Grothendieck topology, then axiom 2 implies that $j_X \colon \Omega \to \Omega$ is monotone. In particular, $j_X(S \cap T) \subseteq j_X(S) \cap j_X(T)$ for any $S, T \in \Omega(X)$ , since $S \cap T \subseteq S, T$ . For the reverse implication, if $f \colon Y \to X$ satisfies $f \in (S \in J)(Y) \cap (T \in J)(Y)$ , then $f^*S \in J(Y)$ and $f^*T \in J(Y)$ , so the remark of two weeks ago shows that $f^*(S \cap T) \in J(Y)$ , i.e. $f \in ((S \cap T) \in J)(Y)$ . We see that $j_X(S \cap T) = j_X(S) \cap j_X(T)$ , showing that $j_X$ is a morphism of meet semilattices. $\qedsymbol$

Subobject classifiers on presheaf categories (topologies 5/6)

Posted on 3 April 2022 by Remy

In the first post of this series, we saw how subobjects of representable presheaves $h_U \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ correspond to sieves on $U$ . Last week, we saw how sieves give a convenient language for defining Grothendieck topologies on a small category. In next week’s (hopefully) final instalment of this series, we will generalise this even further to Lawvere–Tierney topologies on an arbitrary topos. Today’s post defines the last object we need to do this, which we will show generalises the presheaf $\mathbf{Siv}$ from last week:

Definition. Let $\mathscr C$ be a (possibly large) category with a terminal object $\mathbf{1}$ . Then a subobject classifier on $\mathscr C$ is a monomorphism $\mathbf{1} \to \Omega$ in $\mathscr C$ such that for every monomorphism $U \to X$ in $\mathscr C$ , there exists a unique arrow $X \to \Omega$ such that there is a pullback diagram

$\begin{array}{ccc}U & \to & \mathbf{1} \\ \downarrow & & \downarrow \\ X & \to & \Omega.\!\end{array}$

That is, $\mathbf{1} \to \Omega$ is the “universal” monomorphism in $\mathscr C$ , i.e. the pair $(\Omega,\mathbf{1} \hookrightarrow \Omega)$ represents the (possibly large) presheaf $X \mapsto \{\text{monomorphisms } U \hookrightarrow X\}/\cong$ , where $\cong$ denotes isomorphism in the slice category $\mathscr C/X$ . It is an easy exercise to show that any representative $(\Omega, T \hookrightarrow \Omega)$ of this presheaf actually has the form described above, i.e. $T$ is a terminal object (apply the uniqueness property above to the identity monomorphism $X \hookrightarrow X$ , and use the pullback square

$\begin{array}{ccc}T & = & T \\ \shortparallel & & \downarrow \\ T & \to & \Omega\end{array}$

coming from the hypothesis that $T \hookrightarrow \Omega$ is a monomorphism).

Example. If $\mathscr C = \mathbf{Set}$ , then the two-point set $\Omega = \{0,1\}$ with its natural inclusion $\mathbf 1 \to \Omega$ given by $1 \mapsto 1$ is a subobject classifier: the monomorphism $U \subseteq X$ corresponds to the indicator function $\delta_U \colon X \to \Omega$ that is $1$ on $U$ and $0$ on its complement. (In other situations I would denote this by $\mathbf 1_U$ , but that notation was already used in this series to denote the representable presheaf $h_U$ .)

It’s even more natural to take $\Omega$ to be the power set $\{\varnothing,\mathbf 1\}$ of $\mathbf{1}$ . As in the first post of this series, we think of $\mathbf 1$ representing “true” and $\varnothing$ representing “false”. The generalisation of the power set of $\mathbf{1}$ to presheaf categories is the presheaf $\mathbf{Siv}$ of subpresheaves of $h_X$ defined last week:

Lemma. Let $\mathscr C$ be a small category. Then the presheaf $\mathbf{Siv}$ together with the map $\mathbf 1 \to \mathbf{Siv}$ taking the unique section $1 \in \mathbf 1(X)$ to the maximal sieve $h_X \subseteq h_X$ for any $X \in \mathscr C$ is a subobject classifier in $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ .

Proof. Note that the prescribed map $\mathbf 1 \to \mathbf{Siv}$ is a morphism of presheaves, since the inverse image of the maximal sieve $h_X$ under any morphism $f \colon Y \to X$ in $\mathscr C$ is the maximal sieve $h_Y$ . Again using the notation from last week, if $\mathscr G \hookrightarrow \mathscr F$ is any monomorphism of presheaves, we get a morphism of presheaves $\phi \colon \mathscr F \to \Omega$ defined on $X \in \mathscr C$ by

$\begin{align*}\mathscr F(X) &\to \mathbf{Siv}(X) \\s &\mapsto (s \in \mathscr G).\end{align*}$

If $f \colon Y \to X$ is a morphism in $\mathscr C$ , then for any $Z \in \mathscr C$ we have

$\begin{align*}\big(f^*(s \in \mathscr G)\big)(Z) &= \{g \colon Z \to Y\ |\ fg \in (s \in \mathscr G)(Z)\} \\&= \{g \colon Z \to Y\ |\ (fg)^*(s) \in \mathscr G(Z)\} \\&= \{g \colon Z \to Y\ |\ g^*(f^*(s)) \in \mathscr G(Z)\} = (f^*s \in \mathscr G)(Z),\end{align*}$

showing that $f^*\phi(s) = \phi(f^*s)$ , so $\phi$ is indeed a natural transformation. We already noted last week that $(s \in \mathscr G) = h_X$ for $s \in \mathscr F(X)$ if and only if $s \in \mathscr G(X)$ , so $\mathscr G$ is the pullback

$\begin{array}{ccc}\mathscr G & \to & \mathbf{1} \\ \downarrow & & \downarrow \\ \mathscr F & \to & \mathbf{Siv}.\!\end{array}$

Conversely, if $\psi \colon \mathscr F \to \mathbf{Siv}$ is any morphism with this property and $s \in \mathscr F(X)$ , then $s \in \mathscr G(X)$ if and only if $\psi(s) = h_X$ , which together with naturality of $\psi$ gives

$\begin{align*}(s \in \mathscr G)(Y) &= \{f \colon Y \to X\ |\ f^*s \in \mathscr G(Y)\} \\&= \{f \colon Y \to X\ |\ \psi(f^*s) = h_Y\} \\&= \{f \colon Y \to X\ |\ f^*\psi(s) = h_Y\} \\&= \left\{f \colon Y \to X\ |\ \operatorname{id}_Y \in \big(f^*\psi(s)\big)(Y)\right\} \\&= \{f \colon Y \to X\ |\ f \circ \operatorname{id}_Y \in (\psi(s))(Y)\} = (\psi(s))(Y),\end{align*}$

so $\psi(s) = (s \in \mathscr G)$ . $\qedsymbol$

We will discuss some other properties of subobject classifiers in future posts.

Grothendieck topologies (topologies 4/6)

Posted on 28 March 2022 by Remy

This post is the first goal in a series on sieves (subobjects of representable presheaves); I will give another generalisation in the next two posts. In the first post of the series, I defined sieves and gave basic examples, and last week I showed how the sheaf condition on a site can be stated in terms of sieves:

Corollary. Let $\mathscr C$ be a (small) site. For a set of morphisms $\mathscr U = \{U_i \to U\}_{i \in I}$ with the same target, write $S_{\mathscr U} \subseteq h_U$ for the presheaf image of $\coprod_{i\in I} h_{U_i} \to h_U$ . Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if and only if for every covering $\mathscr U = \{U_i \to U\}_{i \in I}$ in $\mathscr C$ , the inclusion $S_{\mathscr U} \hookrightarrow h_U$ induces an isomorphism

$\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).$

Thus, if $\mathscr C$ is a site (a small category with a Grothendieck pretopology), we should be able to obtain the category $\mathbf{Sh}(\mathscr C) \subseteq \mathbf{PSh}(\mathscr C)$ of sheaves purely in terms of sieves. This is the notion of a Grothendieck topology that we describe at the end of this post.

Before giving the definition, note that any morphism $f \colon Y \to X$ in $\mathscr C$ gives a pullback $\mathbf{Siv}(X) \to \mathbf{Siv}(Y)$ taking $S \subseteq h_X$ to its inverse image under $h_f \colon h_Y \to h_X$ (I avoid the word ‘pullback’ here to make sure this is truly a subpresheaf and not a presheaf with a monomorphism to $h_Y$ defined uniquely up to unique isomorphism). Thus, $\mathbf{Siv}$ is itself a presheaf $\mathscr C^{\operatorname{op}} \to \mathbf{Set}$ (it takes values in $\mathbf{Set}$ since $\mathscr C$ is small).

Also note the following method for producing sieves: if $\mathscr F$ is a presheaf, $\mathscr G \subseteq \mathscr F$ a subpresheaf, and $s \in \mathscr F(X)$ a section over some $X \in \mathscr C$ , we get a sieve $(s \in \mathscr G) \in \mathbf{Siv}(X)$ by

$(s \in \mathscr G)(Y) = \left\{f \colon Y \to X\ \big|\ f^*(s) \in \mathscr G(Y)\right\}.$

By the Yoneda lemma, this is just the inverse image of $\mathscr G \subseteq \mathscr F$ along the morphism $h_X \to \mathscr F$ classifying $s$ . Note that $(s \in \mathscr G)$ is the maximal sieve $h_X$ if and only if $s \in \mathscr G(X)$ .

Definition. Let $\mathscr C$ be a small category. Then a Grothendieck topology on $\mathscr C$ consists of a subpresheaf $J \subseteq \mathbf{Siv}$ such that

For all $X \in \mathscr C$ , the maximal sieve $h_X \subseteq h_X$ is in $J(X)$ .
If $S \in J(X)$ and $S' \in \mathbf{Siv}(X)$ with $S \subseteq S'$ , then $S' \in J(X)$ .
If $S \in \mathbf{Siv}(X)$ is a sieve such that $(S \in J) \in J(X)$ , then $S \in J(X)$ (equivalently, then $(S \in J)$ is the maximal sieve $h_X$ ).

The sieves $S \in J(X)$ are called covering sieves. Since $J$ is a presheaf, we see that for any $f \colon Y \to X$ and any covering sieve $S \subseteq h_X$ , the pullback $f^*S \subseteq h_Y$ is covering. Condition 2 says that any sieve containing a covering sieve is covering. In the presence of condition 1, conditions 2 and 3 together are equivalent to the local character found in SGA IV $_1$ , Exp. II, Def. 1.1:

If $S, S' \in \mathbf{Siv}(X)$ with $S \in J(X)$ , such that for every morphism $h_Y \to S$ the inverse image of $S' \subseteq h_X$ along $h_Y \to S \to h_X$ is in $J(Y)$ , then $S' \in J(X)$ .

Indeed, applying this criterion when $S \subseteq S'$ immedately shows $S' \in J(X)$ if $S \in J(X)$ , since the inverse image of $S' \subseteq h_X$ along $h_Y \to S \to h_X$ is the maximal sieve $h_Y$ . Thus the local character implies criterion 2. The local character says that if $(S' \in J)$ contains a covering sieve $S$ , then $S'$ is covering. Assuming criterion 2, the sieve $(S' \in J)$ contains a covering sieve if and only if $(S' \in J)$ is itself covering, so the local character is equivalent to criterion 3.

Remark. One property that follows from the axioms is that $J(X)$ is closed under binary intersection, i.e. if $S, T \in J(X)$ then $(S \cap T) \in J(X)$ . Indeed, if $f \in S(Y)$ for some $f \colon Y \to X$ , then

$f^*(S \cap T) = f^*S \cap f^*T = h_Y \cap f^*T = f^*T \in J(Y),$

so $S \subseteq ((S \cap T) \in J)$ . Axioms 2 and 3 give $(S \cap T) \in J(X)$ .

Example. Let $\mathcal Cov(\mathscr C)$ be a pretopology on the (small) category $\mathscr C$ ; see Tag 00VH for a list of axioms. For each $X \in \mathscr C$ , define the subset $J(X) \subseteq \mathbf{Siv}(X)$ as those $S \subseteq h_X$ that contain a sieve of the form $S_{\mathscr U}$ for some covering $\mathscr U = \{U_i \to X\}$ in $\mathcal Cov(\mathscr C)$ . (See the corollary at the top for the definition of $S_{\mathscr U}$ .) Concretely, this means that there exists a covering $\{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ such that $f_i \in S(U_i)$ for all $i \in I$ , i.e. $X$ is covered by morphisms $f_i \colon U_i \to X$ that are in the given sieve $S$ .

Lemma. The association $X \mapsto J(X)$ is a topology. It is the coarsest topology on $\mathscr C$ for which each $S_{\mathscr U}$ for $\mathscr U \in \mathcal Cov(\mathscr C)$ is a covering sieve.

Proof. We will use the criteria of Tag 00VH. If $S \in J(X)$ , then there exists $\mathscr U = \{U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ with $S_{\mathscr U} \subseteq S$ . If $f \colon Y \to X$ is any morphism in $\mathscr C$ , then $f^*\mathscr U = \{U_i \times_X Y \to Y\}_{i \in I} \in \mathcal Cov(\mathscr C)$ by criterion 3 of Tag 00VH. But $S_{f^*\mathscr U} = f^*S_{\mathscr U}$ , because a morphism $g \colon U \to Y$ factors through $U_i \times_X Y$ if and only if $fg \colon U \to X$ factors through $U_i$ . Thus, $S_{f^*\mathscr U} = f^*S_{\mathscr U} \subseteq f^*S$ , so $f^*S \in J(Y)$ , and $J$ is a subpresheaf of $\mathbf{Siv}$ .

Condition 1 follows immediately from criterion 1 in Tag 00VH, and condition 2 is satisfied by definition. For condition 3, suppose $S \in \mathbf{Siv}(X)$ satisfies $(S \in J) \in J(X)$ . Then there exists $\mathscr U = \{f_i \colon U_i \to X\}_{i \in I} \in \mathcal Cov(\mathscr C)$ with $S_{\mathscr U} \subseteq (S \in J)$ . This means that $f_i \in (S \in J)(U_i)$ for all $i$ , i.e. $f_i^*S \in J(U_i)$ for all $i$ . Thus, for each $i \in I$ there exists $\mathscr V_i = \{g_{ij} \colon V_{ij} \to U_i\}_{j \in J_i}$ in $\mathcal Cov(\mathscr C)$ such that $S_{\mathscr V_i} \subseteq f_i^*S$ , i.e. $f_ig_{ij} \in S(X)$ for all $i \in I$ and all $j \in J_i$ . Thus, if $\mathscr V$ denotes $\{f_ig_{ij} \colon V_{ij} \to X\}_{i \in I, j \in J_i}$ , then we get $S_{\mathscr V} \subseteq S$ . But $\mathscr V$ is a covering by criterion 2 of Tag 00VH, so $S \in J(X)$ .

If $J'$ is any other Grothendieck topology for which each $S_{\mathscr U}$ for $\mathscr U \in \mathcal Cov(\mathscr C)$ is covering, then $J'$ contains $J$ by criterion 2. $\qedsymbol$

To state the obvious (hopefully), the notion of sheaf can therefore be defined on a Grothendieck topology in a way that coincides with the usual notion for a Grothendieck pretopology:

Definition. Let $\mathscr C$ be a small category, and let $J \subseteq \mathbf{Siv}$ be a Grothendieck topology. Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if for any $X \in \mathscr C$ and any $S \in J(X)$ , the map $S \hookrightarrow h_X$ induces an isomorphism

$\operatorname{Hom}(h_X,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S,\mathscr F).$

Thus, a Grothendieck topology is an internal characterisation (inside $\mathbf{PSh}(\mathscr C)$ ) of which morphisms $S \to h_X$ one needs to localise to get $\mathbf{Sh}(\mathscr C,J)$ . In the last two posts, we will generalise this even further to a Lawvere–Tierney topology on an arbitrary topos.

Covering sieves and the sheaf condition (topologies 3/6)

Posted on 20 March 2022 by Remy

In the first post of this series, I explained how subobjects of the constant presheaf (resp. constant sheaf) $\mathbf 1_X$ on a small category (resp. small site) with terminal object $X$ correspond to left closed (resp. local) properties on $\mathscr C$ . In this post, I will explain the main examples that intervene in setting up topoi, and show how to define the sheaf condition using sieves (instead of coverings).

For simplicity, assume $\mathscr C$ is a small category with fibre products.

Definition. Given a set of morphisms $\mathscr U = \{f_i \colon U_i \to U\}_{i \in I}$ with the same target $U \in \mathscr C$ , define the sieve $S_{\mathscr U} \subseteq h_U$ generated by $\mathscr U$ as the sieve on $U$ of those morphisms $V \to U$ that factor through some $f_i \colon U_i \to U$ .

It is in a sense the right ideal in $\operatorname{Hom}(-,U)$ generated by the $f_i$ . What does this look like as a subobject of $h_U$ ?

Example. If $I$ has one element, i.e. $\mathscr U = \{V \to U\}$ , then $S_{\mathscr U}$ is the image of the morphism of representable presheaves $h_V \to h_U$ . In the case where $V \to U$ is already a monomorphism (this is always the case when $\mathscr C$ is a poset, such as $\operatorname{Open}(X)$ for some topological space $X$ ), then $h_V \to h_U$ is itself injective (this is the definition of a monomorphism!), so $S_{\mathscr U}$ is just $h_V$ .

In general, $S_{\mathscr U}$ is the image of the map

$\coprod_{i \in I} h_{U_i} \to h_U$

induced by the maps $U_i \to U$ . Indeed, an element of $h_U(V)$ is a morphism $f \colon V \to U$ , and it comes from some $h_{U_i}(V)$ if and only if $f$ factors through $f_i \colon U_i \to U$ .

This shows that, in fact, every sieve $S \subseteq h_X$ is of this form for some set $\{U_i \to U\}_{i \in I}$ : take as index set (the objects of) the slice category $(h \downarrow S)$ , which as in the previous post gives a surjection $\coprod_{(V,\alpha)} h_V \to S$ . This corresponds to generating an ideal by all its elements.

But we can also characterise $S_{\mathscr U}$ without using the word ‘image’ (which somehow computes its first syzygy):

Lemma. Let $\mathscr U = \{U_i \to U\}$ be a set of morphisms with common target, and $S_{\mathscr U}$ the sieve generated by $\mathscr U$ . Then $S_{\mathscr U}$ is the coequaliser of the diagram

$\coprod_{i,j \in I} h_{U_i \underset U\times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i},$

where the maps are induced by the two projections $I^2 \to I$ .

We will give two proofs, one using the description of coequalisers of sets, and the other using that presheaves are colimits of representable presheaves, as discussed in the previous post.

Proof 1. The diagram

$\begin{array}{ccc}\displaystyle\coprod_{i,j \in I} h_{U_i \underset U\times U_j} & \to & \displaystyle\coprod_{i \in I} h_{U_i} \\ \downarrow & & \downarrow \\ \displaystyle\coprod_{j \in I} h_{U_j} & \to & h_U \end{array}$

is a pullback, by the universal property of fibre products $U_i \times_U U_j$ and since fibre products with a fixed set/presheaf of sets commute with coproducts. Then the same goes for the square

since $S_{\mathscr U} \to h_U$ is a monomorphism. But $\coprod_{i \in I} h_{U_i} \to S_{\mathscr U}$ is an epimorphism (objectwise surjection) by definition, so this square is a pushout as well (in $\mathbf{Set}$ , epimorphisms are regular). $\qedsymbol$

Proof 2. By the previous post, the presheaf $S_{\mathscr U}$ is the colimit over $(V,\alpha) \in (h \downarrow S_{\mathscr U})$ of $h_V$ (see post for precise statement). Let $D \colon (\bullet \rightrightarrows \bullet) \to \mathbf{Set}$ be the diagram $I^2 \rightrightarrows I$ of the two projections, and let $\mathcal I = \bigcup D = (h \downarrow D)^{\operatorname{op}}$ be the category of elements of $D$ , as in this post. There is a natural functor $F \colon \mathcal I \to (h \downarrow S_{\mathscr U})$ taking $(i,j) \in I^2$ to $(U_i \times_U U_j,h_{U_i \times_U U_j} \to S_{\mathscr U})$ and $i \in I$ to $(U_i,h_{U_i} \to S_{\mathscr U})$ , taking the morphisms $i \leftarrow (i,j) \to j$ in $\mathcal I$ to the projections $U_i \leftarrow U_i \times_U U_j \to U_j$ . We claim that $F$ is cofinal, hence the colimit can be computed over $\mathcal I$ instead (see Tag 04E7).

To verify this, we use the criteria of Tag 04E6. If $(V,\alpha) \in (h \downarrow S_{\mathscr U})$ , then by definition the composition $h_V \stackrel\alpha\to S_{\mathscr U} \hookrightarrow h_U$ is given by a morphism $f \colon V \to U$ that is contained in $S_{\mathscr U}(V)$ . Since $S_{\mathscr U}$ is generated by the $U_i$ , this factors through some $V \to U_i$ over $S_{\mathscr U}$ , giving a map $(V,\alpha) \to F(i)$ .

If $(V,\alpha) \to F(i)$ and $(V,\alpha) \to F(j)$ are two such maps, they factor uniquely through $(V,\alpha) \to F(i,j)$ . The general result for $(V,\alpha) \to F(x)$ and $(V,\alpha) \to F(y)$ for $x,y \in \mathcal I$ (either of the form $i$ or of the form $(i,j)$ ) follows since elements of the form $(i,j)$ always map to the elements $i$ and $j$ , showing that the category $((V,\alpha) \downarrow F)$ is weakly connected. $\qedsymbol$

Corollary. Let $S_{\mathscr U}$ as above, and let $\mathscr F$ be a presheaf on $\mathscr C$ . Then

$\operatorname{Hom}(S_{\mathscr U},\mathscr F) \stackrel\sim\to \operatorname{Eq}\left( \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\Big(U_i \underset U\times U_j\Big) \right).$

Proof. By the lemma above, we compute

$\begin{align*}\operatorname{Hom}(S_{\mathscr U},\mathscr F) &\cong \operatorname{Hom}\left(\operatorname{Coeq}\left(\coprod_{i \in I} h_{U_i \underset U \times U_j} \rightrightarrows \coprod_{i \in I} h_{U_i}\right), \mathscr F\right) \\&\cong \operatorname{Eq}\left(\prod_{i \in I} \operatorname{Hom}(h_{U_i},\mathscr F) \rightrightarrows \operatorname{Hom}\Big(h_{U_i \underset U\times U_j},\mathscr F\Big)\right),\end{align*}$

so the result follows from the Yoneda lemma. $\qedsymbol$

Corollary. Let $\mathscr C$ be a (small) site. Then a presheaf $\mathscr F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}$ is a sheaf if and only if for every object $U \in \mathscr C$ and every covering $\{U_i \to U\}_{i \in I}$ in the site, the inclusion $S_{\mathscr U} \to h_U$ induces an isomorphism

$\operatorname{Hom}(h_U,\mathscr F) \stackrel\sim\to \operatorname{Hom}(S_{\mathscr U},\mathscr F).$

Proof. Immediate from the previous corollary. $\qedsymbol$

Thus, the category of sheaves on $\mathscr C$ can be recovered from $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ if we know at which subobjects $S \subseteq h_U$ we should localise (make the inclusion invertible). Next week, we will use this to give a definition of a Grothendieck topology, abstracting and generalising the notion of a site (i.e. Grothendieck pretopology).

Subterminal presheaves and sheaves (topologies 1/6)

Posted on 4 March 2022 by Remy

Grothendieck pretopologies feature prominently in algebraic geometry, but the more beautiful concept of Grothendieck topologies is rarely touched upon. In a series of short posts, I aim to introduce some of these ideas, show how key concepts like the sheaf condition get very nice categorical descriptions in this language, and give examples of why topoi have much better formal properties than sites.

Let $\mathscr C$ be a small category, and write $\mathbf{PSh}(\mathscr C)$ for the functor category $[\mathscr C^{\operatorname{op}},\mathbf{Set}]$ .

Definition. A sieve on an object $X \in \mathscr C$ is a subpresheaf $S \subseteq h_X$ of the representable presheaf $h_X = \operatorname{Hom}(-,X)$ .

Concretely, this means that each $S(U)$ is a set of morphisms $f \colon U \to X$ with the property that if $g \colon V \to U$ is any morphism, then $fg \colon V \to X$ is in $S(V)$ . Thus, this is like a “right ideal in $\operatorname{Hom}(-,X)$ “. Since $\mathscr C$ is small, we see that sieves on $X$ form a set, which we will denote $\mathbf{Siv}(X)$ .

Lemma. Let $\mathscr D$ and $\mathscr C$ be small categories, $X \in \mathscr D$ an object, and $F \colon \mathscr D \to \mathscr C$ a functor. Then there is a pullback map

$F^* \colon \mathbf{Siv}(F(X)) \to \mathbf{Siv}(X)$

defined by

$F^*S(U) = \big\{f \colon U \to X\ \big|\ F(f) \in S(F(U))\big\}.$

If $\mathscr D = \mathscr C/X$ and $F$ is the forgetful functor, then $F^*$ gives a bijection

$\mathbf{Siv}(X) \stackrel\sim\to \mathbf{Siv}(X \stackrel{\operatorname{id}}\to X).$

Proof. If $S$ is a sieve, then so is $F^*S$ since $f \in F^*S(U)$ and $g \in \operatorname{Hom}(V,U)$ implies $F(fg) = F(f)F(g) \in S(F(V))$ , so $fg \in F^*S(V)$ . For the second statement, given a sieve $T$ on $X \to X$ , define the sieve $S$ on $X$ by

$S(U) = \left\{ f \colon U \to X\ \ \left|\ \ \left(\begin{array}{ccccc}\!\!U\!\!\!\!\! & & \!\!\!\!\!\stackrel f\longrightarrow\!\!\!\!\! & & \!\!\!\!\!X\!\! \\ & \!\!\!{\underset{f}{}}\!\!\searrow\!\!\!\! & & \!\!\!\!\swarrow\!\!{\underset{\operatorname{id}}{}}\!\!\!\! & \\[-.3em] & & X.\! & & \end{array}\right) \in T\left(U \stackrel f\to X\right)\right\}\right..$

Then $S$ is a sieve on $X$ , and is the unique sieve on $X$ such that $F^*S = T$ . $\qedsymbol$

Beware that the notation $F^*S$ could also mean the presheaf pullback $S \circ F$ , but we won’t use it as such.

Remark. In particular, it suffices to study the case where $\mathscr C$ has a terminal object, which we will denote by $X$ (in analogy with the small Zariski and étale sites of a scheme $X$ , which have $X$ as a terminal object). We are thus interested in studying the subobjects of the terminal presheaf $\mathbf{1}_X$ . We will do so both in the case of presheaves and in the case of sheaves. Note that $\mathbf 1_X$ is a sheaf: for any set $I$ (empty or not), the product $\prod_{i \in I} \{*\}$ is a singleton, so the diagrams

$\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i,j\in I} \mathscr F\left(U_i \underset U\times U_j\right)$

are vacuously equalisers whenever $\{U_i \to U\}_{i \in I}$ is a covering (or any collection of morphisms).

Definition. A property $\mathcal P$ on a set $A$ is a function $\mathcal P \colon A \to P(\{*\})$ to the power set of a point $\{*\}$ . The property $\mathcal P$ holds for $a \in A$ if $\mathcal P(a) = \{*\}$ , and fails if $\mathcal P(a) = \varnothing$ .

Given a property $\mathcal P$ on the objects of a small category $\mathscr C$ , we say that $\mathcal P$ is left closed if for any morphism $f \colon U \to V$ , the implication $\mathcal P(V) \Rightarrow \mathcal P(U)$ holds. (This terminology is my own. Below, we confusingly prove that these are equivalent to what we described earlier as “right ideals”. This change of orientation arises from the fact that diagrams are drawn in the opposite direction compared to composition of morphisms.)

If $\mathscr C$ is a site (a small category together with a Grothendieck pretopology), we say that $\mathcal P$ is local if it is left closed, and for any covering $\{U_i \to U\}_{i \in I}$ in $\mathscr C$ , if $\mathcal P(U_i)$ holds for all $i \in I$ , then $\mathcal P(U)$ holds.

Lemma. Let $\mathscr C$ be a small category with a terminal object $X$ .

Giving a subpresheaf of $\mathbf 1_X$ is equivalent to giving a left closed property $\mathcal P$ on the objects of $\mathscr C$ .
If $\mathscr C$ is a site, then giving a subsheaf of the presheaf $\mathbf 1_X$ is equivalent to a giving a local property $\mathcal P$ .

A homotopy theorist might say that a local property is a $(-1)$ -truncated sheaf [of spaces] on $\mathscr C$ .

Proof. 1. The terminal presheaf $\mathbf 1_X$ takes on values $\{*\}$ at every $U \in \mathscr C$ , thus any subpresheaf $\mathscr F$ takes on the values $\varnothing$ and $\{*\}$ , hence is a property $\mathcal P$ on the objects of $\mathscr C$ . The presheaf condition means that for every morphism $f \colon U \to V$ , there is a map $f^* \colon \mathscr F(V) \to \mathscr F(U)$ , which is exactly the implication $\mathcal P(V) \Rightarrow \mathcal P(U)$ since there are no maps $\{*\} \to \varnothing$ .

Alternatively, one notes immediately from the definition that a sieve on an object $X \in \mathscr C$ is the same thing as a subcategory of $\mathscr C/X$ which is left closed.

2. Being a subpresheaf translates to a left closed property $\mathcal P$ by 1. Then $\mathscr F$ is a sheaf if and only if, for every covering $\{U_i \to U\}_{i \in I}$ in $\mathscr C$ , the diagram

$\mathscr F(U) \to \prod_{i \in I} \mathscr F(U_i) \rightrightarrows \prod_{i, j \in I} \mathscr F\Big(U_i \underset U\times U_j\Big)$

is an equaliser. If one $\mathscr F(U_i)$ is empty, then so is $\mathscr F(U)$ since $\mathcal P$ is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume $\mathscr F(U_i) = \{*\}$ for all $i \in I$ , i.e. $\mathcal P(U_i)$ holds for all $i \in I$ . Since $\mathcal P$ is left closed, this implies that $\mathscr F(U_i \times_U U_j) = \{*\}$ for all $i, j \in I$ , so the two arrows agree on $\prod_i \mathscr F(U_i)$ , and the diagram is an equaliser if and only if $\mathscr F(U) = \{*\}$ . Running over all coverings $\{U_i \to U\}$ in $\mathscr C$ , this is exactly the condition that $\mathcal P$ is local. $\qedsymbol$

Sites without a terminal object

Posted on 3 April 2017 by Remy

Let $\mathcal C$ be a site with a terminal object $X$ . Then the cohomology on the site is defined as the derived functors of the global sections functor $\Gamma(X,-)$ . But what do we do if the site does not have a terminal object?

The solution is to define $H^i(\mathcal C,-)$ as $\Ext{\mathcal O}{i}(\mathcal O,-)$ , where $\mathcal O$ denotes the structure sheaf if $\mathcal C$ is a ringed site. If $\mathcal C$ is not equipped with a ring structure, we take $\mathcal O$ to be the constant sheaf $\underline{\mathbb Z}$ ; this makes $\mathcal C$ into a ringed site.

Lemma. Let $\mathcal C$ be a site with a terminal object $X$ . Then the above definitions agree, i.e.

$H^i(X,-) = \Ext{\mathcal O}{i}(\mathcal O,-).$

Proof. Note that $\Hom_{\mathcal O}(\mathcal O, \mathscr F) = \Gamma(X, \mathscr F)$ , since any map $\mathcal O(X) \to \mathscr F(X)$ can be uniquely extended to a morphism of (pre)sheaves $\mathcal O \to \mathscr F$ , and conversely every such morphism is determined by its map on global sections. The result now follows since $\Ext{\mathcal O}{i}(\mathcal O, -)$ and $H^i(X,-)$ are defined as the derived functors of $\Hom_{\mathcal O}(\mathcal O,-)$ and $\Gamma(X,-)$ respectively. $\qedsymbol$

Remark. From this perspective, it seems quite magical that for a sheaf $\mathscr F$ of $\mathcal O_X$ -modules on a ringed space $(X,\mathcal O_X)$ , the cohomology groups $\Ext{\mathcal O_X}{i}(\mathcal O_X,\mathscr F)$ and $\Ext{\underline{\Z}}{i}(\underline{\Z},\mathscr F)$ agree. It turns out that this is true in the setting of ringed sites as well; see Tag 03FD.

So why is this useful? Let’s give some examples of sites that do not have a terminal object.

Example. Let $G$ be a group scheme over $k$ . Then we have a stack $BG$ of $G$ -torsors. The objects of $BG$ are pairs $(U,P)$ , where $U$ is a $k$ -scheme and $P$ is a $G$ -torsor over $U$ . Morphisms $(U,P) \to (U',P')$ are pairs $(f,g) \colon (U,P) \to (U',P')$ making the diagram

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commutative. This forces the diagram to be a pullback, since all maps between $G$ -torsors are isomorphisms.

The (large) Zariski site on $BG$ is defined by declaring coverings $\{(U_i, P_i) \to (U,P)\}$ to be families such that $\{U_i \to U\}$ is a Zariski covering (and similarly for the étale and fppf sites).

Now does the category $BG$ have a terminal object? This would be a $G$ -torsor $P_0 \to U_0$ such that every other $G$ -torsor $P \to U$ admits a unique map to it, realising $P$ as the pullback of $P_0$ along $U \to U_0$ . But this object would exactly be the classifying stack $U_0 = BG$ , which does not exist as a scheme (or algebraic space). The fact that a terminal object does not exist is the whole reason we need to define it as a stack in the first place!

Example. Let $X/k$ be a variety in characteristic $p > 0$ ; for simplicity, let’s say $k = \mathbb F_p$ . Then consider the crystalline site of $X/\Spec(\Z/p^n\Z)$ . Roughly speaking, its objects are triples $(U,T,\delta)$ , where $U \to X$ is an open immersion, $U \to T$ is a thickening with a map to $\Spec{\F_p} \to \Spec{\Z/p^n\Z}$ , and $\delta$ is a divided power structure on the ideal sheaf $\mathcal I_U \subseteq \mathcal O_T$ (with a compatibility condition w.r.t. $\Spec{\F_p} \to \Spec{\Z/p^n\Z}$ ). There is a suitable notion of morphisms.

This site does not have a terminal object, basically because there are many thickenings on $U = X$ with the respective compatibilities. (I am admittedly no expert, and it could very well be true that this is not 100% correct. However, I am certain that the crystalline site in general does not have a terminal object.)

Lovely little lemmas

Or maybe ‘amusing observations’

Category Archives: Sites