Finite topological spaces

One of my favourite bits of point set topology is messing around with easy topological spaces. What could be easier than finite topological spaces? The main result (below) is that the category of finite topological spaces is equivalent to the category of finite preorders.

Recall (e.g. from algebraic geometry) the following definition:

Definition. Let X be a topological space. Then the specialisation preorder on (the underlying set of) X is the relation x \leq y if and only if x \in \overline{\{y\}}.

Note that it is indeed a preorder: clearly x \leq x, and if x \leq y and y \leq z, then \{y\} \subseteq \overline{\{z\}}, so x \in \overline{\{y\}} \subseteq \overline{\{z\}}, showing x \leq z. We denote this preorder by X^{\operatorname{sp}}.

Note that the relation x \leq y is usually denoted y \rightsquigarrow x in algebraic geometry, which is pronounced “y specialises to x”.

Definition. Given a preorder (X,\leq), the Alexandroff topology on X is the topology whose opens U \subseteq X are the cosieves, i.e. the upwards closed sets (meaning x \in U and x \leq y implies y \in U).

To see that this defines a topology, note that an arbitrary (possibly empty) union or intersection of cosieves is a cosieve. A subbase for the topology is given by the principal cosieves X_{\geq x} = \{y \in X\ |\ y \geq x\} for any x \in X. We denote the set X with its Alexandroff topology by X^{\operatorname{Alex}}.

Likewise, the closed sets in X are the sieves (or downwards closed sets); for instance the principal sieves X_{\leq x} = \{y \in X\ |\ y \leq x\}. The closure of S \subseteq X is the sieve X_{\leq S} = \bigcup_{s \in S} X_{\leq s} generated by S; for instance the closure of a singleton \{x\} is the principal sieve X_{\leq x}.

Theorem. Let F \colon \mathbf{PreOrd} \to \mathbf{Top} be the functor X \mapsto X^{\operatorname{Alex}}, and G \colon \mathbf{Top} \to \mathbf{PreOrd} the functor Y \mapsto Y^{\operatorname{sp}}.

  1. Let (X,\leq) be a preorder, Y a topological space, and f \colon X \to Y a function. Then f is a monotone function X \to Y^{\operatorname{sp}} if and only if f is a continuous function X^{\operatorname{Alex}} \to Y.
  2. The functors F and G are adjoint: F \dashv G.
  3. The composition GF \colon \mathbf{PreOrd} \to \mathbf{PreOrd} is equal (not just isomorphic!) to the identity functor.
  4. The restriction of FG \colon \mathbf{Top} \to \mathbf{Top} to the category \mathbf{Top}^{\operatorname{fin}} of finite topological spaces is equal to the identity functor.
  5. If Y is a topological space, then Y is T_0 if and only if Y^{\operatorname{sp}} is a poset.
  6. If (X,\leq) is a preorder, then X is a poset if and only if X^{\operatorname{Alex}} is T_0.
  7. The functors F and G give rise to adjoint equivalences

        \begin{align*}F\!:\mathbf{PreOrd}^{\operatorname{fin}} &\leftrightarrows \mathbf{Top}^{\operatorname{fin}}:\!G \\F\!:\mathbf{Pos}^{\operatorname{fin}} &\leftrightarrows \mathbf{Top}^{\operatorname{fin}}_{T_0}:\!G.\end{align*}

Proof. (1) Suppose f \colon X \to Y^{\operatorname{sp}} is monotone, let Z \subseteq Y be a closed subset, and let W = f^{-1}(Z). Suppose b \in W and a \leq b. Since f is monotone and Z is closed, we get f(a) \leq f(b), i.e. f(a) \in \overline{\{f(b)\}} \subseteq Z. We conclude that a \in W, so W is downward closed, hence closed in X^{\operatorname{Alex}}.

Conversely, suppose f \colon X^{\operatorname{Alex}} \to Y is continuous, and suppose a \leq b in X. Then a \in \overline{\{b\}}, so by continuity we get f(a) \in f\left(\overline{\{b\}}\right) \subseteq \overline{\{f(b)\}}, so f(a) \leq f(b).

(2) This is a restatement of (1): the map

    \begin{align*}\operatorname{Hom}_{\mathbf{PreOrd}}\big(X,G(Y)\big) &\stackrel\sim\to \operatorname{Hom}_{\mathbf{Top}}\big(F(X),Y\big) \\f &\mapsto f\end{align*}

is a bijection.

(3) Since \overline{\{x\}} = X_{\leq x}, we conclude that y \in \overline{\{x\}} if and only if y \leq x, so the specialisation preorder on X^{\operatorname{Alex}} is the original preorder on X.

(4) In general, the counit FG(Y) \to Y is a continuous map on the same underlying space, so FG(Y) is finer than Y. Conversely, suppose Z \subseteq FG(Y) is closed, i.e. Z is a sieve for the specialisation preorder on Y. This means that if y \in Z, then x \in \overline{\{y\}} implies x \in Z; in other words \overline{\{y\}} \subseteq Z. If Y and therefore Z is finite, there are finitely many such y, so Z is the finite union

    \[Z = \bigcup_{y \in Z} \overline{\{y\}}\]

of closed subsets of Y. Thus any closed subset of FG(Y) is closed in Y, so the topologies agree.

(5) The relations x \leq y and y \leq x mean x \in \overline{\{y\}} and y \in \overline{\{x\}}. This is equivalent to the statement that a closed subset Z \subseteq Y contains x if and only if it contains y. The result follows since a poset is a preorder where the first statement only happens if x = y, and a T_0 space is a space where the second statement only happens if x = y.

(6) Follows from (5) applied to Y = F(X) since X = G(Y) by (3).

(7) The equivalence \mathbf{PreOrd}^{\operatorname{fin}} \leftrightarrows \mathbf{Top}^{\operatorname{fin}} follows from (3) and (4), and the equivalence \mathbf{Pos}^{\operatorname{fin}} \leftrightarrows \mathbf{Top}^{\operatorname{fin}}_{T_0} then follows from (5) and (6). \qedsymbol

Example. The Alexandroff topology on the poset [1] = \{0 \leq 1\} is the Sierpiński space S = \{0,1\} with topology \{\varnothing, \{1\}, S\}. As explained in this post, continuous maps X \to S from a topological space X to S are in bijection with open subsets U \subseteq X, where f \colon X \to S is sent to f^{-1}(1) \subseteq X (and U \subseteq X to the indicator function \mathbf 1_U \colon X \to S).

Example. Let X = \{x,y\} be a set with two elements. There are 4 possible topologies on X, sitting in the following diagram (where vertical arrows indicate inclusion bottom to top):

    \[{\arraycolsep=-1em\begin{array}{ccccc} & & \{\varnothing,\{x\},\{y\},X\} & & \\ & \ \ / & & \backslash\ \  & \\ \{\varnothing,\{x\},X\} & & & & \{\varnothing,\{y\},X\} \\ & \ \ \backslash & & /\ \  & \\ & & \{\varnothing,X\}.\! & & \end{array}}\]

These correspond to 4 possible preorder relations \{(a,b)\ |\ a \leq b\} \subseteq X\times X, sitting in the following diagram (where vertical arrows indicate inclusion top to bottom):

    \[{\arraycolsep=-1.5em\begin{array}{ccccc} & & \{(x,x),(y,y)\} & & \\ & \ \ / & & \backslash\ \ & \\ \{(x,x),(x,y),(y,y)\} & & & & \{(x,x),(y,x),(y,y)\} \\ & \ \ \backslash & & /\ \ & \\ & & \{(x,x),(x,y),(y,x),(y,y)\}.\!\! & & \end{array}}\]

We see that finer topologies (more opens) have stronger relations (fewer inequalities).

Example. The statement in (4) is false for infinite topological spaces. For instance, if Y is the Zariski topology on a curve, then any set of closed points is downwards closed, but it is only closed if it’s finite. Or if Y is a Hausdorff space, then the specialisation preorder is just the equality relation \Delta_Y \subseteq Y \times Y, whose Alexandroff topology is the discrete topology.

I find the examples useful for remembering which way the adjunction goes: topological spaces generally have fewer opens than Alexandroff topologies on posets, so the continuous map should go X^{\operatorname{Alex}} \to Y.

Lawvere–Tierney topologies (topologies 6/6)

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)

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)

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

  1. For all X \in \mathscr C, the maximal sieve h_X \subseteq h_X is in J(X).
  2. If S \in J(X) and S' \in \mathbf{Siv}(X) with S \subseteq S', then S' \in J(X).
  3. 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)

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

    \[\begin{array}{ccc}\displaystyle\coprod_{i,j \in I} h_{U_i \underset U\times U_j} & \to & \displaystyle\coprod_{i \in I} h_{U_i} \\ \downarrow & & \downarrow \\ \displaystyle\coprod_{j \in I} h_{U_j} & \to & S_{\mathscr U} \end{array}\]

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).

Presheaves are colimits of representables (topologies 2/6)

Last week, I started a series on sieves and Grothendieck topoi. This is a short intermezzo on a well-known lemma from category theory that I will need for next week’s instalment.

Let \mathscr C be a small category. Recall that a presheaf on \mathscr C is a functor \mathscr C^{\operatorname{op}} \to \mathbf{Set}. Examples include the representable presheaves h_X for X \in \mathscr C, given by \operatorname{Hom}(-,X). The Yoneda lemma says that for any presheaf F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set} and any X \in \mathscr C, the map

    \begin{align*}\operatorname{Nat}(h_X,F) &\to F(X) \\\alpha &\mapsto \alpha_X(\operatorname{id}_X)\end{align*}

is an isomorphism. Applying this to F = h_Y shows that the Yoneda embedding

    \begin{align*}h \colon \mathscr C &\to [\mathscr C^{\operatorname{op}},\mathbf{Set}] \\X &\mapsto h_X\end{align*}

is fully faithful.

Given a functor F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set}, write (h \downarrow F) for the comma category whose objects are pairs (X,\alpha) where X \in \mathscr C and \alpha \colon h_X \to F is a morphism (natural transformation) in [\mathscr C^{\operatorname{op}},\mathbf{Set}]. A morphism f \colon (X,\alpha) \to (Y,\beta) is a morphism f \colon X \to Y such that the triangle

(1)   \[\begin{array}{ccccc}\!\!h_X\!\!\!\!\! & & \!\!\!\!\!\stackrel{h_f}\longrightarrow\!\!\!\!\! & & \!\!\!\!\!h_Y\!\! \\ & \!\!\!\!{\underset{\alpha}{}}\!\!\searrow\!\!\!\! & & \!\!\!\!\swarrow\!\!{\underset{\beta}{}}\!\!\!\! & \\[-.3em] & & F\! & & \end{array}\]

of natural transformations commutes, where h_f \colon \operatorname{Hom}(-,X) \to \operatorname{Hom}(-,Y) denotes postcomposition by f. Note that (h \downarrow F) is again a small category, and there is a forgetful functor U \colon (h \downarrow F) \to \mathscr C taking (X,\alpha) to X \in \mathscr C.

By the Yoneda lemma, the category (h \downarrow F) is isomorphic (not just equivalent!) to the category \int F of pairs (X,s) with X \in \mathscr C and s \in F(X) with morphisms f \colon (X,s) \to (Y,t) given by morphisms f \colon X \to Y in \mathscr C such that F(f)(t) = s. It’s convenient to keep both points of view.

Lemma. Let \mathscr C be a small category, and let F \colon \mathscr C^{\operatorname{op}} \to \mathbf{Set} be a functor.

  1. The object F \in [\mathscr C^{\operatorname{op}},\mathbf{Set}] is naturally a cocone under hU \colon (h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}] via the morphisms \alpha \colon h_X \to F.
  2. This cocone makes F the colimit of the diagram hU of representable functors.

In particular, any presheaf on a small category is a colimit of representable presheaves.

Proof. A cocone under hU is a presheaf G with a natural transformation \phi \colon hU \to G to the constant diagram (h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}] with value G. This means every (X,\alpha) \in (h \downarrow F) is taken to a natural transformation \phi_{(X,\alpha)} \colon h_X \to G, such that for any morphism f \colon (X,\alpha) \to (Y,\beta), the square

    \[\begin{array}{ccc} h_X & \stackrel{\phi_{(X,\alpha)}}\longrightarrow & G \\ \!\!\!\!\!\!\!\!\!{h_f}\downarrow & & |\!| \\ h_Y & \underset{\phi_{(Y,\beta)}}\longrightarrow & G\end{array}\]

commutes. By the Yoneda lemma, such a datum corresponds to an association of elements \phi_{(X,s)} \in G(X) for all (X,s) \in \int F such that for every f \colon (X,s) \to (Y,t) with F(f)(t) = s, we have G(f)(\phi_{(Y,t)}) = \phi_{(X,s)}.

(1) To make F a cocone under hU, simply take \phi_{(X,s)} = s \in F(X).

(2) Given any other cocone \phi \colon hU \to G under hU, define the natural transformation \eta \colon F \to G by

    \begin{align*}\eta_X \colon F(X) &\to G(X) \\s &\mapsto \phi_{(X,s)}.\end{align*}

Naturality follows since F(f)(t) = s implies G(f)(\phi_{(Y,t)}) = \phi_{(X,s)}. It is clear that \eta is the unique natural transformation of cocones F \to G under hU, showing that F is the colimit. \qedsymbol

One can also easily rewrite this argument in terms of natural transformations h_X \to G. For instance, the universal cocone hU \to F is the natural transformation \phi \colon hU \to F of functors (h \downarrow F) \to [\mathscr C^{\operatorname{op}},\mathbf{Set}] given on (X,\alpha) \in (h \downarrow F) by \alpha \colon h_X \to F. Naturality of \phi follows at once from (1). But checking that this thing is universal is a bit more tedious in this language.

Example. A standard example where this point of view is useful is simplicial sets. Let \Delta be the category of finite nonempty totally ordered sets, with (weakly) monotone increasing functions as morphisms. A simplicial set is a functor \Delta^{\operatorname{op}} \to \mathbf{Set}, and we often think of them as combinatorial models for topological spaces. The representable ones are the standard n-simplices \Delta^n = \operatorname{Hom}(-,[n]), where [n] is the totally ordered set \{0,\ldots,n\} for n \in \mathbf Z_{\geq 0}.

If X \colon \Delta^{\operatorname{op}} \to \mathbf{Set} is a simplicial set, its value at [n] is called the n-simplices X_n of X. By the Yoneda lemma, this is \operatorname{Hom}(\Delta^n,X). Then the story above is saying that a simplicial set is the colimit over all its n-simplices for n \in \mathbf Z_{\geq 0}. This is extremely useful, as many arguments proceed by attaching simplices one at a time.

Subterminal presheaves and sheaves (topologies 1/6)

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.

  1. Giving a subpresheaf of \mathbf 1_X is equivalent to giving a left closed property \mathcal P on the objects of \mathscr C.
  2. 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

Union of hyperplanes over a finite field


The following lemma is a (presumably well-known) result that Raymond Cheng and I happened upon while writing our paper Unbounded negativity on rational surfaces in positive characteristic (arXiv, DOI). Well, Raymond probably knew what he was doing, but to me it was a pleasant surprise.

Lemma. Let q be a power of a prime p, and let x_0,\ldots,x_n \in \bar{\mathbf F}_q. Then x_0,\ldots,x_n satisfy a linear relation over \mathbf F_q if and only if

    \[\det \begin{pmatrix} x_0 & x_1 & \cdots & x_n \\ x_0^q & x_1^q & \cdots & x_n^q \\ \vdots & \vdots & \ddots & \vdots \\ x_0^{q^n} & x_1^{q^n} & \cdots & x_n^{q^n} \end{pmatrix} = 0.\]

Proof. If \sum_{i=0}^n c_ix_i = 0 for (c_0,\ldots,c_n) \in \mathbf F_q^n - \{0\}, then c_i^{q^j} = c_i for all i,j \in \{0,\ldots,n\} since c_i \in \mathbf F_q. As (-)^q \colon \bar{\mathbf F}_q \to \bar{\mathbf F}_q is a ring homomorphism, we find

    \[\begin{pmatrix} x_0 & x_1 & \cdots & x_n \\ x_0^q & x_1^q & \cdots & x_n^q \\ \vdots & \vdots & \ddots & \vdots \\ x_0^{q^n} & x_1^{q^n} & \cdots & x_n^{q^n} \end{pmatrix}\begin{pmatrix} c_0 \\ c_1 \\ \vdots \\ c_n \end{pmatrix} = 0,\]

so the determinant is zero. Conversely, the union of \mathbf F_q-rational hyperplanes H \subseteq \mathbf P^n_{\mathbf F_q} is a hypersurface Y of degree |\check{\mathbf P}^n(\mathbf F_q)| = q^n + \ldots + q + 1 (where \check{\mathbf P}^n denotes the dual projective space parametrising hyperplanes in \mathbf P^n). Since the determinant above is a polynomial of the same degree q^n + \ldots + q + 1 that vanishes on all \mathbf F_q-rational hyperplanes, we conclude that it is the polynomial cutting out Y, so any [x_0:\ldots:x_n] \in \mathbf P^n(\bar{\mathbf F_q}) for which the determinant vanishes lies on one of the hyperplanes. \qedsymbol

Of course when the determinant is zero, one immediately gets a vector (c_0,\ldots,c_n) \in \bar{\mathbf F}_q^{n+1} - \{0\} in the kernel. There may well be an immediate argument why this vector is proportional to an element of \mathbf F_q^{n+1}, but the above cleverly circumvents this problem.

For concreteness, we can work out what this determinant is in small cases:

  • n=0: a point x_0 \in \bar{\mathbf F}_q only satisfies a linear relation over \mathbf F_q if it is zero.
  • n=1: the polynomial x_0x_1^q-x_0^qx_1 cuts out the \mathbf F_q-rational points of \mathbf P^1.
  • n=2: the polynomial

        \[x_0x_1^qx_2^{q^2}+x_1x_2^qx_0^{q^2}+x_2x_0^qx_1^{q^2}-x_0^{q^2}x_1^qx_2-x_1^{q^2}x_2^qx_0-x_2^{q^2}x_0^qx_1\]

    cuts out the union of \mathbf F_q-rational lines in \mathbf P^2. This is the case considered in the paper.

Algebraic closure of the field of two elements

I think I learned about this from a comment on MathOverflow.

Recall that the field of two elements \mathbf F_2 is the ring \mathbf Z/2\mathbf Z of integers modulo 2. In other words, it consists of the elements 0 and 1 with addition 1+1 = 0 and the obvious multiplication. Clearly every nonzero element is invertible, so \mathbf F_2 is a field.

Lemma. The field \mathbf F_2 is algebraically closed.

Proof. We need to show that every non-constant polynomial f \in \mathbf F_2[x] has a root. Suppose f does not have a root, so that f(0) \neq 0 and f(1) \neq 0. Then f(0) = f(1) = 1, so f is the constant polynomial 1. This contradicts the assumption that f is non-constant. \qedsymbol

Hodge diamonds that cannot be realised

In Paulsen–Schreieder [PS19] and vDdB–Paulsen [DBP20], the authors/we show that any block of numbers

    \[\left(\begin{array}{ccccc} & & h^{n,n} & & \\ & \iddots & & \ddots & \\ h^{n,0} & & & & h^{0,n} \\ & \ddots & & \iddots & \\ & & h^{0,0} & & \end{array}\right) \in \big(\mathbf Z/m\big)^{(n+1)^2}\]

satisfying h^{0,0} = 1, h^{p,q} = h^{n-p,n-q}, and h^{p,q} = h^{q,p} (characteristic 0 only) can be realised as the modulo m reduction of a Hodge diamond of a smooth projective variety.

While preparing for a talk on [DBP20], I came up with the following easy example of a Hodge diamond that cannot be realised integrally, while not obviously violating any of the conditions (symmetry, nonnegativity, hard Lefschetz, …).

Lemma. There is no smooth projective variety (in any characteristic) whose Hodge diamond is

    \[\begin{array}{ccccc} & & 1 & & \\ & 1 & & 1 & \\ 0 & & 1 & & 0 \\ & 1 & & 1 & \\ & & 1.\!\! & & \end{array}\]

Proof. If \operatorname{char} k > 0, we have \sum_{p+q = i}h^{p,q} \geq h_{\operatorname{dR}}^i \geq h_{\operatorname{cris}}^i, with equality for all i if and only if the Hodge–de Rham spectral sequence degenerates and H_{\operatorname{cris}}^i is torsion-free for all i. Because H^2_{\operatorname{cris}} contains an ample class, we must have equality on h^2, hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that h^1_{\operatorname{Weil}}(X) = 2, so \dim \mathbf{Pic}_X^0 = 1 and the same for \dim \mathbf{Alb}_X. Thus, X \to \mathbf{Alb}_X is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption h^{1,1}(X) = 1. \qedsymbol

Remark. In characteristic zero, the Hodge diamonds

    \[\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 1 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}\]

cannot occur for any a \geq 1, by essentially the same argument. Indeed, the only thing left to prove is that the image X \to \mathbf{Alb}_X cannot be a surface. If it were, then X would have a global 2-form; see e.g. [Beau96, Lemma V.18].

This argument does not work in positive characteristic due to the possibility of an inseparable Albanese map. It seems to follow from Bombieri–Mumford’s classification of surfaces in positive characteristic that the above Hodge diamond does not occur in positive characteristic either, but the analysis is a little intricate.

Remark. On the other hand, the nearly identical Hodge diamond

    \[\begin{array}{ccccc} & & 1 & & \\ & a & & a & \\ 0 & & 2 & & 0 \\ & a & & a & \\ & & 1 & & \end{array}\]

is realised by C \times \mathbf P^1, where C is a curve of genus a. This is some evidence that the full inverse Hodge problem is very difficult, and I do not expect a full classification of which Hodge diamonds are possible (even for surfaces this might be out of reach).


References.

[Beau96] A. Beauville, Complex algebraic surfaces. London Mathematical Society Student Texts 34 (1996).

[DBP20] R. van Dobben de Bruyn and M. Paulsen, The construction problem for Hodge numbers modulo an integer in positive characteristic. Forum Math. Sigma (to appear).

[PS19] M. Paulsen and S. Schreieder, The construction problem for Hodge numbers modulo an integer. Algebra Number Theory 13.10, p. 2427–2434 (2019).