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

Simplicial sets

A few weeks ago, I finally struck up the courage to take some baby steps reading Lurie’s Higher topos theory. In a series of posts mostly written for my own benefit, I will untangle some of the basic definitions and provide some easy examples. The first one is one I was already somewhat familiar with: simplicial sets.

Definition. For each n \in \mathbf N, write [n] for the poset 0 \leq \ldots \leq n. The full subcategory of \mathbf{Poset} on these [n] is denoted \Delta, the simplex category. Concretely, it has objects [n] for all n \in \mathbf N, and morphisms

    \[\operatorname{Hom}\big([m],[n]\big) = \left\{f \colon [m] \to [n]\ \Big|\ i \leq j \Rightarrow f(i) \leq f(j)\right\}.\]

A simplicial set is a functor X \colon \Delta^{\operatorname{op}} \to \mathbf{Set}. This can be described rather concretely using the objects X_n = X([n]) and the face and degeneracy maps between them; see e.g. Tag 0169. The category of simplicial sets is usually denoted [\Delta^{\operatorname{op}}, \mathbf{Set}], \mathbf{sSet}, or \mathbf{Set}_{\Delta} (in analogy with cosimplicial sets \mathbf{Set}^\Delta = [\Delta,\mathbf{Set}]).

The representable simplicial set \operatorname{Hom}(-,[n]) is usually denoted \Delta^n or \Delta[n]. Then the Yoneda lemma shows that the functor \mathbf{sSet} \to \mathbf{Set} given by X \mapsto X_n is represented by \Delta^n, i.e.

    \[X_n = \operatorname{Hom}_{\mathbf{sSet}}\big(\Delta^n,X).\]

Definition. The geometric realisation functor | \cdot | \colon \mathbf{sSet} \to \mathbf{Top} is defined as follows: for \Delta^n, the geometric realisation |\Delta^n| is the standard n-simplex

    \[|\Delta^n| := \left\{(x_0,\ldots,x_n) \in \mathbf R^{n+1}\ \Bigg|\ x_i \geq 0, \sum_{i=0}^n x_i = 1\right\} \subseteq \mathbf R^{n+1}.\]

(If no confusion arises, it may also be denoted \Delta^n.) This is functorial in [n]: for a map a \colon [m] \to [n] (equivalently, by the Yoneda lemma, a map a \colon \Delta^m \to \Delta^n) we get a continuous map a \colon |\Delta^m| \to |\Delta^n| by

    \[a(x_0,\ldots,x_m)_j = \sum_{i \in a^{-1}(j)} x_i.\]

For an arbitrary simplicial set X, write

    \[|X| := \underset{\Delta^n \to X}{\operatorname{colim}}\ |\Delta^n|,\]

where the transition map |\Delta^m| \to |\Delta^n| corresponding to a map \Delta^m \to \Delta^n over X is defined via

    \[\operatorname{Hom}_{\mathbf{sSet}}\big(\Delta^m,\Delta^n\big) \stackrel\sim\leftarrow \operatorname{Hom}_\Delta\big([m],[n]\big) \to \operatorname{Hom}_{\mathbf{Top}}\big(|\Delta^m|,|\Delta^n|\big).\]

This is functorial in X, and when X = \Delta^n it coindices with the previous definition because the identity \Delta^n \to \Delta^n is terminal in the index category.

Remark. In a fancier language, | \cdot | is the left Kan extension of the functor [n] \mapsto |\Delta^n| along the Yoneda embedding \Delta \to \mathbf{sSet}. (Those of you familiar with presheaves on spaces will recognise the similarity with the definition of f^{-1}\mathscr F for f \colon X \to Y a continuous map of topological spaces, which is another example of a left Kan extension.)

Remark. It is a formal consequence of the definitions that geometric realisation preserves arbitrary colimits (“colimits commute with colimits”). This also follows because it is a left adjoint to the singular set functor, but we won’t explore this here.

Wisdom. The most geometric way to think about a simplicial set is through its geometric realisation.

For example, we can define the i^{\text{th}} horn \Lambda_i^n in \Delta^n as the union of the images of the maps \Delta^{n-1} \to \Delta^n coming from the face maps \delta^j_n \colon [n-1] \to [n] for j \neq i. Since geometric realisation preserves colimits (alternatively, stare at the definitions), we see that the geometric realisation of \Lambda^i_n is obtained in the same way from the maps |\Delta^{n-1}| \to |\Delta^n|, so it is the n-simplex with its interior and face opposite the i^{\text{th}} vertex removed.

The geometric realisation is a good first approximation for thinking about a simplicial set. However, when thinking about \infty-categories (e.g. in the next few posts), this is actually not the way you want to think about a simplicial set. Indeed, homotopy of simplicial sets (equivalently their geometric realisations) is stronger than equivalence of \infty-categories. (More details later, hopefully.)

Limits as equalisers of products

The first and second corollary below are well-known category theory lemmas. We give a slightly different argument than usual (i.e. we took a trivial result and changed it into something much more complicated).

Here is a lovely little definition:

Definition. Given a small diagram D \colon \mathcal I \to \mathbf{Set} of sets, write \bigcup D for the small category with

    \[\operatorname{ob}\left( \bigcup D \right) = \bigcup_{i \in \operatorname{ob} \mathcal I} D(i),\]

and morphisms

    \[\operatorname{Mor}\big(a_i,b_j\big) = \left\{f \in \operatorname{Mor}(i,j)\ \Big|\ D(f)(a_i) = b_j\right\}\]

for a_i \in D(i) and b_j \in D(j) (where i, j \in \operatorname{ob} \mathcal I), with composition induced by composition of maps D(i) \to D(j).

Note that by the Yoneda lemma, this category is isomorphic (not just equivalent!) to (h \downarrow D)^{\operatorname{op}}, where h \colon \mathcal I^{\operatorname{op}} \to [\mathcal I,\mathbf{Set}] is the Yoneda embedding. Indeed, a_i \in D(i) are in bijection with natural transformations h_i = \operatorname{Mor}_{\mathcal I}(i,-) \to D, and morphisms a_i \to b_j correspond to a morphism f \colon i \to j rendering commutative the associated diagram

    \[\begin{array}{ccccc}h_j\!\!\!\! & & \!\!\!\!\!\!\stackrel{- \circ f}\longrightarrow\!\!\!\!\!\! & & \!\!\!\!h_i \\ & \!\!\!\!\searrow\!\!\!\!\!\!\!\! & & \!\!\!\!\!\!\!\!\swarrow\!\!\!\! \\ & & D.\! & & \end{array}\]

Example 1. If \mathcal I = (\bullet \rightrightarrows \bullet), then a diagram D is a pair of sets S, T with parallel arrows f, g \colon S \rightrightarrows T. Then \bigcup D looks like a ‘bipartite preorder’ where every source object has outgoing valence 2:

    \[\begin{array}{ccc}s_1 & \to & t_1 \\ & \searrow\!\!\!\!\!\!\nearrow & \\ s_2 & & t_2 \\ & \searrow & \\ \vdots & & \vdots \end{array}\]

Example 2. Given a set S, write S^{\operatorname{disc}} for the discrete category on S, i.e. \operatorname{ob}(S^{\operatorname{disc}}) = S and

    \[\operatorname{Mor}(a,b) = \begin{cases}\{\mathbf{1}_a\}, & a = b, \\ \varnothing, & \text{else}.\end{cases}\]

If \mathcal I = I^{\operatorname{disc}} is itself a discrete category, then D is just a collection \mathbf S = \{S_i\ |\ i \in I\} of sets, and

    \[\bigcup D = \left(\bigcup \mathbf S\right)^{\operatorname{disc}}.\]

Remark. Giving a functor F \colon \bigcup D \to \mathscr C is the same thing as giving functors F(i) \colon D(i)^{\operatorname{disc}} \to \mathscr C and natural transformations

    \[F(f) \colon F(i) \to F(j) \circ D(f)^{\operatorname{disc}}\]

of functors D(i)^{\operatorname{disc}} \to \mathscr C for all f \colon i \to j in \mathcal I, such that

    \[F(g \circ f) = \left(F(g) \star \mathbf 1_{D(f)^{\operatorname{disc}}}\right) \circ F(f)\]

for all i \stackrel f\to j \stackrel g\to k in \mathcal I (where \star denotes horizontal composition of natural transformations, as in Tag 003G).

Example 3.  Let \mathcal I be a small category, and consider the diagram D_{\mathcal I} \colon (\bullet \rightrightarrows \bullet) \to \mathbf{Set} given by the source and target maps s, t \colon \operatorname{ar}(\mathcal I) \rightrightarrows \operatorname{ob}(\mathcal I). Then we have a functor

    \[F \colon \bigcup D_{\mathcal I} \to \mathcal I\]

given on objects by

    \begin{align*} \big(f \in \operatorname{ar}(\mathcal I)\big) &\mapsto s(f),\\ \big(i \in \operatorname{ob}(\mathcal I)\big) &\mapsto i \end{align*}

and on morphisms by

    \begin{align*} \big(s \colon f \to s(f)\big) &\mapsto \mathbf 1_{s(f)},\\ \big(t \colon f \to t(f)\big) &\mapsto f. \end{align*}

In terms of the remark above, it is given by the functors F(\operatorname{ar}) \colon \operatorname{ar}(\mathcal I)^{\operatorname{disc}} \to \mathcal I taking f to s(f) and the natural inclusion F(\operatorname{ob}) \colon \operatorname{ob}(\mathcal I)^{\operatorname{disc}} \to \mathcal I, along with the natural transformations

    \begin{align*} F(s)(f) = \mathbf 1_{s(f)} \colon F(\operatorname{ar})(f) &\to F(\operatorname{ob})(s(f)) \\ F(t)(f) = f \colon F(\operatorname{ar})(f) &\to F(\operatorname{ob})(t(f)). \end{align*}

We can now formulate the main result.

Lemma. Let \mathcal I be a small category. Then the functor F \colon \bigcup D_{\mathcal I} \to \mathcal I of Example 3 is cofinal.

Recall that a functor F \colon \mathcal J \to \mathcal I is cofinal if for all i \in \mathcal I, the comma category (i \downarrow F) is nonemptry and connected. See also Tag 04E6 for a concrete translation of this definition.

Proof. Let i \in \operatorname{ob}(\mathcal I). Since F(i) = i, the identity i \to F(i) gives the object (i,\mathbf 1_i) in (i \downarrow F), showing nonemptyness. For connectedness, it suffices to connect any (x,f) (i.e. f \colon i \to F(x)) to the identity (i, \mathbf 1_i) (i.e. \mathbf 1_i \colon i \to F(i)). If x \in \operatorname{ar}(\mathcal I), then the commutative diagram

    \[\begin{array}{ccccccc}i & = & i & = & i & = & i \\ \!\!\!\!\!{\tiny f}\downarrow & & \!\!\!\!\!\!\!\!{\tiny xf}\downarrow & & || & & || \\ s(x) & \overset x\to & t(x) & \overset{xf}\leftarrow & i & = & i \\ || & & || & & || & & || \\ F(x) & \underset{F(t)}\to & F(t(x)) & \underset{F(t)}\leftarrow & F(xf) & \underset{F(s)}\to & F(i) \end{array}\]

gives a zigzag

    \[(x,f) \stackrel t\to (t(x),xf) \stackrel t\leftarrow (xf,\mathbf 1_i) \stackrel s\to (i,\mathbf 1_i)\]

of morphisms in (i \downarrow F) connecting (x,f) to (i,\mathbf 1_i). If instead x \in \operatorname{ob}(\mathcal I), we can skip the first step, and the diagram

    \[\begin{array}{ccccccc} i & = & i & = & i \\ \!\!\!\!\!\!{\tiny f}\downarrow & & || & & || \\ x & \overset{f}\leftarrow & i & = & i \\ || & & || & & || \\ F(x) & \underset{F(t)}\leftarrow & F(f) & \underset{F(s)}\to & F(i) \end{array}\]

gives a zigzag

    \[(x,f) \stackrel t\leftarrow (f,\mathbf 1_i) \stackrel s\to (i,\mathbf 1_i)\]

connecting (x,f) to (i,\mathbf 1_i). \qedsymbol

Corollary 1. Let D \colon \mathcal I^{\operatorname{op}} \to \mathscr C be a small diagram in a category \mathscr C with small products. Then there is a canonical isomorphism

    \[\lim_{\leftarrow} D = \operatorname{Eq}\left( \prod_{i \in \operatorname{ob}(\mathcal I)} D(i) \rightrightarrows \prod_{f \in \operatorname{ar}(\mathcal I)} D(s(i)) \right),\]

provided that either side exists.

Proof. By the lemma, the functor

    \[F \colon \left(\bigcup D_{\mathcal I}\right)^{\operatorname{op}} \to \mathcal I^{\operatorname{op}}\]

is initial. Hence by Tag 002R, the natural morphism

    \[\lim_{\leftarrow} D \to \lim_{\leftarrow} D \circ F\]

is an isomorphism if either side exists. But \left(\bigcup D_{\mathcal I}\right)^{\operatorname{op}} is a category as in Example 1, and it’s easy to see that the limit over a diagram \left(\bigcup D_{\mathcal I}\right)^{\operatorname{op}} \to \mathscr C is computed as the equaliser of a pair of arrows between the products. \qedsymbol

Of course this is not an improvement of the traditional proof, because the “it’s easy to see” step at the end is very close to the same statement as the corollary in the special case where \mathcal I is of the form \bigcup D for some D \colon (\bullet \rightrightarrows \bullet) \to \mathbf{Set}. But it’s fun to move the argument almost entirely away from limits and into the index category.

Corollary 2. Let \mathscr C be a category that has small products and equalisers of parallel pairs of arrows. Then \mathscr C is (small) complete. \qedsymbol

Application of Schur orthogonality

The post that made me google ‘latex does not exist’.

Lemma. Let \epsilon be a finite group of order \Sigma, and write \equiv for the set of irreducible characters of \epsilon. Then

  1.     \[\forall (,) \in \epsilon : \hspace{1em} \sum_{\Xi \in \equiv} \Xi(()\overline\Xi()) = \begin{cases}|C_\epsilon(()|, & \exists \varepsilon \in \epsilon: (\varepsilon = \varepsilon), \\ 0, & \text{else}.\end{cases}\]

  2.     \[\forall \Xi,\underline\Xi \in \equiv : \hspace{1em} \Sigma^{-1}\sum_{\text O)) \in \epsilon} \Xi(\text O)))\overline{\underline\Xi}(\text O))) = \begin{cases}1, & \Xi = \underline\Xi,\\ 0, &\text{else}.\end{cases}\]

Proof. First consider the case \epsilon = 1. This is just an example; it could also be something much better. Then the second statement is obvious, and the first is left as an exercise to the reader. The general case is similar. \qedsymbol

Here is a trivial consequence:

Corollary. Let \mathbf R be a positive integer, and let f \in \mathbf C^\times[\mathbf R] \setminus \{1\}. Then

    \[\sum_{X = 1}^{\mathbf R} f^X = 0.\]

Proof 1. Without loss of generality, f has exact order \mathbf R > 1. Set \epsilon = \mathbf Z/\mathbf {RZ}, let ((,)) = (1,0) \in \epsilon^2, and note that

    \[\nexists \varepsilon \in \epsilon : (\varepsilon = \varepsilon).\]

Part 1 of the lemma gives the result. \qedsymbol

Proof 2. Set \epsilon = \mathbf Z/\mathbf {RZ} as before, let \Xi \colon \epsilon \to \mathbf C^\times be the homomorphism \varepsilon \mapsto f^{3\varepsilon}, and \underline \Xi \colon \epsilon \to \mathbf C^\times the homomorphism \varepsilon \mapsto f^{2\varepsilon}. Then part 1 of the lemma does not give the result, but part 2 does. \qedsymbol

In fact, the corollary also implies the lemma, because both are true (\mathbf 1 \Rightarrow \mathbf 1).