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 be a small category, and write for the functor category .

Definition. A sieve on an object is a subpresheaf of the representable presheaf .

Concretely, this means that each is a set of morphisms with the property that if is any morphism, then is in . Thus, this is like a “right ideal in “. Since is small, we see that sieves on form a set, which we will denote .

Lemma. Let and be small categories, an object, and a functor. Then there is a pullback map

defined by

If and is the forgetful functor, then gives a bijection

Proof. If is a sieve, then so is since and implies , so . For the second statement, given a sieve on , define the sieve on by

Then is a sieve on , and is the unique sieve on such that .

Beware that the notation could also mean the presheaf pullback , but we won’t use it as such.

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

are vacuously equalisers whenever is a covering (or any collection of morphisms).

Definition. A property on a set is a function to the power set of a point . The property holds for if , and fails if .

Given a property on the objects of a small category , we say that is left closed if for any morphism , the implication 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 is a site (a small category together with a Grothendieck pretopology), we say that is local if it is left closed, and for any covering in , if holds for all , then holds.

Lemma. Let be a small category with a terminal object .

1. Giving a subpresheaf of is equivalent to giving a left closed property on the objects of .
2. If is a site, then giving a subsheaf of the presheaf is equivalent to a giving a local property .

A homotopy theorist might say that a local property is a -truncated sheaf [of spaces] on .

Proof. 1. The terminal presheaf takes on values at every , thus any subpresheaf takes on the values and , hence is a property on the objects of . The presheaf condition means that for every morphism , there is a map , which is exactly the implication since there are no maps .

Alternatively, one notes immediately from the definition that a sieve on an object is the same thing as a subcategory of which is left closed.

2. Being a subpresheaf translates to a left closed property by 1. Then is a sheaf if and only if, for every covering in , the diagram

is an equaliser. If one is empty, then so is since is left closed, so the diagram is always an equaliser.

Thus, in the sheaf condition, we may assume for all , i.e. holds for all . Since is left closed, this implies that for all , so the two arrows agree on , and the diagram is an equaliser if and only if . Running over all coverings in , this is exactly the condition that is local.

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 be a power of a prime , and let . Then satisfy a linear relation over if and only if

Proof. If for , then for all since . As is a ring homomorphism, we find

so the determinant is zero. Conversely, the union of -rational hyperplanes is a hypersurface of degree (where denotes the dual projective space parametrising hyperplanes in ). Since the determinant above is a polynomial of the same degree that vanishes on all -rational hyperplanes, we conclude that it is the polynomial cutting out , so any for which the determinant vanishes lies on one of the hyperplanes.

Of course when the determinant is zero, one immediately gets a vector in the kernel. There may well be an immediate argument why this vector is proportional to an element of , but the above cleverly circumvents this problem.

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

• : a point only satisfies a linear relation over if it is zero.
• : the polynomial cuts out the -rational points of .
• : the polynomial

cuts out the union of -rational lines in . This is the case considered in the paper.

Algebraic closure of the field of two elements

Recall that the field of two elements is the ring of integers modulo . In other words, it consists of the elements and with addition and the obvious multiplication. Clearly every nonzero element is invertible, so is a field.

Lemma. The field is algebraically closed.

Proof. We need to show that every non-constant polynomial has a root. Suppose does not have a root, so that and . Then , so is the constant polynomial . This contradicts the assumption that is non-constant.

Hodge diamonds that cannot be realised

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

satisfying , , and (characteristic only) can be realised as the modulo 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

Proof. If , we have , with equality for all if and only if the Hodge–de Rham spectral sequence degenerates and is torsion-free for all . Because contains an ample class, we must have equality on , hence everywhere because of how spectral sequences and universal coefficients work.

Thus, in any characteristic, we conclude that , so and the same for . Thus, is a fibration, so a fibre and a relatively ample divisor are linearly independent in the Néron–Severi group, contradicting the assumption .

Remark. In characteristic zero, the Hodge diamonds

cannot occur for any , by essentially the same argument. Indeed, the only thing left to prove is that the image cannot be a surface. If it were, then 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

is realised by , where is a curve of genus . 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).

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 , write for the poset . The full subcategory of on these is denoted , the simplex category. Concretely, it has objects for all , and morphisms

A simplicial set is a functor . This can be described rather concretely using the objects and the face and degeneracy maps between them; see e.g. Tag 0169. The category of simplicial sets is usually denoted , , or (in analogy with cosimplicial sets ).

The representable simplicial set is usually denoted or . Then the Yoneda lemma shows that the functor given by is represented by , i.e.

Definition. The geometric realisation functor is defined as follows: for , the geometric realisation is the standard -simplex

(If no confusion arises, it may also be denoted .) This is functorial in : for a map (equivalently, by the Yoneda lemma, a map ) we get a continuous map by

For an arbitrary simplicial set , write

where the transition map corresponding to a map over is defined via

This is functorial in , and when it coindices with the previous definition because the identity is terminal in the index category.

Remark. In a fancier language, is the left Kan extension of the functor along the Yoneda embedding . (Those of you familiar with presheaves on spaces will recognise the similarity with the definition of for 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 horn in as the union of the images of the maps coming from the face maps for . Since geometric realisation preserves colimits (alternatively, stare at the definitions), we see that the geometric realisation of is obtained in the same way from the maps , so it is the -simplex with its interior and face opposite the vertex removed.

The geometric realisation is a good first approximation for thinking about a simplicial set. However, when thinking about -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 -categories. (More details later, hopefully.)

A strange contractible space

Here’s a strange phenomenon that I ran into when writing a MathOverflow answer a few years ago.

Lemma. Let be a set endowed with the cofinite topology, and assume is path connected. Then is contractible.

The assumption is for example satisfied when , for then any injection is a path from to . Path connectedness of cofinite spaces is related to partitioning the interval into disjoint closed subsets; see the remark below for some bounds on the cardinalities.

Proof. The result is trivial if is finite, for then both are equivalent to . Thus we may assume that is infinite. Choose a path from some to some . This induces a continuous map . Choose a bijection

and extend to a map by and for all . Then is continuous: the preimage of is if , and if , both of which are closed. Thus is a homotopy from to the constant map , hence a contraction.

I would love to see an animation of this contraction as goes from to … I find especially the slightly more direct argument for given here elusive yet somehow strangely visual.

Remark. If is countable (still with the cofinite topology), then is path connected if and only if . In the finite case this is clear (because then is discrete), and in the infinite case this is a result of Sierpiński. See for example this MO answer of Timothy Gowers for an easy argument.

There’s also some study of path connectedness of cofinite topological spaces of cardinality strictly between and , if such cardinalities exist. See this MO question for some results. In particular, it is consistent with ZFC that the smallest cardinality for which is path connected is strictly smaller than .

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 of sets, write for the small category with

and morphisms

for and (where ), with composition induced by composition of maps .

Note that by the Yoneda lemma, this category is isomorphic (not just equivalent!) to , where is the Yoneda embedding. Indeed, are in bijection with natural transformations , and morphisms correspond to a morphism rendering commutative the associated diagram

Example 1. If , then a diagram is a pair of sets with parallel arrows . Then looks like a ‘bipartite preorder’ where every source object has outgoing valence :

Example 2. Given a set , write for the discrete category on , i.e. and

If is itself a discrete category, then is just a collection of sets, and

Remark. Giving a functor is the same thing as giving functors and natural transformations

of functors for all in , such that

for all in (where denotes horizontal composition of natural transformations, as in Tag 003G).

Example 3.  Let be a small category, and consider the diagram given by the source and target maps . Then we have a functor

given on objects by

and on morphisms by

In terms of the remark above, it is given by the functors taking to and the natural inclusion , along with the natural transformations

We can now formulate the main result.

Lemma. Let be a small category. Then the functor of Example 3 is cofinal.

Recall that a functor is cofinal if for all , the comma category is nonemptry and connected. See also Tag 04E6 for a concrete translation of this definition.

Proof. Let . Since , the identity gives the object in , showing nonemptyness. For connectedness, it suffices to connect any (i.e. ) to the identity ) (i.e. ). If , then the commutative diagram

gives a zigzag

of morphisms in connecting to . If instead , we can skip the first step, and the diagram

gives a zigzag

connecting to .

Corollary 1. Let be a small diagram in a category with small products. Then there is a canonical isomorphism

provided that either side exists.

Proof. By the lemma, the functor

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

is an isomorphism if either side exists. But is a category as in Example 1, and it’s easy to see that the limit over a diagram is computed as the equaliser of a pair of arrows between the products.

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 is of the form for some . But it’s fun to move the argument almost entirely away from limits and into the index category.

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

Application of Schur orthogonality

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

1.

2.

Proof. First consider the case . 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.

Here is a trivial consequence:

Corollary. Let be a positive integer, and let . Then

Proof 1. Without loss of generality, has exact order . Set , let , and note that

Part 1 of the lemma gives the result.

Proof 2. Set as before, let be the homomorphism , and the homomorphism . Then part 1 of the lemma does not give the result, but part 2 does.

In fact, the corollary also implies the lemma, because both are true ().

Graph colourings and Hedetniemi’s conjecture II: universal colouring

In my previous post, I stated the recently disproved Hedetniemi’s conjecture on colourings of product graphs (see this post for my conventions on graphs). In the next few posts, I will explain some of the ideas of the proof from an algebraic geometer’s perspective.

Lemma. Let be a graph. Then there exists an -colouring on such that for every graph and every -colouring on , there is a unique morphism such that .

Proof. By this post, we have the adjunction

(1)

In particular, the identity gives an -colouring under this adjunction. If is any other graph, (1) gives a bijection between morphisms and -colourings of , which by naturality of (1) is given by .

Corollary. To prove Hedetniemi’s conjecture, it suffices to treat the ‘universal’ case , for every and every loopless graph .

Proof. Suppose by contradiction that there is a counterexample , i.e. there are loopless graphs and such that

(2)

Then there exists an -colouring , so the lemma gives a map such that . This forces since an -colouring on induces an -colouring on by pullback. Thus, (2) implies

showing that is a counterexample as well.

Corollary. Hedetniemi’s conjecture is equivalent to the statement that for any loopless graph and any , either or admits an -colouring.

Example. By the final example of my previous post and the proof of the first corollary above, the cases are trivially true. We can also check this by hand:

• If does not have a -colouring, then it has an edge. Then has no edges by construction, since has no edges. See also Example 2 of this post.
• If does not have a -colouring, then it has an odd cycle . We need to produce a -colouring on . Choose identifications and with adjacencies . Consider the map

To show this is a graph homomorphism, we must show that for adjacent we have . If two maps are adjacent, then for adjacent we have . Taking shows that , so

since is odd.

The case is treated in [EZS85], which seems to be one of the first places where the internal Hom of graphs appears (in the specific setting of ).

References.

[EZS85] M. El-Zahar and N. Sauer, The chromatic number of the product of two 4-chromatic graphs is 4. Combinatorica 5.2, p. 121–126 (1985).

Graph colourings and Hedetniemi’s conjecture I: statement of conjecture

The past three posts have been building up to the statement of the recently disproved Hedetniemi’s conjecture. I wanted to make an attempt to write about this, because from a first reading the main ideas of the counterexample seemed very familiar to an algebraic geometer. (More about this in a future post, hopefully.)

Definition. A colouring of a loopless graph with colours is a graph homomorphism . The chromatic number of is the smallest positive integer such that admits a colouring with colours.

Note that if has a loop, then it cannot admit a colouring with any number of colours. In the loopless case, a trivial upper bound is , since is a subgraph of the complete graph on .

Example. We have if and only if has no edges (we say that is discrete), and if and only if contains no odd cycles (we say that is bipartite). Indeed, if you try to produce a -colouring by colouring adjacent vertices opposite colours, either this produces a -colouring or you find an odd cycle.

Conjecture (Hedetniemi). Let and be graphs. Then

Remark. Note that : if is a colouring, then the composition is a colouring of , and similarly for . Thus, it remains to rule out with and .

Example. The case where is easy to check:

• If and , then both and have an edge, hence so does . Then .
• If and , then both and contain an odd cycle. If has an -cycle and an -cycle with and odd, then these give morphisms and . Wrapping around (resp. ) times gives morphisms , , hence to the product: . Thus, does not admit a -colouring since doesn’t.

Thus, if , then .