# 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 .

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. hen 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 .

# Internal Hom in the category of graphs

In this earlier post, I described what products in the category of graphs look like. In my previous post, I gave some basic examples of internal Hom. Today we will combine these and describe the internal Hom in the category of graphs.

Definition. Let and be graphs. Then the graph has vertices , and an edge from to if and only if implies (where we allow as usual).

Lemma. If , , and are graphs, then there is a natural isomorphism

In other words, is the internal Hom in the symmetric monoidal category .

Proof. There is a bijection

So it suffices to show that is a graph homomorphism if and only if is. The condition that is a graph homomorphism means that for any , the functions have the property that implies . This is equivalent to for all and all . By the construction of the product graph , this is exactly the condition that is a graph homomorphism.

Because the symmetric monoidal structure on is given by the categorical product, it is customary to refer to the internal as the exponential graph .

Example 1. Let be the discrete graph on a set . Then is the complete graph with loops on the set . Indeed, the condition for two functions to be adjacent is vacuous since has no edges.

In particular, any function is a graph homomorphism. Under the adjunction above, this corresponds to the fact that any function is a graph homomorphism, since is a discrete graph.

Example 2. Conversely, is discrete as soon as has an edge, and complete with loops otherwise. Indeed, the condition

can only be satisfied if , and in that case is true for all and .

In particular, a function is a graph homomorphism if and only if either or has no edges. Under the adjunction above, this corresponds to the fact that a function is a graph homomorphism to if and only if has no edges, which means either or has no edges.

Example 3. Let be the discrete graph on a set with loops at every point. Then is the -fold power of . Indeed, the condition that two functions are adjacent is that for all , which means exactly that for each of the projections .

In particular, graph homomorphisms correspond to giving graph homomorphisms . Under the adjunction above, this corresponds to the fact that a graph homomorphism is the same thing as graph homomorphisms , since is the -fold disjoint union of .

Example 4. Let be the terminal graph consisting of a single point with a loop (note that we used instead for in this earlier post). The observation above that also works the other way around: . Then the adjunction gives

This is actually true in any symmetric monoidal category with internal hom and identity object . We conclude that a function is a graph homomorphism if and only if has a loop at . This is also immediately seen from the definition: has a loop at if and only if implies .

Example 5. Let and be the complete graphs on and vertices respectively. Then has as vertices all -tuples , and an edge from to if and only if when . For example, for we get an edge between and if and only if and .

# Internal Hom

This is an introductory post about some easy examples of internal Hom.

Definition. Let be a symmetric monoidal category, i.e. a category with a functor that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in is a functor

such that is a left adjoint to for any , i.e. there are functorial isomorphisms

Remark. In the easiest examples, we typically think of as ‘upgrading to an object of ‘:

Example. Let be a commutative ring, and let be the category of -modules, with the tensor product. Then with its natural -module structure is an internal Hom, by the usual tensor-Hom adjunction:

The same is true when is the category of -bimodules for a not necessarily commutative ring .

However, we cannot do this for left -modules over a noncommutative ring, because there is no natural -module structure on for left -modules and . In general, the tensor product takes an -bimodule and a -bimodule and produces an -bimodule . Taking gives a way to tensor a right -module with a left -module, but there is no standard way to tensor two left -modules, let alone equip it with the structure of a left -module.

Example. Let . Then is naturally a set, making it into an internal Hom for :

When is the categorical product , the internal (if it exists) is usually called an exponential object, in analogy with the case above.

Example. Another example of exponential objects is from topology. Let be the category of locally compact Hausdorff topological spaces. Then the compact-open topology makes into an internal Hom of topological spaces. (There are mild generalisations of this beyond the compact Hausdorff case, but for an arbitrary topological space the functor does not preserve colimits and hence cannot admit a right adjoint.)

Example. An example of a slightly different nature is chain complexes: let be a commutative ring, and let be the category of cochain complexes

of -modules (meaning each is an -module, and the are -linear maps satisfying ). Homomorphisms are commutative diagrams

and the tensor product is given by the direct sum totalisation of the double complex of componentwise tensor products.

There isn’t a natural way to ‘endow with the structure of a chain complex’, but there is an internal Hom given by

with differentials given by

Then we get for example

since a morphism is given by an element such that , i.e. , meaning that is a morphism of cochain complexes.

Example. The final example for today is presheaves and sheaves. If is a topological space, then the category of abelian sheaves on has an internal Hom given by

with the obvious transition maps for inclusions of open sets. This is usually called the sheaf Hom. A similar statement holds for presheaves.

# Limits in the category of graphs

This is a first post about some categorical properties of graphs (there might be a few more).

Definition. For us, a graph is a pair where is a set and is a collection of subsets of of size or . An element with is called an edge from to , and a singleton is a loop at (or sometimes an edge from to itself). If , it is customary to write and .

A morphism of graphs is a map such that for all . The category of graphs will be denoted , and will be called the forgetful functor.

Example. The complete graph on vertices is the graph where and is the set of -element subsets of . In other words, there is an edge from to if and only if .

Then a morphism is exactly an -colouring of : the condition for forces whenever and are adjacent. Conversely, a morphism to a graph without loops is exactly an -clique in : the condition that has no loops forces for .

Lemma. The category has and the forgetful functor preserves all small limits.

Proof. Let be a functor from a small category , and let be the limit of the underlying sets, with cone maps . We will equip with a graph structure such that the maps for are morphisms and then show that the constructed is a limit of in .

To equip with an edge set , simply let be the set of of size or such that for all . Then this clearly makes into a graph such that the are graph morphisms for all . Moreover, these maps make into the limit cone over : for any other cone , the underlying maps factor uniquely through by the definition of , and the construction of shows that is actually a morphism of graphs .

Remark. Note however that does not create limits. On top of the construction above, this would mean that there is a unique graph structure on such that is a cone over . However, there are many such structures on , because we can remove edges all we want (on the same vertex set ).

Example. As an example, we explicitly describe the product of two graphs and : by the lemma its vertex set is . The ‘largest graph structure’ such that both projections and are graph morphisms is given by if and only if and and . This corresponds to the structure found in the proof of the lemma.

For a very concrete example, note that the product of two intervals/edges is a disjoint union of two intervals, corresponding to the diagonals in . This is the local model to keep in mind.

The literature also contains other types of product graphs, which all have the underlying set . Some authors use the notation for the categorical product or tensor product we described. The Cartesian product is defined by , so that the product of two intervals is a box. The strong product is the union of the two, so that the product of two intervals is a box with diagonals. There are numerous other notions of products of graphs.

Remark. Analogously, we can also show that has and preserves all small colimits: just equip the set-theoretic colimit with the edges coming from one of the graphs in the diagram.

Example. For a concrete example of a colimit, let’s carry out an edge contraction. Let be a graph, and let be an edge. The only way to contract in our category is to create a loop: let be the one-point graph without edges, and let be the maps sending to and respectively. Then the coequaliser of the parallel pair is the graph whose vertices are , where is the equivalence relation if and only if or , and whose edges are exactly the images of edges in . In particular, the edge gives a loop at the image .

Remark. Note that the preservation of limits also follows since has a left adjoint: to a set we can associate the discrete graph with vertex set and no edges. Then a morphism to any graph is just a set map .

Similarly, the complete graph with loops gives a right adjoint to , showing that all colimits that exist in must be preserved by . However, these considerations do not actually tell us which limits or colimits exist.

# An interesting Noether–Lefschetz phenomenon

The classical Noether–Lefschetz theorem is the following:

Theorem. Let be a very general smooth surface of degree . Then the natural map is an isomorphism.

If is a smooth proper family over some base (usually of finite type over a field), then a property holds for a very general if there exists a countable intersection of nonempty Zariski opens such that holds for for all .

In general, Hilbert scheme arguments show that the locus where the Picard rank is ‘bigger than expected’ is a countable union of closed subvarieties of (the Noether–Lefschetz loci), but it could be the case that this actually happens everywhere (i.e. ). The hard part of the Noether–Lefschetz theorem is that the jumping loci are strict subvarieties of the full space of degree hypersurfaces.

If is a family of varieties over an uncountable field , then there always exists a very general member with . But over countable fields, very general elements might not exist, because it is possible that even when .

The following interesting phenomenon was brought to my attention by Daniel Bragg (if I recall correctly):

Example. Let (the algebraic closure of the field of elements, but the bar is not so visible in MathJax), let (or some scheme covering it if that makes you happier) with universal family of elliptic curves, and let be the family of product abelian surfaces . Then the locus

is exactly the set of -points (so it misses only the generic point).

Indeed, , and every elliptic curve over has . But the generic elliptic curve only has .

We see that the Noether–Lefschetz loci might cover all -points without covering , even in very natural situations.

# Scales containing every interval

This is a maths/music crossover post, inspired by fidgeting around with diatonic chords containing no thirds. The general lemma is the following (see also the examples below):

Lemma. Let be a positive integer, and a subset containing elements. Then every occurs as a difference between two elements .

Proof. Consider the translate . Since both and have size , they have an element in common. If , then for some , so .

Here are some applications to music theory:

Example 1 (scales containing every chromatic interval). Any scale consisting of at least out of the available chromatic notes contains every interval. Indeed, , so the lemma shows that every difference between two elements of the scale occurs.

The above proof in this case can be rephrased as follows: if we want to construct a minor third (which is semitones) in our scale , we consider the scale and its transpose by a minor third. Because , there must be an overlap somewhere, corresponding to an interval of a minor third in our scale.

In fact, this shows that our scale must contain two minor thirds, since you need at least overlaps to get from down to . For example, the C major scale contains two minor seconds (B to C and E to F), at least two major thirds (C to E and G to B), and two tritones (B to F and F to B).

The closer the original key is to its transpose, the more overlaps there are between them. For example, there are major fifths in C major, since C major and G major overlap at 6 notes. Conversely, if an interval occurs many times in a key , that means that the transposition of by the interval has many notes in common with the old key . (Exercise: make precise the relationship between intervals occurring ‘many times’ and transpositions having ‘many notes in common’.)

We see that this argument is insensitive to enharmonic equivalence: it does not distinguish between a diminished fifth and an augmented fourth. Similarly, a harmonic minor scale contains both a minor third and an augmented second, which this argument does not distinguish.

Remark. We note that the result is sharp: the whole-tone scales and have size , but only contain the even intervals (major second, major third, tritone, minor sixth, and minor seventh).

Example 2 (harmonies containing every diatonic interval). Any cluster of notes in a major or minor scale contains every diatonic interval. Indeed, modelling the scale as integers modulo , we observe that , so the lemma above shows that every diatonic interval occurs at least once.

For example, a seventh chord contains the notes¹ of the key. It contains a second between and , a third between and , a fourth between and , etcetera.

Thus, the largest harmony avoiding all (major or minor) thirds is a triad. In fact, it’s pretty easy to see that such a harmony must be a diatonic transposition of the sus4 (or sus2, which is an inversion) harmony. But these chords may contain a tritone, like the chord B-F-G in C major.

Example 3. If you work with your favourite -tone tuning system, then any scale consisting of at least of those notes contains every chromatic interval available in this tuning.