# 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.