Internal Hom

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

Definition. Let (\mathscr C, \otimes) be a symmetric monoidal category, i.e. a category \mathscr C with a functor \otimes \colon \mathscr C \times \mathscr C \to \mathscr C that is associative, unital, and commutative up to natural isomorphism. Then an internal Hom in \mathscr C is a functor

    \[\mathbf{Hom}(-,-) \colon \mathscr C\op \times \mathscr C \to \mathscr C\]

such that -\otimes Y is a left adjoint to \mathbf{Hom}(Y,-) for any Y \in \mathscr C, i.e. there are functorial isomorphisms

    \[\operatorname{Hom}(X \otimes Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]

Remark. In the easiest examples, we typically think of \mathbf{Hom}(Y,Z) as ‘upgrading \operatorname{Hom}(Y,Z) to an object of \mathscr C‘:

Example. Let R be a commutative ring, and let \mathscr C = \mathbf{Mod}_R be the category of R-modules, with \otimes the tensor product. Then \mathbf{Hom}(M,N) = \operatorname{Hom}_R(M,N) with its natural R-module structure is an internal Hom, by the usual tensor-Hom adjunction:

    \[\operatorname{Hom}_R(M \otimes_R N, K) \cong \operatorname{Hom}_R(M, \mathbf{Hom}(N, K)).\]

The same is true when \mathscr C =\!\ _R\mathbf{Mod}_R is the category of (R,R)-bimodules for a not necessarily commutative ring R.

However, we cannot do this for left R-modules over a noncommutative ring, because there is no natural R-module structure on \operatorname{Hom}_R(M,N) for left R-modules M and N. In general, the tensor product takes an (A,B)-bimodule M and a (B,C)-bimodule N and produces an (A,C)-bimodule M \otimes_B N. Taking A = C = \mathbf Z gives a way to tensor a right R-module with a left R-module, but there is no standard way to tensor two left R-modules, let alone equip it with the structure of a left R-module.

Example. Let \mathscr C = \mathbf{Set}. Then \mathbf{Hom}(X,Y) = \operatorname{Hom}(X,Y) = Y^X is naturally a set, making it into an internal Hom for (\mathscr C, \times):

    \[\operatorname{Hom}(X \times Y, Z) \stackrel\sim\to \operatorname{Hom}(X, \mathbf{Hom}(Y,Z)).\]

When \otimes is the categorical product \times, the internal \mathbf{Hom}(X,Y) (if it exists) is usually called an exponential object, in analogy with the case \mathscr C = \mathbf{Set} above.

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

Example. An example of a slightly different nature is chain complexes: let R be a commutative ring, and let \mathscr C = \mathbf{Ch}(\mathbf{Mod}_R) be the category of cochain complexes

    \[\ldots \to C^{i-1} \to C^i \to C^{i+1} \to \ldots\]

of R-modules (meaning each C^i is an R-module, and the d^i \colon C^i \to C^{i+1} are R-linear maps satisfying d \circ d = 0). Homomorphisms f \colon C \to D are commutative diagrams

    \[\begin{array}{ccccccc}\ldots & \to & C^i & \to & C^{i+1} & \to & \ldots \\ & & \!\!\!\!\! f^i\downarrow & & \downarrow f^{i+1}\!\!\!\!\!\!\! & & \\ \ldots & \to & D^i & \to & D^{i+1} & \to & \ldots,\!\!\end{array}\]

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 \operatorname{Hom}(C, D) with the structure of a chain complex’, but there is an internal Hom given by

    \[\mathbf{Hom}(C, D)^i = \prod_{m \in \mathbf Z} \operatorname{Hom}(C_m, D_{m+i}),\]

with differentials given by

    \[d^if = d_D f - (-1)^i f d_C.\]

Then we get for example

    \[\operatorname{Hom}(R[0], \mathbf{Hom}(C, D)) \cong \operatorname{Hom}(C, D),\]

since a morphism R[0] \to \mathbf{Hom}(C, D) is given by an element f \in \mathbf{Hom}(C, D)^0 such that df = 0, i.e. d_Df = f d_C, meaning that f is a morphism of cochain complexes.

Example. The final example for today is presheaves and sheaves. If X is a topological space, then the category \mathbf{Ab}(X) of abelian sheaves on X has an internal Hom given by

    \[\mathbf{Hom}(\mathscr F, \mathscr G)(U) = \operatorname{Hom}(\mathscr F|_U, \mathscr G|_U),\]

with the obvious transition maps for inclusions V \subseteq U 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 G = (V, E) where V is a set and E \subseteq \mathcal P(V) is a collection of subsets of V of size 1 or 2. An element \{x,y\} \in E with x \neq y is called an edge from x to y, and a singleton \{x\} \in E is a loop at x (or sometimes an edge from x to itself). If G = (V,E), it is customary to write V(G) = V and E(G) = E.

A morphism of graphs f \colon G \to H is a map f \colon V(G) \to V(H) such that f(e) \in E(H) for all e \in E(G). The category of graphs will be denoted \mathbf{Grph}, and V \colon \mathbf{Grph} \to \mathbf{Set} will be called the forgetful functor.

Example. The complete graph K_n on n vertices is the graph (V,E) where V = \{1,\ldots,n\} and E = {V \choose 2} is the set of 2-element subsets of V. In other words, there is an edge from x to y if and only if x \neq y.

Then a morphism G \to K_n is exactly an n-colouring of G: the condition f(e) \in E(K_n) for e \in E(G) forces f(x) \neq f(y) whenever x and y are adjacent. Conversely, a morphism f \colon K_n \to G to a graph G without loops is exactly an n-clique in G: the condition that G has no loops forces f(x) \neq f(y) for x \neq y.

Lemma. The category \mathbf{Grph} has and the forgetful functor V \colon \mathbf{Grph} \to \mathbf{Set} preserves all small limits.

Proof. Let D \colon \mathscr J \to \mathbf{Grph} be a functor from a small category \mathscr J, and let V = \lim V \circ D be the limit of the underlying sets, with cone maps f_j \colon V \to V(D(j)). We will equip V with a graph structure G = (V,E) such that the maps G \to D(j) for j \in \mathscr J are morphisms and then show that the constructed G is a limit of D in \mathbf{Grph}.

To equip V with an edge set E, simply let E be the set of e \subseteq \mathcal P(V) of size 1 or 2 such that f_j(e) \in E(D(j)) for all j \in \mathscr J. Then this clearly makes G = (V,E) into a graph such that the f_j \colon G \to D(j) are graph morphisms for all j \in \mathscr J. Moreover, these maps make G into the limit cone over D: for any other cone g_i \colon H \to D(j), the underlying maps V(g_i) \colon V(H) \to V(D(j)) factor uniquely through g \colon V(H) \to V by the definition of V, and the construction of E(G) shows that g \colon V(H) \to V is actually a morphism of graphs g \colon H \to G. \qedsymbol

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

Example. As an example, we explicitly describe the product G \times H of two graphs G and H: by the lemma its vertex set is V(G \times H) = V(G) \times V(H). The ‘largest graph structure’ such that both projections p \colon G \times H \to G and q \colon G \times H \to H are graph morphisms is given by e \in E(G \times H) if and only if |e| \in \{1,2\} and p(e) \in E(G) and q(e) \in E(H). 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 G = H = K_2 is a disjoint union of two intervals, corresponding to the diagonals in \{1,2\} \times \{1,2\}. This is the local model to keep in mind.

The literature also contains other types of product graphs, which all have the underlying set V(G) \times V(H). Some authors use the notation G \times H for the categorical product or tensor product we described. The Cartesian product G \square H is defined by E(G \square H) = (E(G) \times V(H)) \cup (V(G) \times E(H)), so that the product of two intervals is a box. The strong product G \boxtimes H 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 \mathbf{Grph} has and V 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 G be a graph, and let e = \{x,y\} be an edge. The only way to contract e in our category is to create a loop: let * be the one-point graph without edges, and let f_x, f_y \colon * \to G be the maps sending * to x and y respectively. Then the coequaliser of the parallel pair f_x, f_y \colon * \rightrightarrows G is the graph H whose vertices are V(G)/\sim, where \sim is the equivalence relation a \sim b if and only if a = b or \{a,b\} = \{x,y\}, and whose edges are exactly the images of edges in G. In particular, the edge \{x,y\} gives a loop at the image z = [x] = [y] \in V(H).

Remark. Note that the preservation of limits also follows since V has a left adjoint: to a set S we can associate the discrete graph S^{\operatorname{disc}} with vertex set S and no edges. Then a morphism S^{\operatorname{disc}} \to G to any graph G is just a set map S \to V(G).

Similarly, the complete graph with loops gives a right adjoint to V, showing that all colimits that exist in \mathbf{Grph} must be preserved by V. 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 X \subseteq \mathbf P^3_{\mathbf C} be a very general smooth surface of degree d \geq 4. Then the natural map \Pic(\mathbf P^3) \to \Pic(X) is an isomorphism.

If \mathscr X \to S is a smooth proper family over some base S (usually of finite type over a field), then a property \mathcal P holds for a very general X = \mathscr X_s if there exists a countable intersection U = \bigcap_i U_i \subseteq S of nonempty Zariski opens U_i such that \mathcal P holds for X_s for all s \in U.

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

If \mathscr X \to S is a family of varieties over an uncountable field k, then there always exists a very general member \mathscr X_s with s \in S(k). But over countable fields, very general elements might not exist, because it is possible that \bigcup Z_i(k) = S(k) even when \bigcup Z_i \neq S.

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

Example. Let k = \bar{\mathbf F}_p (the algebraic closure of the field of p elements, but the bar is not so visible in MathJax), let S = \mathcal A_1 = \mathcal M_{1,1} (or some scheme covering it if that makes you happier) with universal family \mathscr E \to S of elliptic curves, and let \mathscr X = \mathscr E \times_S \mathscr E be the family of product abelian surfaces E \times E. Then the locus

    \[NL(S) = \left\{s \in S\ \big| \ \operatorname{rk} \Pic(\mathscr X_s) > 3\right\}\]

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

Indeed, \Pic(E \times E) \cong \Pic(E) \times \Pic(E) \times \End(E), and every elliptic curve E over k has \operatorname{rk} \End(E) \geq 2. But the generic elliptic curve only has \End(E) = \mathbf Z. \qedsymbol

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