Internal Hom

Posted on 31 January 2020 by Remy

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

Posted on 23 January 2020 by Remy

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

Posted on 20 January 2020 by Remy

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.

Lovely little lemmas

Or maybe ‘amusing observations’

Monthly Archives: January 2020

Internal Hom

Limits in the category of graphs

An interesting Noether–Lefschetz phenomenon