Archive for April 2023

Yoneda Lemma

Posted on Saturday, April 8, 2023

Let us take a look at Yoneda Lemma, which might be the most trivial yet the hardest part of Category theory (and algebraic geometry). I would not be drawing any commutative diagrams.


Take a (small locally presumably poset) category $\mathcal{C}$ and hom-functors $h$ on it to ${\bf Set}^C$. So if we have a set of morphism $mor(A,B)$ ($\pi \colon A \rightarrow B$) for $A,B \in \mathcal{C}$, I can construct a functor to ${\bf Set}^C$ out of set of morphism which I write as $H(A, B)$. One does this for every object inside $\mathcal{C}$; in this way, we get many sets of morphisms to form $H(A, X)$. We now find the normal (representation) isomorphism of this functor
$$\xi \colon F \rightarrow Hom(A,X)$$
and this means that an object $A$ is determined up to isomorphism by the pair $(\xi, F)$. We can also say $F$ is the $Hom(Hom(A,X))$. 

Yoneda Lemma states that any information about the local category is encoded in ${\bf Set}^C$. The set of the morphism becomes the objects for ${\bf Set}^C$, and morphism is given by the natural representation of the functor. So any functor in $\mathcal{C}$ can be sent to its functor category ${\bf Set}^C$, which sends $A$ to $h$. Note that we did not say if $h$ is a covariant or contravariant functor, the result is the same for either.

The philosophy of Yoneda Lemma is also encaptured in this video, essentially meaning why only one view is wrong. Another good exercise is to realize how this is a universal property and why taking maps to and from $A$ is important to understand a category.

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Bochner's Tube Theorem

Posted on Sunday, April 2, 2023

Let us say we have an analytic function $f(z)$ where $z \in \mathbb{C}$ defined in a tube $T$

$$T = \{ z \in \mathbb{C}, z = a+ib, b \in \mathcal{C}, a \in \mathbb{R}^n \}$$
where $\mathcal{C}$ is a convex cone at the origin. Given this, we can prove that some $f'(z)$ analytic continuation of $f(z)$ is defined in a similar tube. For this, we say that there exists a connected domain $G \subset \mathcal{R}^n$, which coincides with the boundary values of  $f(z)$ and $f'(z)$. Then it implies that $f(z)$ and $f'(z)$ are the analytical continuations of each other and are analytic around the domain $G$. This is also known as the edge of the wedge problem.

Now we state the classical tube theorem. We also would make use of Malgrange–Zerner theorem. 

Theorem 1. For every connected domain $G \subset \mathbb{R}^n$, there exists a holomorphic envelope $H(G)$ which contains $G$ as its subdomain.

Theorem 2 (Tube Theorem). For every connected domain $G \subset \mathbb{R}^n$, there is a tube given by 
$$T(G) \{ z \in \mathbb{C}^n, Im(z) \in G \}$$
then the holomorphic envelope of the tube $T(G)$ is given by
$$H(T(G)) = T ( Co\ G)$$
where $Co\ G$ is the convex hull of $G$.

This tube theorem (and generalizations like the double cone theorem and Dyson's theorem) provide more insightful results in QFT. See Borchers1961. Timelike tube theorem can also be seen as a quantum generalization of Holmgren's uniqueness theorem, which also deals with analyticity. For more details on the timelike tube theorem, see Witten and Strohmaier & Witten.

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