# Categories and Presheaves (Thursday, September 24)

## Regular Functions vs Holomorphic Functions

Recall that we defined the *regular functions* $\OO_X(U)$ on an open set $U\subset X$ an affine variety as the set of functions $\phi: U\to k$ such that $\phi$ is locally a fraction, i.e. for all $p\in U$ there exists a neighborhood of $p$, say $U_p \subset U$, such that $\phi$ restricted to $U_p$ is given by $g_p \over f_p$ for some $f_p, g_p \in A(X)$.

We proved that on a distinguished open set $D(f) = V(f)^c$, we have $\OO_X(D(f)) = A(X)_f$.
An important example was that $\OO_X(X) = A(X)$.

:::{.question}
If $X$ is a variety over $\CC$, does $A(X) = \Hol(X)$? 
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:::{.answer}
The answer is no, since taking $\AA^1/\CC \cong \CC = X$ we obtain $A(X) = \CC[x]$ but for example $e^z \in \Hol(X)$.
On the other hand, if you require that $f\in \Hol(X)$ is meromorphic at $\infty$, i.e. $f({1\over z})$ is meromorphic at zero, then you do get $\CC[z]$.
This is an example of GAGA!

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> Review: what is a category?

> Review: what is a presheaf?