Relations between two definitions of non-archimedean analytic spaces

Question

I begin to learn some non-archimedean geometry recently, and find that there are two different definitions of analytic spaces in the literature.

Let us fix a non-archimedean complete valuation field $k$ (the valuation is not necessarily non-trivial) The first definition is as in TEMKIN: An analytic space over $k$ consists of a locally Hausdorff topological space $X$, a net $\tau$ on $X$(regarded as a category), and a functor from $\tau$ to the category of $k$-affinoid spaces sending morphisms to affinoid domain embeddings, such that the underlying topological spaces of $U\subset X$ and $\tau(U)$ are naturally identified.

The second definition I found is in Berkovich's original book: A $k$-analytic space is a locally ringed space $X$ with an equivalence class of collections of pairs $(U_i,\phi_i)$, where $U_i\subset X$ is open, $\phi_i$ is an open immersion of $U_i$ into a $k$-affinoid space subject to the usual compatibility conditions as in the definition of a mainfold.

I'm not pretty sure what is the connection between these two definitions, I guess that the first definition axiomatizes analytic domains in the second definition, but I haven't been able to make this precise.

Answer 1

Let me give more a few more details than in nfdc23's comment. The most general definition of a analytic space is the one that you find in Berkovich's IHES paper. One requirement is that for every point $x$ and every open subset $U$ containing $x$, there exist finitely many affinoid domains $V_1,\dotsc,V_n$ of $U$ that all contain $x$ and whose union is a neighborhood of $x$. (The way I picture it is that of a brick wall, where some points lie on the edges of several bricks.)

You could require something stronger, namely that every point has a basis of affinoid neighborhoods (i.e. you can always take $n=1$ above). Such a space is called a good space and this is the class of spaces that is defined in the red book.

To see a typical example of a non-good space, consider the bi-disk $\mathcal{M}(k\{T,U\})$ and, inside it, the space $X$ that is the union of the two affinoid domains defined by $\{|T|=1\}$ and $\{|U|=1\}$. One can prove that the Gauss point has no affinoid neighborhood in $X$. On the other hand, it is important to have these kinds of spaces in the theory. For instance, this one appear as the generic fiber of the (non-affine) formal scheme $Spf(k^\circ\{T,U\})\setminus \{(0,0)\}$.