Which (co)limits exist in the category of adic spaces ? Also, can we impose adjectives such as "noetherian" or "quasi-compact", etc., to get more (co)limits ? I know that finite fibre products of affinoids exist within the category, but I have not been able to find references indicating that other (co)limits also exist, and I am not yet familiar enough with adic spaces to be able to test out a few examples on my own to see if other (co)limits should exist.
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1$\begingroup$ The Monsky-Washnitzer and the overconvergent realizations by Vezzani mentions in Appendix A that inverse limits in the category of adic spaces don't exist, but also that there's a notion of "being similar to an inverse limit". $\endgroup$– EmilyCommented Sep 2, 2021 at 0:20
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1$\begingroup$ @Emily What about (countable) filtered colimits ? I've seen a few, particularly the adic affine line, which is isomorphic to the "union" of adic closed discs with increasing radii, but I can not tell if say, only colimits of a filtered diagram of closed immersions would remain inside the category of adic spaces, or if there are more general colimits (incidentally, this also raises the question of how closed immersions should be defined categorically). $\endgroup$– Dat Minh HaCommented Sep 3, 2021 at 1:12
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1$\begingroup$ I've found however something possibly useful: there are some references mentioning that the situation for fibre products is actually quite delicate. Firstly, there's this basic positive result for finite type morphisms: Huber's Étale Cohomology of Rigid Analytic Varieties and Adic Spaces states in Prop. 1.2.2 that the fibre product of two morphisms of adic spaces $f\colon X\to Z$ and $g\colon Y\to Z$ in the category of adic spaces exists if 1) $f$ is locally of finite type or 2) $f$ is locally of weakly finite type and $g$ is adic. $\endgroup$– EmilyCommented Sep 3, 2021 at 3:41
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1$\begingroup$ Then, Kedlaya–Liu's Relative p-adic Hodge theory: Foundations reads in Remark 8.2.5: The categories of locally v-ringed spaces and preadic spaces admit fibred products. However, it is unknown whether the fibred product of adic spaces (over an adic space) is again an adic space. A counterexample would necessarily involve nonnoetherian Banach rings thanks to Proposition 2.4.16; on the other hand, the example of perfectoid spaces (§8.3) shows that failure of the noetherian property alone is not sufficient. $\endgroup$– EmilyCommented Sep 3, 2021 at 3:41
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1$\begingroup$ @Emily Yes my assertion that general finite pullbacks exist in the category of adic spaces was a mistake :p Finite pullbacks of affinoids, however, do exist (in particular, one has $\mathbf{Spa}(B, B^+) \times_{\mathbf{Spa}(A, A^+)} \mathbf{Spa}(C, C^+) \cong \mathbf{Spa}\left(B \otimes_A C, B^+ \otimes_{A^+} C^+\right)^{\wedge}$, and there is also a terminal object, namely $\mathbf{Spa}(\mathbb{Z}, \mathbb{Z})$, and so finite products of affinoids also exist ... $\endgroup$– Dat Minh HaCommented Sep 3, 2021 at 4:01
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