Sheafification is needed in limits and colimits of condensed abelian groups? If I have a functor $T: i \mapsto T\_i$ from an index category to condensed abelian groups the limit and colimit of this functor are just $S \mapsto \lim\_i T\_i (S)$ and $S \mapsto \text{colim}\_i T\_i (S)$ or sheafification is needed for it to be a condensed abelian group?


In particular if I have a map $\phi: T \rightarrow Q$ of condensed sets the kernel and the cokernel are just $S \mapsto \ker \phi\_S$ and $S \mapsto \text{coker} \phi\_S$?


Thank you!



Limits and colimits in the category of condensed abelian groups
I'll ignore the set-theoretic issues since I don't understand them well enough to say anything about them.


As with any site, the limits (and in particular, the kernel) of sheaves may be computed pointwise. On the other hand, the colomits are usually not computed pointwise (cokernels included). For example, an exact sequence of condensed Abelian groups $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $
gives a long exact sequence of cohomology
$$
0 \rightarrow A(S) \rightarrow B(S) \rightarrow C(S) \rightarrow H^1(S,A) \rightarrow \dotsc
$$


This means that the cokernels should be given by the pointwise quotient if the map $H^1(S,A) \rightarrow H^1(S,B)$ is injective (this happens, for example, when $H^1(S,A)=0$).


