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A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).
4
votes
Accepted
Solid tensor product of pro-discrete space with Laurent series
This is not true in general, the most important observation being that it fails already when $V$ is discrete. In that case $V\otimes^{\blacksquare} \mathbb Z((T))$ is just the usual algebraic tensor p …
9
votes
Accepted
Countable sum $\bigoplus_{n=0}^\infty\mathbb Z_p$ as a topological group
The sum $M=\bigoplus_{\mathbb N} \mathbb Z_p$ is not first-countable, but it is Cauchy complete. More precisely, $M$ maps isomorphically to $\varprojlim_{U\subset M} M/U$ where $U$ runs over open subg …
16
votes
Accepted
6-functor formalism for topological stacks
Such computations are indeed not formal; and virtually impossible if one tries to compute directly from the definitions.
The starting point here is that $f$ is actually cohomologically smooth. More pr …
7
votes
Structure of a profinite group as a condensed set with an action of an open subgroup
Yes, that is true. I'm not sure I'm explaining the step that's confusing you, but you are asking about a special case of the statement that the functor $S\mapsto\underline{S}$ from profinite sets to c …