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Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits, can it be done?

I've made some naive attempts replace cauchy sequence with sum and taking the limit to a functor $F:\mathbb N^{opp}\to \mathbb R-Mod$ given by:

$X\leftarrow X\times X\leftarrow X\times X\times X\leftarrow ...$

where the arrows are $\nabla\times 1_X\times ...$ sending $(x_1,x_2,x_3,..)\to (x_1+x_2,x_3,...)$

But this feels like its going in the exact opposite direction, which itself is either interesting or me being an idiot.

None the less, I'm not getting anywhere. If anybody knows, please fill me in.

Regards, Tobias

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