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An abelian group object in the category of condensed sets.
The category of condensed abelian groups enjoys excellent categorical properties for homological algebra:
It is an abelian category that admits all small limits and colimits;
In this category, filtered colimits and infinite products are exact. The latter property is rather rare.
It has enough compact projective objects: free condensed abelian groups on extremally disconnected compact Hausdorff topological spaces generate all condensed abelian groups under small colimits and their corepresentable functors reflect isomorphisms;
The previous property implies that condensed abelian groups have the same exactness properties as the category of abelian groups.
I have added hyperlinking to “exact” and to “preserves isomorphisms”.
What exactly is being fixed here? The two definitions are obviously equivalent.
From the edit one sees that Anonymous is worried that
might not be equivalent to
However, the definition offered at condensed object looks like it makes this true by fiat. The actual size issue that jbian is cautioning against in that comment is not being addressed there either.
This makes me come back to asking where this and the other condensed entries are meant to be headed:
If they are just meant to record what has already been defined in the literature, then add the reference.
If they are meant to develop uncharted territory, then make this clear in the text.
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