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I added to directed colimit the -directed version, for some regular cardinal .
We should maybe also add to directed set the -directed version. What we currently descrribe there is just the -directed version.
Accordingly then I also added to compact object the definition of the variant of -compact objects.
At small object previously it mentioned "-filtered colimits". I now made that read "-directed colimits".
I hope that's right. If not, do we need to beware of the differene?
Well, there is a difference. Which is correct at small object I don't know. But many filtered colimits are not directed.
‘We should maybe also add to directed set the -directed version.’
It's already there. But you could rearrange it, now that you need it to be more prominent.
Wait a second, isn't at least for the EXISTENCE of filtered colimits enough to have the existence of directed?
I'd think so, filtered is more general than directed.
It seems I see authors use either filtered or directed when defining kappa-compact objects, though.
Thanks. I quickly copied that statement to here.
I find the fact that all colimits can be constructed from coproducts and coequalizers more than a convenient coincidence. It's a recipe for constructing colimits (well, you need more details than just the theorem, but the usual proof has these). I think that I see how this works for filtered colimits too. Very nice.
But I guess that you (Urs) mean that one would normally say that a category has all colimits rather than saying that a category has all coproducts and binary coequalisers. So similarly one should say that a category has all filtered colimits (or whatever one wants to say) rather than that it has all directed colimits.
Probably somebody should write filtered colimit. OK, I started it, with material taken from filtered category; I added an Idea section to the latter to make up for its loss.
Thanks for the filtered colimit page. i was thinking about creating that, but never found the energy. There are plenty of links on our pages that read "filtered colimit" but point to filtered category.
I didn't realise that you were doing that! Possibly I have now changed them all.
This is a perfect example of a case where redirects could have helped. If you'd added [[!redirects filtered colimit]]
to filtered category, then all of these links would have worked as soon as I'd created the new page.
To quote myself above, quoting A&R:
a functor preserves k-directed colimits iff it preserves k-filtered ones.
Category theorists say "preserve" where other people apparently say "commutes with".
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