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The entries on ind-object and flat functor both say that a certain category is the free cocompletion under filtered limits. There is however no link given between the two ideas. Clearly the two notions are closely linked if coming from different areas and intiuitions, and the discussion in Borceuxβs Handbook, I think, discusses flatness without referring to SGA4 and its discussion of ind-objects.
I have given links under the βrelated conceptsβ part of both entries, but am not sure of how to express the close links in an accurate and succint way. Can anyone help?
Let π be a small category and let Ind(π) be the free ind-completion (however constructed). Then there is a canonical functor R:Ind(π)β[πop,Set] extending the Yoneda embedding Y:πβ[πop,Set]. The theorem is that R is fully faithful and its essential image is the full subcategory of flat functors πopβSet. If we construct Ind(π) as the category of ind-objects, then R is a kind of βrealisationβ functor, taking a diagram X:π₯βπ to the filtered colimit colimπ₯YX.
I was about to edit the entry, but now Mike has locked it.
Notice that at accessible category the equivalence is stated.
Iβve added links between the appropriate sections.
I gave your new paragraph a numbered Remark-environment and added a sentence:
For more equivalent characterizations see at accessible category β Definition.
Hm, since I was just editing compactly generated (infinity,1)-category: at accessible category maybe we want to be more fine-tuned with the two possibly different cardinality bounds.
Hm, since I was just editing compactly generated (infinity,1)-category: at accessible category maybe we want to be more fine-tuned with the two possibly different cardinality bounds.
That looks great. Thanks. Like Urs I noticed some need for that βfine tuningβ but as I am using these entries to relearn this stuff, I was not that confident about the fix.
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