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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThe 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 Borceux's Handbook, I think, discusses flatness without referring to SGA4 and its discussion of ind-objects. I have given links under the 'related concepts' part of both entries, but am not sure of how to express the close links in an accurate and succint way. Can anyone help?

   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 Borceux’s Handbook, I think, discusses flatness without referring to SGA4 and its discussion of ind-objects.

   I have given links under the ’related concepts’ part of both entries, but am not sure of how to express the close links in an accurate and succint way. Can anyone help?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexLet $\mathcal{C}$ be a small category and let $\mathbf{Ind} (\mathcal{C})$ be the free ind-completion (however constructed). Then there is a canonical functor $R : \mathbf{Ind} (\mathcal{C}) \to [\mathcal{C}^{op}, \mathbf{Set}]$ extending the Yoneda embedding $Y : \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]$. The theorem is that $R$ is fully faithful and its essential image is the full subcategory of flat functors $\mathcal{C}^{op} \to \mathbf{Set}$. If we construct $\mathbf{Ind} (\mathcal{C})$ as the category of ind-objects, then $R$ is a kind of "realisation" functor, taking a diagram $X : \mathcal{J} \to \mathcal{C}$ to the filtered colimit $\operatorname{colim}_\mathcal{J} Y X$.

   Let $\mathcal{C}$ be a small category and let $\mathbf{Ind} (\mathcal{C})$ be the free ind-completion (however constructed). Then there is a canonical functor $R : \mathbf{Ind} (\mathcal{C}) \to [\mathcal{C}^{op}, \mathbf{Set}]$ extending the Yoneda embedding $Y : \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]$. The theorem is that $R$ is fully faithful and its essential image is the full subcategory of flat functors $\mathcal{C}^{op} \to \mathbf{Set}$. If we construct $\mathbf{Ind} (\mathcal{C})$ as the category of ind-objects, then $R$ is a kind of “realisation” functor, taking a diagram $X : \mathcal{J} \to \mathcal{C}$ to the filtered colimit $\operatorname{colim}_\mathcal{J} Y X$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI was about to edit the entry, but now Mike has locked it. Notice that at _[[accessible category]]_ the equivalence is stated.

   I was about to edit the entry, but now Mike has locked it.

   Notice that at accessible category the equivalence is stated.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI've added links between the appropriate sections.

   I’ve added links between the appropriate sections.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI gave your new paragraph a numbered Remark-environment and added a sentence: > For more equivalent characterizations see at _[accessible category -- Definition](accessible+category#definition)_.

   I gave your new paragraph a numbered Remark-environment and added a sentence:

   For more equivalent characterizations see at accessible category – Definition.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, 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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, 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.

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeFeb 15th 2014
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThat looks great. Thanks. Like Urs I noticed some need for that 'fine tuning' but as I am using these entries to relearn this stuff, I was not that confident about the fix.

   That looks great. Thanks. Like Urs I noticed some need for that ’fine tuning’ but as I am using these entries to relearn this stuff, I was not that confident about the fix.