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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 4th 2010
   • (edited Jun 4th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexThe definition over at [[accessible functor]] doesn't agree with the one I read earlier today. Neither the domain nor codomain has to be accessible. We simply require that the domain has all k-filtered colimits and that F preserves them (according to Cisinski Ast308 definition 1.2.2). Is this usage nonstandard? Can I add a note that "Some authors don't require that the domain and codomain be accessible categories." Perhaps we could come up with the following definition: A category C is weakly k-accessible if it has all k-filtered colimits? I don't know if the name would be confusing or not.

   The definition over at accessible functor doesn’t agree with the one I read earlier today. Neither the domain nor codomain has to be accessible. We simply require that the domain has all k-filtered colimits and that F preserves them (according to Cisinski Ast308 definition 1.2.2). Is this usage nonstandard? Can I add a note that “Some authors don’t require that the domain and codomain be accessible categories.” Perhaps we could come up with the following definition: A category C is weakly k-accessible if it has all k-filtered colimits? I don’t know if the name would be confusing or not.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 4th 2010
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   Author: Urs
   Format: MarkdownItexYes, I think it's standard that this is not standard. :-) Compare also HTT, remark 5.4.2.6. > Can I add a note that "Some authors don't require that the domain and codomain be accessible categories." I suppose it would be good if you did that.

   Yes, I think it’s standard that this is not standard. :-) Compare also HTT, remark 5.4.2.6.

   Can I add a note that “Some authors don’t require that the domain and codomain be accessible categories.”

   I suppose it would be good if you did that.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 4th 2010
   • (edited Jun 4th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexAh, and Cisinski doesn't require that k be regular. Why should we require that?

   Ah, and Cisinski doesn’t require that k be regular. Why should we require that?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJun 5th 2010
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   Author: Mike Shulman
   Format: MarkdownItexI think I remember that if $\kappa$ is not regular, then $\kappa$-filteredness more or less reduces to $\lambda$-filteredness for some $\lambda$ which is regular, so that regularity not really an assumption but a WLOG. Maybe $\lambda = cf(\kappa)$? I can't remember exactly how this goes, though, and I could be remembering wrong. All the applications I've ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

   I think I remember that if $\kappa$ is not regular, then $\kappa$-filteredness more or less reduces to $\lambda$-filteredness for some $\lambda$ which is regular, so that regularity not really an assumption but a WLOG. Maybe $\lambda = cf(\kappa)$? I can’t remember exactly how this goes, though, and I could be remembering wrong.

   All the applications I’ve ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

5. • CommentRowNumber5.
   • CommentAuthorHarry Gindi
   • CommentTimeJun 5th 2010
   • (edited Jun 5th 2010)
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   Author: Harry Gindi
   Format: MarkdownItex> All the applications I've ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories? To be honest, I don't know. However, he never defines accessible categories in Ast308. What he does define is the "accessible part" (my term, not his) $Acc_\kappa(\mathcal{C})$ to be the full subcategory of $\kappa$-accessible presheaves.

   All the applications I’ve ever seen of accessible functors have been between accessible categories, although of course the bare definition makes sense more generally. Does Cisinski have a reason to consider accessible functors between non-accessible categories?

   To be honest, I don’t know. However, he never defines accessible categories in Ast308. What he does define is the “accessible part” (my term, not his) $Acc_\kappa(\mathcal{C})$ to be the full subcategory of $\kappa$-accessible presheaves.