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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeAug 30th 2013
   • (edited Aug 30th 2013)
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   Author: Karol Szumiło
   Format: MarkdownItexIt is well known that small categories $J$ such all $J$-indexed colimits commute with finite limits in the category of sets are precisely the filtered categories. If we replace "finite limits" with "finite products" we get "sifted" instead of "filtered". What happens if we replace it with "pullbacks"? Obviously, an example of such a non-filtered category is $J = \varnothing$. More generally, coproducts of filtered categories are OK. Are there some more interesting examples? Is there a complete characterization?

   It is well known that small categories $J$ such all $J$-indexed colimits commute with finite limits in the category of sets are precisely the filtered categories. If we replace “finite limits” with “finite products” we get “sifted” instead of “filtered”. What happens if we replace it with “pullbacks”? Obviously, an example of such a non-filtered category is $J = \varnothing$. More generally, coproducts of filtered categories are OK. Are there some more interesting examples? Is there a complete characterization?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeAug 30th 2013
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   Author: Zhen Lin
   Format: MarkdownItexYou might want to look at [Adámek, Borceux, Lack, Rosický, _A classification of accessible categories_]. First notice that the class of limits that can be computed using pullbacks includes the class of finite connected limits, so it is the same to ask your question with "finite connected limits" replacing "pullbacks". The class of all $J$ such that $J$-colimits preserve finite connected colimits is precisely the class of coproducts of filtered categories: see Example 1.3(iv). The problem with looking at just pullbacks is that it is not a "sound doctrine": see Example 2.3(vii).

   You might want to look at [Adámek, Borceux, Lack, Rosický, A classification of accessible categories]. First notice that the class of limits that can be computed using pullbacks includes the class of finite connected limits, so it is the same to ask your question with “finite connected limits” replacing “pullbacks”. The class of all $J$ such that $J$-colimits preserve finite connected colimits is precisely the class of coproducts of filtered categories: see Example 1.3(iv). The problem with looking at just pullbacks is that it is not a “sound doctrine”: see Example 2.3(vii).

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 2nd 2013
   • (edited Sep 2nd 2013)
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   Author: Karol Szumiło
   Format: MarkdownItexThanks, that's a really useful reference. Let me see whether I understand the conclusions. In the notation of the paper my question is "which categories are PB-filtered?" Clearly, PB (the doctrine of pullbacks) is contained in FINCL (finite connected categories). FINCL-filtered categories are exactly coproducts of (classical) filtered categories and they are in particular PB-filtered. If PB were sound, then we could conclude that the converse holds too. So far so good, but now comes a part where I get a little lost: doesn't Remark 2.7 essentially say that the converse implication holds if and only if PB is sound? And if so, wouldn't it be possible to construct an example of a PB-filtered category that is not a coproduct of filtered ones? For the record: the reason I'm asking is that I suspect that sharp maps of simplicial sets might be closed under PB-filtered colimits (though I don't know for sure). If PB-filtered categories were exactly coproducts of filtered ones this would be nothing new, but otherwise it might be interesting. In that case I'd like to get my hands on an explicit example to try and test my suspicion.

   Thanks, that’s a really useful reference. Let me see whether I understand the conclusions. In the notation of the paper my question is “which categories are PB-filtered?” Clearly, PB (the doctrine of pullbacks) is contained in FINCL (finite connected categories). FINCL-filtered categories are exactly coproducts of (classical) filtered categories and they are in particular PB-filtered. If PB were sound, then we could conclude that the converse holds too. So far so good, but now comes a part where I get a little lost: doesn’t Remark 2.7 essentially say that the converse implication holds if and only if PB is sound? And if so, wouldn’t it be possible to construct an example of a PB-filtered category that is not a coproduct of filtered ones?

   For the record: the reason I’m asking is that I suspect that sharp maps of simplicial sets might be closed under PB-filtered colimits (though I don’t know for sure). If PB-filtered categories were exactly coproducts of filtered ones this would be nothing new, but otherwise it might be interesting. In that case I’d like to get my hands on an explicit example to try and test my suspicion.

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeSep 2nd 2013
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   Author: Zhen Lin
   Format: MarkdownItexThe soundness of $\mathbb{D}$ has to do with whether or not it is easy to characterise the class of $\mathbb{D}$-filtered categories. As I said, if $\mathcal{C}$ has pullbacks, then it has finite connected limits, and if a functor $\mathcal{C} \to \mathcal{D}$ preserves pullbacks, then it also preserves finite connected limits. So for the purposes of your question it _is_ possible to replace PB by FINCL.

   The soundness of $\mathbb{D}$ has to do with whether or not it is easy to characterise the class of $\mathbb{D}$-filtered categories. As I said, if $\mathcal{C}$ has pullbacks, then it has finite connected limits, and if a functor $\mathcal{C} \to \mathcal{D}$ preserves pullbacks, then it also preserves finite connected limits. So for the purposes of your question it is possible to replace PB by FINCL.

5. • CommentRowNumber5.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 2nd 2013
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   Author: Karol Szumiło
   Format: MarkdownItexI see, apparently I didn't read your original answer carefully enough. But I don't really get your argument. How do you go from pullbacks to finite connected limits? For example, how do you get equalizers? I know how to obtain equalizers from pullbacks and binary products, but I don't see how to do it using pullbacks alone. In fact [[connected limit]] has a theorem that pullbacks and equalizers imply finite connected limits which would be an unnecessarily weak statement if we could do without equalizers.

   I see, apparently I didn’t read your original answer carefully enough. But I don’t really get your argument. How do you go from pullbacks to finite connected limits? For example, how do you get equalizers? I know how to obtain equalizers from pullbacks and binary products, but I don’t see how to do it using pullbacks alone. In fact connected limit has a theorem that pullbacks and equalizers imply finite connected limits which would be an unnecessarily weak statement if we could do without equalizers.

6. • CommentRowNumber6.
   • CommentAuthorZhen Lin
   • CommentTimeSep 2nd 2013
   • (edited Sep 2nd 2013)
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   Author: Zhen Lin
   Format: MarkdownItexAh, sorry – I thought pullbacks were enough. But what I said still works because equalisers can be constructed from pullbacks and "weak coequalisers". In the presence of a terminal object this is well-known. So, for example, suppose $F : \mathcal{C} \to \mathcal{D}$ preserves pullbacks; then $F_{/ 1} : \mathcal{C}_{/ 1} \to \mathcal{D}_{/ F 1}$ preserves finite limits, and the projection $\mathcal{D}_{/ F 1} \to \mathcal{D}$ preserves pullbacks _and_ equalisers, hence, $F : \mathcal{C} \to \mathcal{D}$ preserves finite connected limits.

   Ah, sorry – I thought pullbacks were enough. But what I said still works because equalisers can be constructed from pullbacks and “weak coequalisers”. In the presence of a terminal object this is well-known. So, for example, suppose $F : \mathcal{C} \to \mathcal{D}$ preserves pullbacks; then $F_{/ 1} : \mathcal{C}_{/ 1} \to \mathcal{D}_{/ F 1}$ preserves finite limits, and the projection $\mathcal{D}_{/ F 1} \to \mathcal{D}$ preserves pullbacks and equalisers, hence, $F : \mathcal{C} \to \mathcal{D}$ preserves finite connected limits.

7. • CommentRowNumber7.
   • CommentAuthorKarol Szumiło
   • CommentTimeSep 2nd 2013
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   Author: Karol Szumiło
   Format: MarkdownItexOK, I understand it now. Thanks for your help!

   OK, I understand it now. Thanks for your help!