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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 2nd 2010
   • (edited Aug 2nd 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexThe definition of a $\kappa$-compact object as an object co-representing a $\kappa$-accessible functor is very slick, but it's nearly always true that we use the following properties of compact objects: We say $X$ is $\kappa$-compact if for any $\kappa$-filtered functor $F:I\to C$ (i.e. the category $I$ is $\kappa$-filtered), any morphism $X\to colim F$ factors as $X\to F(i) \to colim F$, and any two factorizations $X\to F(i)\to colim F$ and $X\to F(i')\to colim F$ admit a majorant factorization $X\to F(i'')\to colim F$, that is, a factorization $X\to F(i'')\to colim F$ along with a commutative diagram induced by maps $i\to i'' \leftarrow i'$ $$\begin{matrix}X&\to&F(i)\\ \downarrow&&\downarrow\\ F(i')&\to&F(i'')\\ \end{matrix}$$ such that $$F(i)\to colim F=F(i)\to F(i'')\to colim F$$ and $$F(i')\to colim F=F(i')\to F(i'')\to colim F$$ are the natural triangles. I think these should be mentioned in the nLab article [[compact object]] with proof that in fact, the above properties constitute an elementary definition (i.e. not relying on Yoneda) of a compact object. However, I don't know how to carry out the proof (although I think I have one direction).

   The definition of a $\kappa$-compact object as an object co-representing a $\kappa$-accessible functor is very slick, but it’s nearly always true that we use the following properties of compact objects:

   We say $X$ is $\kappa$-compact if for any $\kappa$-filtered functor $F:I\to C$ (i.e. the category $I$ is $\kappa$-filtered), any morphism $X\to colim F$ factors as $X\to F(i) \to colim F$, and any two factorizations $X\to F(i)\to colim F$ and $X\to F(i')\to colim F$ admit a majorant factorization $X\to F(i'')\to colim F$, that is, a factorization $X\to F(i'')\to colim F$ along with a commutative diagram induced by maps $i\to i'' \leftarrow i'$

   $$\begin{matrix}X&\to&F(i)\\ \downarrow&&\downarrow\\ F(i')&\to&F(i'')\\ \end{matrix}$$

   such that

   $$F(i)\to colim F=F(i)\to F(i'')\to colim F$$

   and

   $$F(i')\to colim F=F(i')\to F(i'')\to colim F$$

   are the natural triangles.

   I think these should be mentioned in the nLab article compact object with proof that in fact, the above properties constitute an elementary definition (i.e. not relying on Yoneda) of a compact object. However, I don’t know how to carry out the proof (although I think I have one direction).

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 3rd 2010
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   Author: DavidRoberts
   Format: MarkdownItexWell put it in the page, and flag that it is incomplete ;) That's what I'd do.

   Well put it in the page, and flag that it is incomplete ;) That’s what I’d do.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 3rd 2010
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   Author: Mike Shulman
   Format: MarkdownItexI thought the proof of that was basically an unraveling of the Yoneda version of the definition in terms of the construction of colimits in Set. But I agree, it should be at the page.

   I thought the proof of that was basically an unraveling of the Yoneda version of the definition in terms of the construction of colimits in Set. But I agree, it should be at the page.

4. • CommentRowNumber4.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 3rd 2010
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItex@Mike: It depends on proving that filtered colimits in Set are given by the quotient construction described in Lang's algebra (points of the filtered colimit are given by equivalence classes of pairs (x,U) and so on. I have never seen a conceptual proof of this equivalence, apart from the direct construction.

   @Mike: It depends on proving that filtered colimits in Set are given by the quotient construction described in Lang’s algebra (points of the filtered colimit are given by equivalence classes of pairs (x,U) and so on. I have never seen a conceptual proof of this equivalence, apart from the direct construction.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeAug 3rd 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItex@Harry: It depends on what you think of as a conceptual proof. The original context of lots of that stuff was refinements of covers and/or zeroing in to a point in a sheafification process. There the classical proof has very neat interpretations in terms of germs of functions at a point. But as I said, it depends on your meaning of `conceptual'. Sometimes historical asides can help, sometimes they hinder!

   @Harry: It depends on what you think of as a conceptual proof. The original context of lots of that stuff was refinements of covers and/or zeroing in to a point in a sheafification process. There the classical proof has very neat interpretations in terms of germs of functions at a point. But as I said, it depends on your meaning of ‘conceptual’.

   Sometimes historical asides can help, sometimes they hinder!

6. • CommentRowNumber6.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 3rd 2010
   • (edited Aug 3rd 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItex@Tim: Sure, I know about the applications/context, but I was wondering if there was a slick way to end up with the construction in question without appealing to the fact that the construction satisfies the universal property. In Adamek-Rosicky, exercise 1.a. is to show precisely that filtered colimits in Sets can be computed this way. If that exercise is simply verifying the universal property, then I find that a little bit disappointing.

   @Tim: Sure, I know about the applications/context, but I was wondering if there was a slick way to end up with the construction in question without appealing to the fact that the construction satisfies the universal property. In Adamek-Rosicky, exercise 1.a. is to show precisely that filtered colimits in Sets can be computed this way. If that exercise is simply verifying the universal property, then I find that a little bit disappointing.

7. • CommentRowNumber7.
   • CommentAuthorTim_Porter
   • CommentTimeAug 3rd 2010
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexAs the colimit is specified by the universal property, the proof is never going to be that different, but might be stood on its head. I do not have their book so cannot comment in detail, but here is an idea. Start with the universal property and reverse their discussion, if possible, to get the construction. I suspect that is more conceptual. You have to have the `inclusions' and there has to be compatibility, now invoke the cofiltering idea etc. That should give a neat approach. I suspect you are hoping for a proof which shows: the colimit MUST be this because of the universal property and what I said, if it works, should be almost that.

   As the colimit is specified by the universal property, the proof is never going to be that different, but might be stood on its head. I do not have their book so cannot comment in detail, but here is an idea. Start with the universal property and reverse their discussion, if possible, to get the construction. I suspect that is more conceptual. You have to have the ‘inclusions’ and there has to be compatibility, now invoke the cofiltering idea etc. That should give a neat approach. I suspect you are hoping for a proof which shows: the colimit MUST be this because of the universal property and what I said, if it works, should be almost that.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeAug 3rd 2010
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   Author: Mike Shulman
   Format: MarkdownItexOne way to see that all colimits in Set are given by equivalence classes of pairs is to construct colimits using coproducts and coequalizers, and observe that coproducts are disjoint unions and coequalizers are quotients by equivalence relations. For a general colimit, the equivalence relation is only given by a generating relation, but in the filtered case I expect Lang's description is just giving a more explicit presentation of that equivalence relation.

   One way to see that all colimits in Set are given by equivalence classes of pairs is to construct colimits using coproducts and coequalizers, and observe that coproducts are disjoint unions and coequalizers are quotients by equivalence relations. For a general colimit, the equivalence relation is only given by a generating relation, but in the filtered case I expect Lang’s description is just giving a more explicit presentation of that equivalence relation.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2012
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   Author: Urs
   Format: MarkdownItexI have added to the _[Properties](http://ncatlab.org/nlab/show/compact+object#Properties)_-section at _[[compact object]]_ the statement that $\kappa$-small colimits of $\kappa$-compact objects are still $\kappa$-compact. (Just for completeness.)

   I have added to the Properties-section at compact object the statement that $\kappa$-small colimits of $\kappa$-compact objects are still $\kappa$-compact. (Just for completeness.)

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)
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    Author: Urs
    Format: MarkdownItexat _[[compact object]]_ it says that in a topos the compact objects are the [[Kuratowski finite objects]]. Is this statement really correct? If $X$ is Kuratowski finite in the topos $C$, then $C_{/X} \to C$ is a [[proper geometric morphism]], hence $C_{/X}$ is compact, as a topos over $C$. I see that this is a condition similar to $C(X,-)$ commuting with all filtered colimits. But is it really equivalent?

    at compact object it says that in a topos the compact objects are the Kuratowski finite objects.

    Is this statement really correct?

    If $X$ is Kuratowski finite in the topos $C$, then $C_{/X} \to C$ is a proper geometric morphism, hence $C_{/X}$ is compact, as a topos over $C$. I see that this is a condition similar to $C(X,-)$ commuting with all filtered colimits. But is it really equivalent?

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeApr 27th 2012
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    Author: Mike Shulman
    Format: MarkdownItexHmm, well, by internalizing the statement from Set, we should be able to show that an object is "internally compact" iff it is K-finite. But maybe internal compactness is not the same as external compactness.

    Hmm, well, by internalizing the statement from Set, we should be able to show that an object is “internally compact” iff it is K-finite. But maybe internal compactness is not the same as external compactness.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> But maybe internal compactness is not the same as external compactness. I am guessing this may be precisely the problem. Because $X \in C$ is compact in the sense that every jointly epi family has a finite jointly epi subfamily if $C_{/X} \to Set$ is a proper geometric morphism, hence if $C_{/X}$ is an absolutely compact topos, not relatively over $C$.

    But maybe internal compactness is not the same as external compactness.

    I am guessing this may be precisely the problem.

    Because $X \in C$ is compact in the sense that every jointly epi family has a finite jointly epi subfamily if $C_{/X} \to Set$ is a proper geometric morphism, hence if $C_{/X}$ is an absolutely compact topos, not relatively over $C$.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2012
    • (edited Apr 27th 2012)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have expanded the Idea-section at _[[compact object]]_ a bit, both to be more instructive for the newcomer, as well such as to highlight better that there are related but different formalizations of "finite".

    I have expanded the Idea-section at compact object a bit, both to be more instructive for the newcomer, as well such as to highlight better that there are related but different formalizations of “finite”.

14. • CommentRowNumber14.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeFeb 21st 2013
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    Author: IngoBlechschmidt
    Format: MarkdownItexI believe that constructively, the compact objects in Set are not the Kuratowski-finite sets (as stated at _[[compact object]]_), but the Bishop-finite sets (those sets admitting a bijection with some $[n]$), because... * ...I'm unable to show that Kuratowski-finite sets $X$ are compact: Let $f:X \to \colim_{d \in D} F(d)$ with $D$ a filtered category. Then one can of course find a single index $d_0 \in D$ such that $f(x)$ lies in $F(d_0)$ for every $x \in X$, but I don't see how to define a _map_ $X \to F(d_0)$ (because of missing uniqueness). * ...conversely, I think that every set is the colimit of all Bishop-finite sets mapping into it; thus a compact set is (as a retract of a Bishop-finite set) Bishop-finite. Instead, the Kuratowski-finite sets are those sets $X$ for which the functor $Hom(X, -)$ commutes with filtered colimits _of monomorphisms_. Do you agree?

    I believe that constructively, the compact objects in Set are not the Kuratowski-finite sets (as stated at compact object), but the Bishop-finite sets (those sets admitting a bijection with some $[n]$), because…

    • …I’m unable to show that Kuratowski-finite sets $X$ are compact: Let $f:X \to \colim_{d \in D} F(d)$ with $D$ a filtered category. Then one can of course find a single index $d_0 \in D$ such that $f(x)$ lies in $F(d_0)$ for every $x \in X$, but I don’t see how to define a map $X \to F(d_0)$ (because of missing uniqueness).

    • …conversely, I think that every set is the colimit of all Bishop-finite sets mapping into it; thus a compact set is (as a retract of a Bishop-finite set) Bishop-finite.

    Instead, the Kuratowski-finite sets are those sets $X$ for which the functor $Hom(X, -)$ commutes with filtered colimits of monomorphisms. Do you agree?

15. • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2013
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    Author: Mike Shulman
    Format: MarkdownItexI think the last time I thought about this, it wasn't even obvious to me what the right constructive definition of "filtered" is. What definition are you using?

    I think the last time I thought about this, it wasn’t even obvious to me what the right constructive definition of “filtered” is. What definition are you using?

16. • CommentRowNumber16.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeFeb 22nd 2013
    • (edited Mar 21st 2013)
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    Author: IngoBlechschmidt
    Format: MarkdownItexThanks, that's an important point! I used the definition at _[[filtered category]]_, "This can be rephrased in more elementary terms by saying that: [...]", which I believe is equivalent to saying that every Kuratowski-finite diagram admits a cocone. Edit: Actually, this is equivalent to saying that every _Bishop_-finite diagram admits a cocone.

    Thanks, that’s an important point! I used the definition at filtered category, “This can be rephrased in more elementary terms by saying that: […]”, which I believe is equivalent to saying that every Kuratowski-finite diagram admits a cocone.

    Edit: Actually, this is equivalent to saying that every Bishop-finite diagram admits a cocone.

17. • CommentRowNumber17.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 24th 2013
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    Author: TobyBartels
    Format: MarkdownItexOnly the Bishop-finite sets are finitely *presented*. The Kuratowski-finite sets are finitely *generated*. And compact objects are supposed to be finitely presented. Of course, this depends on what one means by ‘finitely’. But the intuition that finitely presented objects are amenable to combinatorial analysis says to me that this should be taken strictly. (This goes into Mike\'s question of what should be the definition of ‘filtered category’.) It's true that the K-finite sets are the compact sets; that is, they are the compact discrete spaces. But of course the relationship between compact objects and compact spaces is indirect.

    Only the Bishop-finite sets are finitely presented. The Kuratowski-finite sets are finitely generated. And compact objects are supposed to be finitely presented.

    Of course, this depends on what one means by ‘finitely’. But the intuition that finitely presented objects are amenable to combinatorial analysis says to me that this should be taken strictly. (This goes into Mike's question of what should be the definition of ‘filtered category’.)

    It’s true that the K-finite sets are the compact sets; that is, they are the compact discrete spaces. But of course the relationship between compact objects and compact spaces is indirect.

18. • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 24th 2013
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    Author: TobyBartels
    Format: MarkdownItexIt looks like [I am to blame](http://ncatlab.org/nlab/revision/diff/compact+object/14) for this K-finite business. So I\'ll fix it now.

    It looks like I am to blame for this K-finite business. So I'll fix it now.

19. • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2013
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    Author: Mike Shulman
    Format: MarkdownItexThat makes sense; thanks Toby! I bet the compact space / compact object confusion was also partly to blame.

    That makes sense; thanks Toby! I bet the compact space / compact object confusion was also partly to blame.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
    • (edited Feb 25th 2013)
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    Author: Urs
    Format: MarkdownItexI haven't been following the constructive set-theoretic aspects here, but what Ingo said in #14 about preserving filtered colimits of _monos_ resonates with that "compact space / compact object"-subtlety: last time that I thought about this, I came to the conclusion that generally in an $\infty$-topos we are to consider a tower of notions of compactness where we add qualifiers about the truncation of the objects and morphisms in the filtered colimits involved. Doing this also brings the notion of "compact topological space" back in line with the categorical notion of "compact object". I have a note on this in [dcct, section 3.6.4 page 224](http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf#page=224).

    I haven’t been following the constructive set-theoretic aspects here, but what Ingo said in #14 about preserving filtered colimits of monos resonates with that “compact space / compact object”-subtlety:

    last time that I thought about this, I came to the conclusion that generally in an $\infty$-topos we are to consider a tower of notions of compactness where we add qualifiers about the truncation of the objects and morphisms in the filtered colimits involved.

    Doing this also brings the notion of “compact topological space” back in line with the categorical notion of “compact object”. I have a note on this in dcct, section 3.6.4 page 224.

21. • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2013
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    Author: Mike Shulman
    Format: MarkdownItexWhen I first read the Elephant, I was quite confused that there was the notion of 'proper map' which captured the usual topological meaning of 'compact', but also the notion of 'tidy map' which looked much more categorically natural. It all made much more sense once I realized they were just the cases $n=-1$ and $n=0$ of a notion of '$n$-compact map', in the same way that open maps and locally connected maps are the cases $n=-1$ and $n=0$ of the notion of 'locally $n$-connected map'.

    When I first read the Elephant, I was quite confused that there was the notion of ’proper map’ which captured the usual topological meaning of ’compact’, but also the notion of ’tidy map’ which looked much more categorically natural. It all made much more sense once I realized they were just the cases $n=-1$ and $n=0$ of a notion of ’$n$-compact map’, in the same way that open maps and locally connected maps are the cases $n=-1$ and $n=0$ of the notion of ’locally $n$-connected map’.

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 25th 2013
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    Author: Urs
    Format: MarkdownItexI suppose this is all clear to you, but is there a source where this is all nicely written out? Back when I did the computations linked to above I needed to know: given an $\infty$-topos over a site of definition $(C,J)$ and given an object in $C$ which is (para-)compact as seen by $(C,J)$, how do I characterize its compactness as a property in the $\infty$-topos (without remembering the site and its covers)? Unless I made a mistake back then, I found that characterization and it is clear how it is the beginning of a tower as you indicate. But I have never seen this discussed elsewhere.

    I suppose this is all clear to you, but is there a source where this is all nicely written out? Back when I did the computations linked to above I needed to know: given an $\infty$-topos over a site of definition $(C,J)$ and given an object in $C$ which is (para-)compact as seen by $(C,J)$, how do I characterize its compactness as a property in the $\infty$-topos (without remembering the site and its covers)?

    Unless I made a mistake back then, I found that characterization and it is clear how it is the beginning of a tower as you indicate. But I have never seen this discussed elsewhere.

23. • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 25th 2013
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    Author: Mike Shulman
    Format: MarkdownItexNo, it's only an intuition for me: I've never seen the details written down.

    No, it’s only an intuition for me: I’ve never seen the details written down.

24. • CommentRowNumber24.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMar 19th 2013
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    Author: IngoBlechschmidt
    Format: MarkdownItexThank you for your comments! I added some remarks at _[[compact object]]_, _[[compact space]]_, _[[filtered category]]_ and _[[finite set]]_ summarizing (the 1-category theory aspects of) this discussion.

    Thank you for your comments! I added some remarks at compact object, compact space, filtered category and finite set summarizing (the 1-category theory aspects of) this discussion.

25. • CommentRowNumber25.
    • CommentAuthorTobyBartels
    • CommentTimeMar 20th 2013
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    Author: TobyBartels
    Format: MarkdownItexGreat!

    Great!

26. • CommentRowNumber26.
    • CommentAuthorTim_Porter
    • CommentTimeJan 28th 2014
    • (edited Jan 28th 2014)
    • PermaLink
    Author: Tim_Porter
    Format: MarkdownItexI just noticed that at [[compact object]] we have the line: >For C a topos, X is compact if and it stops there. Doh!

    I just noticed that at compact object we have the line:

    For C a topos, X is compact if

    and it stops there. Doh!

27. • CommentRowNumber27.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeJan 29th 2014
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    Author: IngoBlechschmidt
    Format: MarkdownItexThanks for bringing this issue to attention. I think that the discussion above never converged to a definitive answer. At least we know (from #10, #11, #12): If $X \in C$ is a Kuratowski-finite object and $C$ is a compact topos (over $Set$), then $X$ is a compact object in $C$.

    Thanks for bringing this issue to attention. I think that the discussion above never converged to a definitive answer. At least we know (from #10, #11, #12): If $X \in C$ is a Kuratowski-finite object and $C$ is a compact topos (over $Set$), then $X$ is a compact object in $C$.

28. • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeJan 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexDid you have a look at the discussion I mentioned in #20, section 3.6.4, p. 282 of [dcct](http://arxiv.org/abs/1310.7930)? (Maybe you did and it's not what you are looking for. Just checking.)

    Did you have a look at the discussion I mentioned in #20, section 3.6.4, p. 282 of dcct?

    (Maybe you did and it’s not what you are looking for. Just checking.)

29. • CommentRowNumber29.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexUrs, In section 3.6.4 of dcct, what is the conclusion? I see an introductory paragraph giving some motivation and ideas of what could be the case (phrases like "in a suitable sense"), then some results about compact 1-toposes/compact objects in 1-toposes, then a statement about (higher-)paracompactness for representables. Perhaps I got my hopes unreasonably up, or perhaps it is addressed later, but I would have liked to have seen a definition of an $n$-compact object of an $(\infty,1)$-topos (or whatever prefixes the general notion needs).

    Urs,

    In section 3.6.4 of dcct, what is the conclusion? I see an introductory paragraph giving some motivation and ideas of what could be the case (phrases like “in a suitable sense”), then some results about compact 1-toposes/compact objects in 1-toposes, then a statement about (higher-)paracompactness for representables. Perhaps I got my hopes unreasonably up, or perhaps it is addressed later, but I would have liked to have seen a definition of an $n$-compact object of an $(\infty,1)$-topos (or whatever prefixes the general notion needs).

30. • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexRight, so prop. 3.6.61 says that an object that is compact in the sense of "compact topological space" does -- while not in general distributing over all filtered colimits and hence not being categorically compact -- does distribute over filtered colimits whose structure maps are monos (-1-truncated). So there is a hierarchy stretching from "geometrically compact" at $n = -1$ to "categorically compact" at $n = \infty$.

    Right, so prop. 3.6.61 says that an object that is compact in the sense of “compact topological space” does – while not in general distributing over all filtered colimits and hence not being categorically compact – does distribute over filtered colimits whose structure maps are monos (-1-truncated). So there is a hierarchy stretching from “geometrically compact” at $n = -1$ to “categorically compact” at $n = \infty$.

31. • CommentRowNumber31.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 3rd 2014
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    Author: TobyBartels
    Format: MarkdownItexThat sounds like an excellent perspective, Urs!

    That sounds like an excellent perspective, Urs!

32. • CommentRowNumber32.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 3rd 2014
    • (edited Feb 3rd 2014)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexWait -- is this supposed to be an iff characterization of compact topological spaces? (I'm skeptical.)

    Wait – is this supposed to be an iff characterization of compact topological spaces? (I’m skeptical.)

33. • CommentRowNumber33.
    • CommentAuthorMarc Hoyois
    • CommentTimeFeb 3rd 2014
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    Author: Marc Hoyois
    Format: MarkdownItexI'm also curious as to how the notions Urs is considering (in terms of existence of certain refinements of certain hypercovers) relate to Lurie's hierarchy of higher coherence (from DAG VII), where 0-coherent means quasi-compact, 1-coherent means quasi-compact and quasi-separated, etc. Lurie proves that the global section functor of an n-coherent (∞,1)-topos preserves filtered colimits of (n-1)-truncated objects. What's unfortunate is that this doesn't hold for $n=\infty$, at least with Lurie's definition of ∞-coherent (= n-coherent for all n).

    I’m also curious as to how the notions Urs is considering (in terms of existence of certain refinements of certain hypercovers) relate to Lurie’s hierarchy of higher coherence (from DAG VII), where 0-coherent means quasi-compact, 1-coherent means quasi-compact and quasi-separated, etc. Lurie proves that the global section functor of an n-coherent (∞,1)-topos preserves filtered colimits of (n-1)-truncated objects. What’s unfortunate is that this doesn’t hold for $n=\infty$, at least with Lurie’s definition of ∞-coherent (= n-coherent for all n).

34. • CommentRowNumber34.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexI'm curious to see the formal rendition of >we may say an object is compact of height n if every hypercover of height n over the object is refined by a “finite hypercover” in a suitable sense. Since I'm interested in compact differentiable $n$-stacks, rather than anything dealing with representables.

    I’m curious to see the formal rendition of

    we may say an object is compact of height n if every hypercover of height n over the object is refined by a “finite hypercover” in a suitable sense.

    Since I’m interested in compact differentiable $n$-stacks, rather than anything dealing with representables.

35. • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexTodd, I talk about just one direction. Marc, thanks, I need to look at that.

    Todd, I talk about just one direction. Marc, thanks, I need to look at that.

36. • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2014
    • (edited Feb 3rd 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexMarc, still haven't had a chance to look at it, but from what you say Lurie's definition is what we had called "compact of height $n$" at _[[compact topos]]_, by extrapolation from Moerdijk-Vermeulen's articles cited there. David, to every differentiable $n$-stack $X$ is associated, by [[differential cohesion]], its "[[etale topos]]" $Sh_(X)$, which is a structured $\infty$-topos in Lurie's sense. So you may say that $X$ is $n$-compact if $Sh(X)$ "compact of height $n$" or whatever one calls it. Back then what made me consider the result discussed above is that I needed a "gros topos" version of this. For instance for the discussion of stable compex vector bundles in smooth stacks one wants to know that $$ Hom(X, \mathbf{B}U) = Hom(X, \underset{\to}{lim}_n \mathbf{B}U(n)) \simeq \underset{\to}{lim}_n Hom(X, \mathbf{B}U(n)) \,. $$ That's what the result we discussed gives, and that's what I needed it for. It was directly motivated by the version in petit toposes, though: since filtered colimits of (-1)-truncated objects necessarily have (-1)-truncated structure maps, that suggested in the "gros" cohesive context (where (-1)-truncated objects are uninteresting) to look for distributivity of filtered colimits with monomorphic structure maps.

    Marc, still haven’t had a chance to look at it, but from what you say Lurie’s definition is what we had called “compact of height $n$” at compact topos, by extrapolation from Moerdijk-Vermeulen’s articles cited there.

    David, to every differentiable $n$-stack $X$ is associated, by differential cohesion, its “etale topos” $Sh_(X)$, which is a structured $\infty$-topos in Lurie’s sense. So you may say that $X$ is $n$-compact if $Sh(X)$ “compact of height $n$” or whatever one calls it.

    Back then what made me consider the result discussed above is that I needed a “gros topos” version of this. For instance for the discussion of stable compex vector bundles in smooth stacks one wants to know that

    $$Hom(X, \mathbf{B}U) = Hom(X, \underset{\to}{lim}_n \mathbf{B}U(n)) \simeq \underset{\to}{lim}_n Hom(X, \mathbf{B}U(n)) \,.$$

    That’s what the result we discussed gives, and that’s what I needed it for.

    It was directly motivated by the version in petit toposes, though: since filtered colimits of (-1)-truncated objects necessarily have (-1)-truncated structure maps, that suggested in the “gros” cohesive context (where (-1)-truncated objects are uninteresting) to look for distributivity of filtered colimits with monomorphic structure maps.

37. • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexMarc, sorry I am slow these days with little quiet time. I looked at _[[Spectral Schemes]]_, but maybe I am looking at the wrong version or something. What's the page you are looking at? I see lemma 3.21,which is something about quasi-compact objest distributing over very special filtered colimits. But where is that statement which you mention in #33? Sorry.

    Marc, sorry I am slow these days with little quiet time. I looked at Spectral Schemes, but maybe I am looking at the wrong version or something. What’s the page you are looking at?

    I see lemma 3.21,which is something about quasi-compact objest distributing over very special filtered colimits. But where is that statement which you mention in #33? Sorry.

38. • CommentRowNumber38.
    • CommentAuthorMarc Hoyois
    • CommentTimeFeb 3rd 2014
    • PermaLink
    Author: Marc Hoyois
    Format: MarkdownItex@Urs: My bad, I forgot that this statement is hidden away in DAG XIII, Prop. 2.3.9. In any case, the n-coherence hierarchy is not the same as the compact of height n-1 hierarchy. Already for 1-topoi, the fact that $\Gamma$ preserves filtered colimits is not enough for the topos to be coherent. But it seems believable that n-coherence can be expressed in terms of refinements of hypercovers. Proposition 3.6.64 in dcct would then be a special case of Lurie's proposition.

    @Urs: My bad, I forgot that this statement is hidden away in DAG XIII, Prop. 2.3.9. In any case, the n-coherence hierarchy is not the same as the compact of height n-1 hierarchy. Already for 1-topoi, the fact that $\Gamma$ preserves filtered colimits is not enough for the topos to be coherent. But it seems believable that n-coherence can be expressed in terms of refinements of hypercovers. Proposition 3.6.64 in dcct would then be a special case of Lurie’s proposition.

39. • CommentRowNumber39.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 4th 2014
    • (edited Feb 4th 2014)
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItex@Urs, ok, I've added links to [[compact topos]] at the pages [[compact object]] and [[compact object in an (∞,1)-category]] under 'related concepts'. I also only need this for a diagram involving monomorphisms, but was worrying about something more general that I can ignore in practice.

    @Urs,

    ok, I’ve added links to compact topos at the pages compact object and compact object in an (∞,1)-category under ’related concepts’. I also only need this for a diagram involving monomorphisms, but was worrying about something more general that I can ignore in practice.

40. • CommentRowNumber40.
    • CommentAuthorMarc Hoyois
    • CommentTimeFeb 4th 2014
    • PermaLink
    Author: Marc Hoyois
    Format: MarkdownItexOn further thought, the representably paracompact condition is far from implying coherence: a 1-coherent Hausdorff topological space is finite!

    On further thought, the representably paracompact condition is far from implying coherence: a 1-coherent Hausdorff topological space is finite!

41. • CommentRowNumber41.
    • CommentAuthorjesse
    • CommentTimeAug 4th 2016
    • PermaLink
    Author: jesse
    Format: MarkdownItexA couple years ago, Tim Porter noted that: > I just noticed that at [[compact object]] we have the line: > > For C a topos, X is compact if > and it stops there. Doh! I've filled this in with parts of Remark D3.3.12 in the Elephant, added a reference, and also a remark in the 'Subtleties and different meanings' section on the fact that 'compact' and 'finitely-presented' in the Elephant are distinct notions (the latter being the notion of 'compact' which appears on the page, and the former being a coherent object without stability assumptions).

    A couple years ago, Tim Porter noted that:

    I just noticed that at compact object we have the line:

    For C a topos, X is compact if

    and it stops there. Doh!

    I’ve filled this in with parts of Remark D3.3.12 in the Elephant, added a reference, and also a remark in the ’Subtleties and different meanings’ section on the fact that ’compact’ and ’finitely-presented’ in the Elephant are distinct notions (the latter being the notion of ’compact’ which appears on the page, and the former being a coherent object without stability assumptions).