About the cartesian closedness of the category of all small diagrams
Dear categorists, I have three questions, the first one is a mathematical question, the second one a bibliographical question and the last one is a speculative question. 1) Let K be a complete, cocomplete and cartesian closed category. Consider the category DK of all small diagrams over K. The objects are all small diagrams F:I-->K from a small category I to K. And a map from (F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural transformation mu:F-->Gf. DK is complete and cocomplete and I would like to know if it is cartesian closed as well. 2) My question was initially posted in https://mathoverflow.net/q/266597/24563. From MathOverflow, I now know that the functor DK-->Cat forgetting K is a fibred category. Since then, I browsed the Borceux book's chapter devoted to fibred categories (Vol.2 Chap.8). Is there other reference you could recommend me ? 3) I also would like to know what is known about the link between locally presentability and fibred category. Googling these terms or looking them up in MathSciNet together gives nothing relevant. Actually, my motivation is to know whether D(DeltaTop) and D(SimplicialSet) are locally presentable and cartesian closed (DeltaTop is the category of Delta-generated spaces, and SimplicialSet the category of simplicial sets). Therefore I would like to conclude this email with a speculative question: is there a general philosophy to deduce from the properties of the fibers of a fibred category E-->B the same property on E ? In the case of DK-->Cat, the fiber over I is the well-known category of I-shaped diagrams over K... Philippe Gaucher. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
This is something of an answer to question 2). I was very influenced by Streicher, T. ‘Fibred categories `a la B´enabou’. http://www.mathematik.tu-darmstadt.de/∼streicher/ (1999) 1–85. to see how useful fibrations and cofibrations of categories for giving the abstract background to constructions that occurred commonly in my work with Higgins and Loday on nonabelian colimit constructions in homotopy theory. So Sivera and I put these in Appendix B of the book "Nonabelian Algebraic Topology" (EMS 2011) (NAT). Note that Higgins pioneered in effect the use of the functor Ob: Groupoids \to Sets as a cofibration of categories, (see his Notes on Categroies and Groupoids" TAC Reprint) and Section B.3 of NAT gives suitable general results as background to say colimits of groupoids, knowing them for sets. There is more sophisticated material in the above lectures which I have not managed to use. Any advice on this could be useful! Ronnie Brown ------ Original Message ------ From: "gaucher" <gaucher@irif.fr> To: categories@mta.ca Sent: 13/04/2017 15:13:46 Subject: categories: About the cartesian closedness of the category of all small diagrams
Dear categorists,
I have three questions, the first one is a mathematical question, the second one a bibliographical question and the last one is a speculative question.
1) Let K be a complete, cocomplete and cartesian closed category. Consider the category DK of all small diagrams over K. The objects are all small diagrams F:I-->K from a small category I to K. And a map from (F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural transformation mu:F-->Gf. DK is complete and cocomplete and I would like to know if it is cartesian closed as well.
2) My question was initially posted in https://mathoverflow.net/q/266597/24563. From MathOverflow, I now know that the functor DK-->Cat forgetting K is a fibred category. Since then, I browsed the Borceux book's chapter devoted to fibred categories (Vol.2 Chap.8). Is there other reference you could recommend me ?
3) I also would like to know what is known about the link between locally presentability and fibred category. Googling these terms or looking them up in MathSciNet together gives nothing relevant. Actually, my motivation is to know whether D(DeltaTop) and D(SimplicialSet) are locally presentable and cartesian closed (DeltaTop is the category of Delta-generated spaces, and SimplicialSet the category of simplicial sets). Therefore I would like to conclude this email with a speculative question: is there a general philosophy to deduce from the properties of the fibers of a fibred category E-->B the same property on E ? In the case of DK-->Cat, the fiber over I is the well-known category of I-shaped diagrams over K...
Philippe Gaucher.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Philippe On 14 Apr 2017, at 12:13 AM, gaucher <gaucher@irif.fr<mailto:gaucher@irif.fr>> wrote: 1) Let K be a complete, cocomplete and cartesian closed category. Consider the category DK of all small diagrams over K. The objects are all small diagrams F:I-->K from a small category I to K. And a map from (F:I-->K) to (G:J-->K) is a functor f:I-->J together with a natural transformation mu:F-->Gf. DK is complete and cocomplete and I would like to know if it is cartesian closed as well. Yes it is. The internal hom of F : I --> K and G : J --> K is H : [I,J] --> K defined by Hr = end over i in I of [Fi,Gri] where [I,J] is internal hom in Cat and, for h, k in K, [h,k] is internal hom in K. 3) I also would like to know what is known about the link between locally presentability and fibred category. I suspect that, if T : K^{op} --> LFP is a functor, where K is locally finitely presentable (lfp) and LFP is the 2-category of lfp categories and right adjoint functors which preserve filtered colimits, then the ``Grothendieck fibration construction'' El(T) --> T gives an lfp category El(T). I can't think of a quick reason but others may think of one, a reference, or a counterexample. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
For a fibration of ccc's over a ccc one knows that the total category is again ccc and this structure is preserved (could be in Bart Jacob's book). As to (1) even if K = Set we know that the Set^C are ccc's (actually toposes) but reindexing in general doesn't preserve the ccc's structure since in Set^C the exponentials are not computed pointwise (unless C is discrete). It already goes wrong when C is the ordinal 2. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
As follows from Ross Street's argument a fibration between ccc's may be cartesian closed even if it is not a fibration of cartesian closed categories (instantiating K by Set). Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
For a fibration of ccc's over a ccc one knows that the total category is again ccc and this structure is preserved (could be in Bart Jacob's book).
This is wrong! Let EE be the free topos (with nno). Then the fibration Fam(EE) over Set is a fibered ccc but if it were a cartesian closed functor between toposes then EE would have small sums which it hasn't. It is not clear at all which fibrations of ccc's over ccc's have the property that they are ccc-preserving functors between ccc's. It seems as if such fibrations have to be internally complete. That's what I definitely overlooked. Sorry for that, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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gaucher -
Ronnie -
Ross Street -
Thomas Streicher