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.
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