Composition of Fibrations and Quantification
Dear All We know that if p and q are fibrations, then their composition p.q is a fibration. But what about quantification … that is if reindexing along every morphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances? Thanks for any thoughts Neil [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Some thoughts: * The result about composition of fibrations holds in any 2-category with comma objects and 2-pullbacks, not just Cat. (Think of the Chevalley criterion for fibrations.) * By duality on 2-cells it thus also applies to opfibrations, and hence to bifibrations. * It is bifibration structure that gives you the left adjoints you ask for. * For the right adjoints, look at the dual 2-category, where your fibrations become bifibrations. Hence it seems to me that your conjectures are all true, and even generalize widely. Steve.
On 6 Jun 2014, at 10:47, Neil Ghani <neil.ghani@strath.ac.uk> wrote:
Dear All
We know that if p and q are fibrations, then their composition p.q is a fibration.
But what about quantification … that is if reindexing along every morphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances?
Thanks for any thoughts Neil
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Neil, Steve, Steve: what you say about fibrations and opfibrations is completely correct. So if E ---> D is a bifibration, and D ---> C is a bifibration, then the composite E ---> C is a bifibration. It is moreover easy to check that if the individual parts of this satisfy Beck-Chevalley, then so too will the composite. Thus the composite of fibrations with sums is again a fibration with sums. For products, though, I think the situation is a bit more complex. I could not see how to derive the result by a simple duality. From scribbling on the back of an envelope, it does appear to be true that: - if E -p--> D and D ---q---> C are fibrations with products, then so too is E ---qp---> C The argument I have is basically the following. Given e in E with pe = d and qd = c, and some f: c ---> c' in C, we wish to describe the Pi of e along f; we'll write this as f_*(e). Well first we form f_*(d) in D, its pullback g : f^*f_*(d) ----> f_*(d) along f, and the counit k: f^*f_*(d) ---> d in the fibre over c. Using these data in D, we now form in E the pullback k^*(e) of e along k and then the Pi along g, so yielding e' = g_*k^*(e). Now pe' = f_*(d) and q(f_*(d)) = c' and it's now not so hard to check that e' is in fact the Pi of e along f, as desired. For the checking in the last step, I seemed to need a few times the Beck-Chevalley condition for products. So I doubt one could get away without that. I have not tried to verify whether the Pi's of the composite so defined themselves satisfy the Beck-Chevalley condition, but I would be amazed if this were not the case. Finally, one might ask the following question. Suppose that E ----> D and D ----> C are fibrations with products and sums, and that in each case the sums and products satisfy the distributivity axiom ("type-theoretic axiom of choice"). Does the composite fibration, which by the above again has sums and products, also satisfy the distributivity axiom? I do not know the answer to this, but I rather suspect so. It would be a largeish diagram chase. It would be nice to know if there was a more abstract reason why this is true. Richard On Sun, Jun 8, 2014, at 11:58 PM, Steve Vickers wrote:
Some thoughts:
* The result about composition of fibrations holds in any 2-category with comma objects and 2-pullbacks, not just Cat. (Think of the Chevalley criterion for fibrations.)
* By duality on 2-cells it thus also applies to opfibrations, and hence to bifibrations.
* It is bifibration structure that gives you the left adjoints you ask for.
* For the right adjoints, look at the dual 2-category, where your fibrations become bifibrations.
Hence it seems to me that your conjectures are all true, and even generalize widely.
Steve.
On 6 Jun 2014, at 10:47, Neil Ghani <neil.ghani@strath.ac.uk> wrote:
Dear All
We know that if p and q are fibrations, then their composition p.q is a fibration.
But what about quantification … that is if reindexing along every morphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances?
Thanks for any thoughts Neil
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Adding to Steve's most appropriate comments. One also wants to have the so-called Beck-Chevalley conditions. Recall that a fibration has internal sums iff it is a bifibration where cocartesian arrows are stable under pullbacks along cartesian arrows. This property is easily seen to be preserved by composition. But P has internal products iff P^op has internal sums. However, we don't have (P \circ Q)^op = P^op \circ Q^op in particular because since the right composite doesn't exist. (If P is a fibration over BB then P^op still is a fibration over BB and not over BB^op). Still I guess one can check directly that composition preserves the property of having small products. Maybe it's even in Bart Jacob's book? Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear All We know that if p and q are fibrations, then their composition p.q is a fibration. But what about quantification … that is if reindexing along every morphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances? Thanks for any thoughts Neil Dear Neil, Instead of quantification, I will refer to the structure you ask about as products (in the case of right adjoints) and coproducts (in the case of left adjoints) for a fibration, which is a more standard terminology in fibred category theory. Notice that as Thomas pointed out, one considers such adjoints subject to Beck-Chevalley conditions, i.e., all products/coproducts are required to be given pointwise. With regard to products in a composite fibration, it probably helps if we start by recalling a simple result about ordinary products and fibrations. *PROP* *[1]*: Let p: E -> B be a fibration, and B admit (finite) products. The following are equivalent i) E admits (finite) products and p preserves them ii) p has fibred (finite) products Here ii) means every fibre has (finite) products, preserved by reindexing. To formulate the equivalent result for a composite of fibrations, the key is to regard such a composite gadget as a fibration between fibrations. Namely, given fibrations F: E -> D and b: D -> B, consider the composite fibration t = bF: E -> B. Now, F is a fibred functor F: t -> b (over B), that is, a morphism in the 2-category Fib/B Benabou proved the following equivalent: a) F is a fibration in Fib/B b) F is a fibration in Cat Item i) means that every fibre F_I:E_I -> D_I is a fibration and cartesian morphsims of such fibrations are stable under reindexing. Hence, the given fibration F can be profitably viewed as a fibration between the fibrations t and b. Now we can reproduce the first result: *PROP*: Assume the base fibration b has products. The following are equivalent i) t has products and F preserves them ii) F has fibred products Just as in the Cat case, ii) means that every fibre fibration F_I: E_I -> D_I has products, and these later are stable under reindexing. It is what results from spelling out ‘fibration with products’ in the 2-category Fib/B. All the relevant definitions can be found in Jacobs’ book, I think. One can specialize the products relative to certain classes of morphisms, e.g., projections from cartesian products. Playing with the various dualities on bases and fibres, one gets similar results for coproducts. With regards to proofs, I will just note that ALL the results above, namely, existence of products, fibration in Cat = fibration in Fib/B, and the fact that the composite of 2 fibrations is again a fibration are direct consequences of the lifting/factorization of adjunctions in the 2-fibration cod:Fib -> Cat *[2]* References: *[1]* *Gray, John W.* Fibred and cofibred categories. 1966 *Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) *pp. 21--83*Springer, New York* *[2] **Hermida, Claudio*. Some properties of *Fib* as a fibred 2-category. *J. Pure Appl. Algebra* *134 *(1999), no. 1, 83--109. Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Claudio Hermida -
Neil Ghani -
Richard Garner -
Steve Vickers -
Thomas Streicher