Re: Composition of Fibrations and Quantification

[Claudio Hermida <claudio.hermida@gmail.com>]

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



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