a note on sources of my text of fibered categories

a note on sources of my text of fibered categories

[Thomas Streicher]

> Renewed study of these ideas, in light of the later exposition by
> Thomas Streicher, should lead to further applications (for example
> to the solution of problems posed in my 1972 Perugia Notes.)

Thanks for mentioning this text. But since I think ownership of ideas
is an important issue I want to point out that the aim of this text
was to exhibit part of the ideas and results of Jean B'enabou's almost
single handed approach to Fibered Categories as a foundation of
Category Theory over most general base categories. I also tried to explain
some work by J.-L. Moens (his 1982 Thesis) where I think I have added
a bit of additional material. As far as B'enabou's work is concerned
my sources were Roisins notes from B'enabou's 1980 Louvain-la-Neuve
lectures. My exposition evolved over the years and I have integrated
additional material and made corrections as I learnt from Jean for
which I am very grateful to him.

At one place I have referred to a fibred version of the Special
Adjoint Functor Theorem which one can find in Par'e and Schumacher's
text or alternatively in J.Celeyrette's These d'Etat from 1974 under
supervision of B'enabou.

The references in my text are not exhaustive at all. It's certainly a
mistake not to have formally referred to work by Grothendieck and Giraud.
But I was not using too much these original sources.

Moreover, I have used results from a paper by Mamuka Jibladze. I have
not given the precise reference but made clear in the title of the
appendix that it is Mamuka's result.

The aim of these notes was not to document precisely who did
what. However, on the first page I clearly stated whose work
influenced me!

I am also aware that in Bill's "Perugia Notes" one can find the idea
that fibered or indexed categories are suitable for doing category
theory over a base topos. But as I said it wasn't my intention to
document the history of ideas but rather to write up the view of
things as I learnt it from Jean's work and private communications.

Thomas Streicher



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Re: a note on sources of my text of fibered categories

[Ronnie Brown]

I would like to say that I found Thomas' notes on fibred categories very
useful in showing me the importance of these ideas as a general view of
some key processes in the book
"Nonabelian algebraic topology: crossed complexes, filtered spaces,
cubical homotopy groyupoids" RB, P.J. Higgins, R. Sivera, EMS Tract
vol15 (2011).

In homotopy theory identifications in low dimensions can strongly affect
higher dimensional homotopy invariants.  One way of dealing with this is
to have algebraic  homotopy invariants which have structure in a range
of dimensions. Then we have  forgetful functors from dimension n to
lower dimensions.  In the cases we deal with, these functors are
bifibrarions, and so general properties of these are helpful  in
formalising  a number of elementary but useful calculational facts.
This general approach is given in Appendix B of the above book (of which
EMS allow a pdf to be available on my web page).

As a simple example, the functor Ob from groupoids to sets is a
bifibration. The general theory  shows how this is useful in calculating
colimits of groupoids in  applications of the fundamental groupoid
version of the Seifert-van Kampen Theorem. This model is also useful in
higher dimensions.

I would like to be shown  applications of the more advanced theory of
fibred and cofibred categories to these areas!

Ronnie Brown
http://pages.bangor.ac.u/~mas010/nonab-a-t.html


On 02/10/2014 08:37, Thomas Streicher wrote:
>> Renewed study of these ideas, in light of the later exposition by
>> Thomas Streicher, should lead to further applications (for example
>> to the solution of problems posed in my 1972 Perugia Notes.)
> Thanks for mentioning this text. But since I think ownership of ideas
> is an important issue I want to point out that the aim of this text
> was to exhibit part of the ideas and results of Jean B'enabou's almost
> single handed approach to Fibered Categories as a foundation of
> Category Theory over most general base categories. I also tried to explain
> some work by J.-L. Moens (his 1982 Thesis) where I think I have added
> a bit of additional material. As far as B'enabou's work is concerned
> my sources were Roisins notes from B'enabou's 1980 Louvain-la-Neuve
> lectures. My exposition evolved over the years and I have integrated
> additional material and made corrections as I learnt from Jean for
> which I am very grateful to him.
>
> At one place I have referred to a fibred version of the Special
> Adjoint Functor Theorem which one can find in Par'e and Schumacher's
> text or alternatively in J.Celeyrette's These d'Etat from 1974 under
> supervision of B'enabou.
>
> The references in my text are not exhaustive at all. It's certainly a
> mistake not to have formally referred to work by Grothendieck and Giraud.
> But I was not using too much these original sources.
>
> Moreover, I have used results from a paper by Mamuka Jibladze. I have
> not given the precise reference but made clear in the title of the
> appendix that it is Mamuka's result.
>
> The aim of these notes was not to document precisely who did
> what. However, on the first page I clearly stated whose work
> influenced me!
>
> I am also aware that in Bill's "Perugia Notes" one can find the idea
> that fibered or indexed categories are suitable for doing category
> theory over a base topos. But as I said it wasn't my intention to
> document the history of ideas but rather to write up the view of
> things as I learnt it from Jean's work and private communications.
>
> Thomas Streicher
>
>



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]