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Stream: learning: questions

Topic: learning about fibred categories

David Egolf (Jan 21 2022 at 20:35):

I'd like to learn a little bit about fibre categories, at least enough to understand the "Grothendieck construction" and the "category of elements". I have the book by Jacobs Categorical Logic and Type Theory which looks great.

Are there any other resources you would recommend for learning about fibre categories?

Joe Moeller (Jan 21 2022 at 20:51):

This is an unusual answer, but the first source I learned about it from is Christina Vasilakopoulou's thesis.

Mike Shulman (Jan 21 2022 at 20:58):

Weird synchronicity -- I feel like this is the third or fourth time I've heard this question in the past week or two. Two references that I've given are volume 2 chapter 8 of Handbook of categorical algebra, and chapter B1 of Sketches of an Elephant, . But I don't really feel like I have a good answer to this question; there are a number of modern "introduction to category theory" textbooks, but do any of them reach the Grothendieck construction?

Joe Moeller (Jan 21 2022 at 20:59):

I think Riehl talks about the category of elements.

Joe Moeller (Jan 21 2022 at 21:00):

Johnson and Yau dedicate two chapters to it I think.

Joe Moeller (Jan 21 2022 at 21:06):

Something to keep in mind @David Egolf is that you either talk at the level of presheaves and the category of elements, or you must talk about 2-categorical things. This will skew what sources will talk about it.

Patrick Nicodemus (Jan 21 2022 at 21:21):

Jacobs' book is definitely a good source on the grothendieck construction, it's fundamental for him

David Egolf (Jan 21 2022 at 21:23):

Thanks everyone, those look great!

Joe Moeller said:

Something to keep in mind David Egolf is that you either talk at the level of presheaves and the category of elements, or you must talk about 2-categorical things. This will skew what sources will talk about it.

That is good to know. 2-categorical things still scare me, so that will limit which sources I work from.

Mike Shulman (Jan 21 2022 at 22:46):

Johnson and Yau is a good reference, thanks for pointing that out Joe. Reading their earlier chapters might be a good way to get over a fear of 2-categories. (-:

Bryce Clarke (Jan 21 2022 at 22:52):

Chapter 12 in Barr and Wells - Category Theory for Computing Science covers Fibrations and the Grothendieck construction. It is where I first learnt about it.

Bryce Clarke (Jan 21 2022 at 22:54):

It is a fairly gentle introduction which from memory does not use 2-categories

Paolo Perrone (Jan 21 2022 at 22:57):

What is your background?

John Baez (Jan 21 2022 at 23:44):

I think we're talking about "fibred categories", not "fibre categories", right?

David Egolf (Jan 22 2022 at 00:06):

John Baez said:

I think we're talking about "fibred categories", not "fibre categories", right?

Yes. I've updated the title.

David Egolf (Jan 22 2022 at 00:15):

Paolo Perrone said:

What is your background?

I have an engineering background. I've self-taught myself some basics of category theory working from Seven Sketches and RIehl's Category Theory in Context (as well as few other math things like introductory real analysis, introductory abstract algebra, and introductory topology).

John Baez told me that a category I'm interested in for thinking about observations in the context of imaging (see #practice: applied ct > spans and images) is the "category of elements of a presheaf category". That's one reason I'd like to learn what a"category of elements" is.

So, a lot of these resources will be a bit hard for me. But some of them look approachable!

John Baez (Jan 22 2022 at 01:16):

I'll just tell you: given a functor $F: C^{\mathrm{ op}} \to \mathsf{Set}$, an element of $F$ is an element of some set $F(c)$, and a morphism from the element $x \in F(c)$ to the element $x' \in F(c')$ is a morphism $f : c' \to c$ such that

$F(f)(x) = x'$

Elements of $F$ and the morphisms between them form the category of elements of $F$.

John Baez (Jan 22 2022 at 01:20):

You can see that you've got a special case of this in your comment here.

If $\mathrm{Open}(Y)$ is the poset of open subsets of your topological space $Y$ there's a functor $F: \mathrm{Open}(Y)^{\textrm{op}} \to \mathrm{Set}$ such that $F(U)$ is the set of functions $f: U \to H$. I'll let you guess what this functor does on morphisms - it does the only imaginable thing.

John Baez (Jan 22 2022 at 01:22):

You can check that the category you cooked up from this situation is the category of elements of $F$.

David Egolf (Jan 22 2022 at 01:23):

Thanks for explaining that!
I'm taking a little break from this today, but I'll give this some proper consideration later this weekend.

John Baez (Jan 22 2022 at 01:25):

A functor $F: \mathrm{C}^{\mathrm{op}} \to \mathrm{Set}$ is called a presheaf, and your example is one of the basic examples that made people invent presheaves. An even more typical example would be if you let $F(U)$ consist of continuous functions $f: U \to H$, assuming $H$ has some topology. Then you'd actually be using the topology on $Y$ in a real way.

Jon Sterling (Jan 22 2022 at 09:48):

Hi all, I have been working on some (slightly idiosyncratic) lecture notes on the topic of fibered category theory. I caution you because it is very provisional and I have not yet given much motivation or examples, as I am first trying to work out the theory in a way that I like. http://www.jonmsterling.com/math/lectures/categorical-foundations.html

Fawzi Hreiki (Jan 22 2022 at 09:56):

Another good set of notes is these by Streicher

Jon Sterling (Jan 22 2022 at 10:33):

Yes, I really like Thomas's notes too. They are a very important source for me.

Fawzi Hreiki (Jan 22 2022 at 10:52):

I think the first step though is to internalise the Grothendieck construction at the level of presheaves of sets.

If you’re interested in algebraic/differential geometry, then you can try and translate many geometric constructions on schemes/manifolds to their categories of (affine) elements.

Jon Sterling (Jan 22 2022 at 11:27):

That's a good idea; I am not a geometer, but from what little I know of scheme theory the category of elements is very natural for geometry because one is considering at all times not just global points, but generalized points. The category of elements makes this explicit.

David Egolf (Jan 23 2022 at 16:16):

Thanks again, everyone.