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Stream: learning: questions

Topic: FS in Cat using connected functors and discrete bifibrations

Rubén Maldonado (Jan 30 2025 at 23:42):

In the nLab article on factorization systems, there is a series of examples of factorization systems in the category Cat. My focus is on Example 6.11, where the left class consists of connected functors and the right class consists of discrete bifibrations.

From the same reference, I understand that a functor  

$$F:C\to D$$

is connected if the induced functor 

$$\pi_1(F):\pi_1(C)\to \pi_2(D)$$

on the fundamental groupoids is essentially surjective and full. However, the article does not provide an explicit description of how to factor a functor using this factorization system.

I would greatly appreciate it if someone could explain how to explicitly construct such a factorization for a given functor 

$$F:C\to D.$$

Any insights or references would be very helpful!

Mike Shulman (Jan 31 2025 at 00:07):

Note that this is not on the main nLab but "Joyal's Catlab", which was mostly written by Joyal. You might want to ask him.

John Baez (Jan 31 2025 at 06:33):

Btw, for any beginners out there, "$\pi_2(D)$" should read "$\pi_1(D)$".

Rubén Maldonado (Jan 31 2025 at 22:31):

Thanks for catching that! It has been corrected

Kevin Carlson (Feb 03 2025 at 18:19):

There's a standard not-very-explicit way to construct orthogonal factorization systems in locally presentable categories which Joyal knows very, very well and may be relying on here. I'm always a bit unsure of a good reference for this in the case of orthogonal factorization systems, but in the more general case of weak factorization systems it's called the small object argument and is used constantly in model category theory. The upshot is that there's no reason to believe there should be a particularly nice formula for this factorization system, which I've never heard mentioned anywhere else.

Mike Shulman (Feb 03 2025 at 20:20):

It's not even obvious to me that these two classes of maps are a prefactorization system, i.e. each is the class of maps that's orthogonal to the other.