What is...a concrete category?

[@MathForLife]

What is the "inverse" for a forgetful functor?

[@VisualMath]

Great question! In some sense the answer is "obvious". Take for example Forget from GROUP to SET. As there are many groups on the same set (for example Z/4Z and Z/2ZxZ/2Z on a four element set), Forget can not be invertible since there is information loss. However, it feels like there should be an inverse. That is why you are asking, and I totally agree! So you might try to make that work, giving up a few properties of what an "inverse" should be. You will end up with the definition of an adjoint functor (covered in later videos), which is a version of an inverse. In this sense the inverse of Forget is Free, the functor that assigns a set to the free group on this set. In short: Forget is not invertible, but the "inverse" is the free functor. That is (kind of) a cool statement - no wonder that "free things" appear everywhere in mathematics ;-)

[@MathForLife]

@@VisualMath yes, exactly, that's why I put inverse in quotes. I am just trying better understand the motivation / visual picture of an adjoint functor. Your videos help a lot!

[@VisualMath]

I am planing to make videos on the motivation behind adjoint functors, but this reply is not supposed to be a cliffhanger ;-), so: I have two different (but, of course equivalent) explanations to offer why adjoint functors make sense and appear everywhere: - The first explanation is the one you have in mind, the keyword being the “inverse” in quotation marks. Namely, they are not really inverses but instead of GF=id_C (isomorphism), GF\cong id_C (equivalence) it is now GF \to id_C in a certain controlled way (plus the opposite equations). The resulting functors F and G are then “inverses” in the sense that one can recover the hom-spaces: hom_D(X,FY)=hom_C(GX,Y). Here F:C\to D and G goes in the opposite direction. However, I find that still a bit puzzling. Without knowing tons of examples where this actually happens, this explanation feels a bit ad hoc in my opinion. A better one, which works without knowing any example is: - Adjoint functors generalize planar isotopies. How could that work? Well, the equations the two maps GF \to id_C and id_D \to GF have to satisfy are the zigzag relations in terms of string diagrams: en.wikipedia.org/wiki/String_diagram#Example I find the second explanation very cute and it totally makes sense for me. However, there is no free lunch and, I guess, one needs to know string diagrams and one needs a certain amount of love for diagrams before appreciating the second explanation.