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Stream: learning: questions

Topic: Example of universal property

Ralph Sarkis (Oct 17 2020 at 14:41):

I was wondering if anyone had an example of a universal property which: 0) involves only very basic mathematics (stuff taught early courses in uni like groups, rings, metric spaces or vector spaces), 1) is not a (co)limit, 2) arises as a terminal object in a comma category, 3) does not involve a forgetful functor.

I am giving a lecture on universal properties next week (after having taught (co)limits) and I am building a list of other kinds of universal properties. I have already developed the constructions of the free monoid generated by a set, the abelianization of a group and a basis of a vector space (equivalently, the free vector space). However, these all arise as initial objects in a comma category involving a forgetful functor into Set. Hence, I am looking for something a bit different. Last time I taught this class, I showed them how the evaluation function $A^X \times X \rightarrow A$ was a universal morphism, but it did not seem to click with my students. Any ideas?

Fawzi Hreiki (Oct 17 2020 at 19:25):

If you're willing to forgive that the inclusion of the integers into the reals is actually a forgetful functor, one of my favourite examples is the adjoint triple given by (ceiling $\dashv$ inclusion $\dashv$ floor).

Fawzi Hreiki (Oct 17 2020 at 19:36):

Another good example is a natural numbers object. Every first year mathematics student learns how useful induction is, but most of the time, they don't realise that this is just a special case of recursion, which is much much more fundamental.

Fawzi Hreiki (Oct 17 2020 at 19:45):

A nice exercise, after defining what a natural numbers object is, is proving that taking the product with $\mathbb{N}$ gives a right adjoint to the inclusion $\mathscr{C}^{\circlearrowleft} \hookrightarrow \mathscr{C}$ (where the domain is the category of endomorphisms in $\mathscr{C}$).

John Baez (Oct 17 2020 at 21:17):

Since Ralph seems to be teaching an introductory class, I hasten to add that you can define the universal property of the natural numbers in the category Set without getting into anything "fancy".

Fawzi Hreiki (Oct 17 2020 at 21:33):

Yeah, beyond the category of sets, the next most accessible examples are some simple presheaf categories (e.g. endomaps of sets, sets through discrete time, graphs, etc...)

Ralph Sarkis (Oct 17 2020 at 21:36):

Thanks Fawzi, I will think a bit more about this adjoint triple, but we have already covered the natural numbers as an initial $(\mathbf{1}+-)$--algebra. (We are doing natural transformations in the lecture after that, so I cannot discuss functor categories yet.)

Dan Doel (Oct 17 2020 at 21:37):

What about the extended natural numbers as a final $(1 + -)$ coalgebra?

Fawzi Hreiki (Oct 17 2020 at 21:38):

The reason I like adjoint triples as examples is that you see the contrast between the left and right adjoints

Fawzi Hreiki (Oct 17 2020 at 21:43):

Another nice adjoint string is (connected components $\dashv$ discrete $\dashv$ points $\dashv$ codiscrete)

Fawzi Hreiki (Oct 17 2020 at 21:43):

Either with topological spaces over sets, or something more combinatorial

Morgan Rogers (he/him) (Oct 18 2020 at 09:38):

Ralph Sarkis said:

I was wondering if anyone had an example of a universal property which: 1) is not a (co)limit, 2) arises as a terminal object in a comma category

Umm... Aren't these contradictory requirements?

Morgan Rogers (he/him) (Oct 18 2020 at 09:42):

Ralph Sarkis said:

(We are doing natural transformations in the lecture after that, so I cannot discuss functor categories yet.)

As a bridging example which you should be able to present even before presenting functor categories in full is a category of the form $$[M^{\mathrm{op},\mathbf{Set}]$$ of right actions of a monoid $M$; you can drop the op and use left actions if you prefer, or even use a group if you insist. The forgetful functor has left and right adjoints, but since you're looking for non-forgetful things, there is another adjoint triple whose middle functor is the functor from $\mathbf{Set}$ sending a set to the trivial action of the monoid on that set: this is the global sections geometric morphism, if that means anything to you :wink:

Ralph Sarkis (Oct 18 2020 at 10:28):

[Mod] Morgan Rogers said:

Umm... Aren't these contradictory requirements?

Indeed, I meant not a (co)limit in the base category.

Ralph Sarkis (Oct 18 2020 at 10:28):

Thanks, I'll look into this triple as well.