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Stream: learning: questions

Topic: Why a “universal” construction?

Nick Smith (May 31 2021 at 02:56):

I’m learning about universal constructions and universal properties for the first time. My biggest issue right now is simply the terminology. Why are we using the word “universal”? It seems like a more precise word would be “unique”, because a universal construction appears to be a unique object identified by its unique property. Were it not for tradition, could we simply be using the terms “unique object”, “unique property”, and “unique morphism” instead? Or am I missing some bigger picture here that justifies the adopted name?

Fintan Halpenny (May 31 2021 at 06:15):

The universal comes from how we talk about that object in terms of the other objects. For example, the terminal object would be defined as:
For all (universal quantification here :)) objects in X in C, the object T in C is terminal if there is exactly one morphism from X to T.

Fintan Halpenny (May 31 2021 at 06:16):

I still need to brush up on universal construction but that's what I've read when it clicked for me :grinning_face_with_smiling_eyes:

Henry Story (May 31 2021 at 06:57):

could we simply be using the terms “unique object”, “unique property”,

Not really, as there can be many other objects that are universal in a given regard. So for example there are infinitely many terminal objects in Set (all the singletons), but these are all isomorphic.

Fawzi Hreiki (May 31 2021 at 08:21):

The definition of the adjective 'universal' in the Oxford dictionary is 'done by or involving all the people in the world or in a particular group' e.g. universal suffrage

Fawzi Hreiki (May 31 2021 at 08:22):

So 'universal' just refers to something which relates to everything else in its domain in a particular way, which is just what a universal property is

Fawzi Hreiki (May 31 2021 at 08:23):

Also, consider 'universal quantification'. Plus it also just sounds cool.

Fawzi Hreiki (May 31 2021 at 08:29):

'Unique' is most definetly incorrect - the natural numbers are the universal iterative system but they are certainly not the unique iterative system (otherwise they'd be quite useless since their job is to parametrise iteration in other systems)

Nick Smith (May 31 2021 at 08:48):

there can be many other objects that are universal in a given regard. So for example there are infinitely many terminal objects in Set (all the singletons), but these are all isomorphic.

Yes, this is true. When I say "unique" I mean "unique up to unique isomorphism", as is the standard caveat.

Nick Smith (May 31 2021 at 08:49):

the natural numbers are the universal iterative system but they are certainly not the unique iterative system

I have no idea what this means :smile:

Fawzi Hreiki (May 31 2021 at 08:49):

Universal just means initial or terminal in some appropriate category right

Fawzi Hreiki (May 31 2021 at 08:50):

But there are more objects than just the initial and terminal ones. So universal ≠ unique.

Fawzi Hreiki (May 31 2021 at 08:51):

To take a simpler example, the product of A and B is the universal object with maps to A and B, but it is definitely not the unique one with such maps.

Nick Smith (May 31 2021 at 08:52):

It's the unique object that has maps to A and B and for which every other object having maps A to B has a unique morphism to it (satisfying the commutative diagram).

Fawzi Hreiki (May 31 2021 at 08:53):

Well yes. That’s the definition of universal. But it’s a convoluted way of saying universal if you have to reiterate the full definition every time.

Fawzi Hreiki (May 31 2021 at 08:53):

‘Universal group’ is much cleaner than ‘unique category containing a group such that for another other category containing a group there exists a product preserving functor … etc’

Nick Smith (May 31 2021 at 08:53):

Yeah but I'm just saying you can call it "unique" in the above sense :P because there's only one object (up to isomorphism) satisfying those conditions.

Nick Smith (May 31 2021 at 08:55):

So I'm wondering whether it would be sensible to (in an alternate universe) have just used the word "unique" in place of "universal" in the first place :stuck_out_tongue: purely to aid my comprehension when I began to learn about these constructions

Fawzi Hreiki (May 31 2021 at 08:57):

Mathematicians like short descriptions. Your proposed use of ‘unique’ isn’t an alternative name, it’s just unpacking the definition of universal which can become quite cumbersome in practice

Fawzi Hreiki (May 31 2021 at 08:58):

Category theorists have a very refined intuition for what it means for something to be ‘universal’ even when not spelled out fully

Nick Smith (May 31 2021 at 09:00):

I don't see how the word "universal" shortens the description though. Am I missing something? You seem to be saying that for those who know what a "universal construction" is, the term conveys information that doesn't need to be re-explained. But that's true of every concept that you can come up with a name for, and it equally applies to the term "unique construction" if that were used and disseminated instead.

Fawzi Hreiki (May 31 2021 at 09:02):

So the product of A and B would be the ‘unique object with maps to A and B’?

Fawzi Hreiki (May 31 2021 at 09:03):

If you replace ‘universal’ with ‘unique’ wholesale, the statements become false.

Nick Smith (May 31 2021 at 09:03):

The product of A and B would be a "unique construction", the details of which you'd need to define as always.

Chad Nester (May 31 2021 at 09:09):

FWIW you could also just call them "limits".

Chad Nester (May 31 2021 at 09:11):

Personally I think that we should call terminal and initial objects the "leewardmost" and "windwardmost" points, respectively.

Nick Smith (May 31 2021 at 09:15):

Is every universal construction a limit or a colimit? It's suprisingly hard to find an answer to this on the internet. The wiki pages don't state it.

Chad Nester (May 31 2021 at 09:16):

If we talk about "universal arrows" from an object to a functor, then yes! These turn out to be initial objects in a comma category. (I like Daniele Turi's notes as an introduction to category theory, and they have this early on: http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf)

Chad Nester (May 31 2021 at 09:18):

I think the best way to deal with mathematical terminology is not to worry too much about why it is the way it is. Maybe it used to make sense, maybe it never made sense, but it usually doesn't stand up to any kind of serious linguistic interrogation.

Fawzi Hreiki (May 31 2021 at 09:22):

Chad’s answer is technically correct but misleading. Not all universal constructions are (co)limits in the more usual way of thinking about it.

Nick Smith (May 31 2021 at 09:22):

Unfortunately I'm the kind of person who needs his concepts and terminologies to be justified :sweat_smile: it's the only way things make sense to me. I don't absorb things that I find to be arbitrary very well, including weird names! The greek letters always slow me down too.

Fawzi Hreiki (May 31 2021 at 09:23):

For example, an exponential object in a category is not a colimit or limit of any diagram in that same category.

Chad Nester (May 31 2021 at 09:28):

That is true, you have to be willing to work in a different category to obtain it as a limit.

Chad Nester (May 31 2021 at 09:36):

It might be worth bringing up that the uniqueness of universal properties is sort of complicated (to me, anyway). It's like how we can define a "there exists a unique" quantifier in predicate logic as in:
$\exists ! x . P(x) \Leftrightarrow (\exists x . P(x)) \wedge (\forall x,y . P(x) \wedge P(y) \Rightarrow x = y)$.

Chad Nester (May 31 2021 at 09:37):

The right hand side kind of tells you about the two ways we use universal properties. First, on the left, is that something satisfying that property exists, so we can give it a name and use it in further constructions. Second, on the right, is that if two things have that property they must be equal, so if we want to show two morphisms are equal we can show that they both satisfy the universal property.

Chad Nester (May 31 2021 at 09:38):

I find this helpful, but I guess it is slightly off-topic.

Nick Smith (May 31 2021 at 09:40):

The uniqueness part hasn't confused me (thus far) :smile: . I think it's cool!

Nick Smith (May 31 2021 at 09:42):

Going back to earlier though:

So the product of A and B would be the ‘unique object with maps to A and B’? If you replace ‘universal’ with ‘unique’ wholesale, the statements become false.

Isn't it nonsensical to talk about "the universal object with maps to A and B" as well? This doesn't seem like a synonym for "the limit of the diagram consisting of objects A and B", because you haven't even made a choice between limit and colimit or even mentioned what the diagram should be.

Fawzi Hreiki (May 31 2021 at 09:49):

Usually the direction of the universality (i.e. terminal or initial) is obvious from the context since the other direction will either obviously not exist or be trivial.

Fawzi Hreiki (May 31 2021 at 09:49):

But that's something that just takes some exposure to examples to get used to.

Henry Story (May 31 2021 at 10:14):

I just came across this long Twitter thread on universality and coherence that seems relevant https://twitter.com/amar_hh/status/1399275808926142464

I'll use Matteo's tweet as an excuse for a thread on universality & coherence: two ideas that are quite basic and pervasive in mathematics at all levels, notwithstanding the categorical mysticism. I'll try to keep it simple! 1/ https://twitter.com/mattecapu/status/1398901228856348678

- Amar Hadzihasanovic (@amar_hh)

Morgan Rogers (he/him) (May 31 2021 at 10:37):

Nick Smith said:

Unfortunately I'm the kind of person who needs his concepts and terminologies to be justified :sweat_smile: it's the only way things make sense to me. I don't absorb things that I find to be arbitrary very well, including weird names! The greek letters always slow me down too.

Wasn't @Fawzi Hreiki's dictionary definition of universal a good enough explanation? I hope it's clear to you by this point that "universal" means something distinct from "unique", in that a universal property doesn't just define it uniquely up to unique isomorphism, but specifies its relationship with all other entities of the same form. I could describe the terminal object in the category of sets uniquely up to unique isomorphism by formally describing the property of a set to have a unique element, but this is not the same as describing its universal property: the same description gives me uniqueness up to unique isomorphism of a 1-element set in the opposite of the category of sets, or in the core (maximal groupoid) of the category of Sets, but a 1-element set is no longer a terminal object in these other categories.

Jules Hedges (May 31 2021 at 10:54):

Nick Smith said:

Unfortunately I'm the kind of person who needs his concepts and terminologies to be justified :sweat_smile: it's the only way things make sense to me. I don't absorb things that I find to be arbitrary very well, including weird names! The greek letters always slow me down too.

You may be in the wrong field, pretty much all mathematical terminology is a weird hodgepodge of historical accidents

As exhibit A I give you the lambda calculus, which is named after a copy editor's mistake

Nick Smith (May 31 2021 at 11:09):

@Jules Hedges I'm not in the field of mathematics, I'm in the field of software development :stuck_out_tongue:. In software development, good names and documentation are essential! I'm only here to apply category theory to language design, for the purpose of aiding software development. :innocent:

Jules Hedges (May 31 2021 at 11:09):

Fair enough!

Jules Hedges (May 31 2021 at 11:12):

It may be worth pointing out that category theory is probably one of the few fields of mathematics whose original foundational papers were written in English, so our terminology has never been translated

Tom de Jong (May 31 2021 at 11:12):

Jules Hedges said:

As exhibit A I give you the lambda calculus, which is named after a copy editor's mistake

This story seems to be due to Henk Barendregt, but according to Dana Scott, Church's student, Barendregt's account is false. See https://math.stackexchange.com/questions/64468/why-is-lambda-calculus-named-after-that-specific-greek-letter-why-not-rho-calc for more info and links.
(But your points still stands: according to Scott, Church's reasoning for picking lambda was "eeny, meeny, miny, moe".)

Nick Smith (May 31 2021 at 11:14):

Wasn't @Fawzi Hreiki's dictionary definition of universal a good enough explanation?

Evidently not, else we would not still be talking about this :stuck_out_tongue:. Anyway, thanks for attempting to clarify. My understanding is growing, and I might just continue chipping away at these textbooks to grow it further.

Jens Hemelaer (May 31 2021 at 11:15):

I like the terminology "universal", but agree with @Nick Smith that it is a bit vague.

For example, "the universal object $x$ with maps $a \to x$ and $b \to x$" could mean, in theory:

• the coproduct $a \sqcup b$ (which is the initial such object); or
• the terminal object $1$ (which is the terminal such object).

In practice, what is meant is the first. It might be better to use the more precise terminology initial/terminal or colimit/limit, especially for more complicated diagrams.

Morgan Rogers (he/him) (May 31 2021 at 11:25):

I'd like to add to my comments above that the "up to unique isomorphism" part is an abbreviation of "up to unique isomorphism commuting with the data defining the property". For example, there are two distinct ways to express a two element set as a coproduct of its one-element subsets, and the universal property ensures there is a unique isomorphism transforming one into the other, which is not the identity morphism in this case.

That is, two-element sets are not defined uniquely up to unique isomorphism (there is a pair of isomorphisms between any pair of two element sets) but they do have the universal property of the coproduct of two one-element sets (in the category of sets, of course).

Jason Erbele (May 31 2021 at 17:22):

Where it comes to naming the construction, it seems to me that singling out uniqueness is at least as arbitrary as singling out the universal quantifier in statements of the form "For all gizmos there exists a unique gadget such that..." – uniqueness is third place in the logical order; it doesn't even come up until after you have described a universal situation and postulated the existence of something in that universal situation.

Jason Erbele (May 31 2021 at 18:23):

Okay, this is why I should let myself wake up a bit before responding to questions like this... I wrote the logical form incorrectly, though universal quantification still comes first, with existence and uniqueness secondary. Colloquially:
"Every gizmo factors through an [insert name here] gizmo in a unique way,"
where "[insert name here]" is commonly called universal. The [insert name here] gizmos are not the unique things here, but gizmos are the things that can have what is commonly referred to as a universal property.

Mike Shulman (May 31 2021 at 22:57):

Jens Hemelaer said:

For example, "the universal object $x$ with maps $a \to x$ and $b \to x$" could mean, in theory:

• the coproduct $a \sqcup b$ (which is the initial such object); or
• the terminal object $1$ (which is the terminal such object).

Occasionally I have seen people say "universal" and "couniversal". But I could never remember which was which. (-:

John Baez (May 31 2021 at 23:31):

Others have already answered, but I can't resist doing it too:

@Nick Smith wrote:

Why are we using the word “universal”? It seems like a more precise word would be “unique”, because a universal construction appears to be a unique object identified by its unique property.

First of all, objects identified by universal properties are usually not unique: they are unique up to isomorphism, and more importantly unique up to a unique isomorphism with some special property.

(If this sounds repetitive and confusing, wait until we get to 2-category theory. Then the story gets even longer!)

Second of all, I would say "uniqueness up to unique isomorphism" is just one aspect of what makes us want to call something a universal property.

In a category, universal properties are usually ways of characterizing an object uniquely up to unique isomorphism in terms of maps out of it or maps into it. These are called "left" universal properties and "right" universal properties, respectively.

John Baez (May 31 2021 at 23:38):

Is every universal construction a limit or a colimit?

As mentioned already: no.

John Baez (May 31 2021 at 23:44):

By the way, sometimes we weaken the concept of universal property, by removing some of the uniqueness, and talk about "versal properties". So, "universal - unique = versal".

John Baez (May 31 2021 at 23:44):

For a good example, check out the concept of weak pullback.

John Baez (May 31 2021 at 23:44):

This is like the concept of pullback, but with the word "unique" removed at one point.

Nick Smith (Jun 01 2021 at 06:01):

I accept all of the above comments and am now content with the choice of word “universal” 🦄 thanks all