Adjoint functors Puns

I forgot to mention that F: Mod_R \to Set is the forgetful functor which assigns each Module its underlying set.

Let X \in Ob(Mod_R) and Y \in Ob(Set). Let G: Set \to Mod_R.

Then

• left adjoint to F is clear for me, i think: Because then F has a left adjoint G : Set → Mod R, which assigns to each set X the free R-module generated by the elements of X.

• Why exists there no right adjoint to F? Thought: There exists more than one generating set for a module X.

Is there a better explanation of this that I can find anywhere else. I think I understand what the adjoint of a matrix is but I don't get what it has to do with optimization either

Could someone give me a reasonably non-technical account of adjoint functors? What exactly are they, and why do they 'arise everywhere'? I've been getting into category theory and I find the whole arrow-theoretic aspect of it very confusing, particularly in relation to adjoint functors.

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So, I have been studying Category Theory for a while and things eventually begin to take a shape in my mind. As was promised, adjunctions become household items.

• I can tell that a forgetful functor may have an adjoint called «free», (of which an example is the list, also known as the free monoid functor), and possibly also an adjoint called «cofree» (not sure about examples).
• I can tell that a diagonal functor may have adjoints called «sum» and «product», or generally inward and outward limits.
• I can tell that there may be an adjunction between a product and an exponential object, of which an example is the currying phenomenon in the functional programming paradigm.

That is to say, I understand several examples of adjunctions.

I also understand that adjunctions give rise to universal arrows. For example, there are universal arrows π₁, π₂: x ← x × y → y from the diagonal functor and in₁, in₂: x → x + y ← y to the diagonal functor Δ: C → C² (also called «cones»). The definition of universal arrow (after Saunders Mac Lane, section III.1) only mentions one functor though, so there is some asymmetry inherent to it that must induce some orientation between the adjoint functors from which universal arrows arise. If I know what universal arrows arise from an adjunction — specifically, are they «from a functor» or «to a functor» — I should be able to tell whether this functor is left or right in this adjunction.

What I do not understand is how to tell which functor of a given adjunction is «left» and which is «right», other than to memorize specific cases. Whence these titles? What makes a functor definitely «left» or «right»? Can I tell by looking at a functor that it must be a left or right adjoint in some adjunction?

Rant incoming.

The vast* majority of quadratics with integer coefficients aren't even factorable in the rationals. If someone wants to find the roots of a polynomial, trying to factorize is generally the worst way to do it. I understand that students need a foundational understanding of factoring to move onto more advanced concepts that they might see in calculus or complex analysis, but spending months on "factoring by grouping" vs "factoring by sight" vs "factoring with the x-box method" etc is silly.

Even if someone insists on factoring a quadratic in the wild using the above methods, they would be wise to first check that b^(2) - 4ac is a perfect square before they embark on what is probably a fruitless endeavor. And even if the discriminant is a perfect square, then at that point they've already done the majority of the work in factoring via the quadratic formula so there's no need to learn the other methods that are drilled into them, despite never understanding how the methods work.

I've had someone try to tell me that students need to know how to factor so they'll be prepared for factoring bigger polynomials in college and beyond... as if mathematicians just factor bigger and bigger polynomials as their day job. Throughout my math career including my math degree, I've only had one singular problem where I necessarily had to factor a quadratic in the course of solving; in that instance, the only method that could have worked was to find the roots via the quadratic formula and then write the factored version from there.

Being unfactorable is the norm but I've tutored multiple students who told me they thought factoring was just as valid and reliable a method as the quadratic formula for finding roots. And I don't blame them for thinking that, given there is so much emphasis on factoring while ignoring the reality that it's basically useless as a method.

</rant>

*I just ran simulations that uniformly chose integers from -50 to 50 for the coefficients of a quadratic. ~95% couldn't be factored in the rationals. When I did the same while keeping the leading coefficient equal to 1, it was still about 95% unfactorable. I ran another where I kept the coefficients between -10 and 10 and 85% were unfactorable.

What is meant by "predicate functor logic is a way of algebraizing first order logic?" How can something be algebraic/mathematical if it does not quantize over anything?

I have seen that given a pair of adjoint functors, one can construct a simplicial object by iterating the composition of adjoint functors (and using the unit and the counit morphisms). This construction is apparently called the "bar construction". I have heard that such a construction gives a 'resolution' of the original object. Is there an implicit, natural model structure and does this construction give a fibrant-cofibrant replacement of an object in that model structure?

What are some applications of the bar constructions?

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

I guess the concept didn't work

Alot of great jokes get posted here! However just because you have a joke, doesn't mean it's a dad joke.

THIS IS NOT ABOUT NSFW, THIS IS ABOUT LONG JOKES, BLONDE JOKES, SEXUAL JOKES, KNOCK KNOCK JOKES, POLITICAL JOKES, ETC BEING POSTED IN A DAD JOKE SUB

Try telling these sexual jokes that get posted here, to your kid and see how your spouse likes it.. if that goes well, Try telling one of your friends kid about your sex life being like Coca cola, first it was normal, than light and now zero , and see if the parents are OK with you telling their kid the "dad joke"

I'm not even referencing the NSFW, I'm saying Dad jokes are corny, and sometimes painful, not sexual

So check out r/jokes for all types of jokes

r/unclejokes for dirty jokes

r/3amjokes for real weird and alot of OC

r/cleandadjokes If your really sick of seeing not dad jokes in r/dadjokes

Punchline !

Edit: this is not a post about NSFW , This is about jokes, knock knock jokes, blonde jokes, political jokes etc being posted in a dad joke sub

Edit 2: don't touch the thermostat

Do your worst!

How the hell am I suppose to know when it’s raining in Sweden?

Ants don’t even have the concept fathers, let alone a good dad joke. Keep r/ants out of my r/dadjokes.

But no, seriously. I understand rule 7 is great to have intelligent discussion, but sometimes it feels like 1 in 10 posts here is someone getting upset about the jokes on this sub. Let the mods deal with it, they regulate the sub.

I've recently started learning Category Theory and have finally gotten around to learning about adjunctions.

It's been said that things like products, coproducts, and exponentials can be stated in terms of adjunctions. I'm having a hard time understanding how one can get the uniqueness conditions required for these constructions from the unit-counit definition of an adjunction.

For example. A product is a special object such that given any other object that has a pair of morphisms p : c -> a and q : c -> b then there must exist a unique morphism h : c -> axb such that p = fst . h and q = snd . h where fst : axb -> a and snd : axb -> b. How can one prove this statement given the unit and counits for the diagonal-product functor adjunction?

P.S: I'm not a mathematician but a Haskell programmer so I'm apt to not understanding the math speak.

Mathematical puns makes me number

They were cooked in Greece.

I'm surprised it hasn't decade.

Now that I listen to albums, I hardly ever leave the house.

He lost May

My title is vague, but that’s the best way to summarize what I’m thinking.

I’m a new math grad student finishing up a course on representations of finite groups. This is my first taste of rep theory and I’m enthralled.

My first specific question: why are only certain categories studied in association with representations? The big ones seem to be groups, associative algebras, and Lie algebras. Was representation theory of, say, rings ever investigated? Why or why not? Besides the obvious answer that we get important results in the three categories that I listed. Was this known beforehand or were there failed attempts at further generalization?

Second, even restricted to finite groups, representations seem to have a lot of important properties. The most striking one to me is this notion of induced representation - that a representation on any subgroup extends uniquely to that of the whole group. And of course it has many desirable properties like Frobenius reciprocity. Does this induction functor generalize to other categories, perhaps with a more abstract characterization? In other words, are there other functors which have these nice properties that induction does? I imagine any reasonable answer would have to involve adjoint functors (given the Frobenius formula).

Two muffins are in an oven, one muffin looks at the other and says "is it just me, or is it hot in here?"

Then the other muffin says "AHH, TALKING MUFFIN!!!"

Don't you know a good pun is its own reword?

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

And now I’m cannelloni

Because she wanted to see the task manager.

And boy are my arms legs.

But that’s comparing apples to oranges

Put it on my bill

Heard they've been doing some shady business.

but then I remembered it was ground this morning.

Edit: Thank you guys for the awards, they're much nicer than the cardboard sleeve I've been using and reassures me that my jokes aren't stale

Edit 2: I have already been made aware that Men In Black 3 has told a version of this joke before. If the joke is not new to you, please enjoy any of the single origin puns in the comments

BamBOO!

They’re on standbi

A play on words.

Calcium, nickel, neon

My daughter, Chewbecca, not so much.