weirdvic

[weirdvic]

autistic guy with autistic anxiety is so good at inventing brand new social rules that may or may not exist specifically for the purpose of getting scared of breaking them. meanwhile somehow collecting a 100% strike rate on breaking the social rules that do exist. it’s a special talent

[weirdvic]

Ive tried to explain the difference between analysis and algebra to my friends that analysis was made prodolenently by series people and category theory a major part of algebra was made by a guy who went on to be a monk. Do you have any better way to explain this to laymen why keeping the mystique intact

I would say that analysis is the organic fleshy part of mathematics. It's mushy. It's soft. It decays. It's relatable. Ultimately, if we want to explore space, it holds us back. But, it is inarguably human, and we would lose something central were we to overlook it.

Algebra, in contrast, is crystalline and metallic. It's the mathematics of the machine mind. When we meet aliens will they speak of convergence? Maybe yes, maybe no. It really depends on whether their culture also obsesses on always getting to the end, perhaps they are happy to study only the journey. But they will know of symmetry. That is certain. They will recognise our rings and our groups. Algebra is universal in this way. It carries us far beyond the limits of our human imagination.

[weirdvic]

the other day i saw a meme that said everyone knows link is trans but nobody knows which way and. yeah. no matter who you are i love you beautiful beautiful

traditional happy pride month from everyone's favorite transgender video game characters

[weirdvic]

Tired of watching utter garbage. I'm rewatching the realest show there is: Shoujo Kakumei Utena!

[weirdvic]

okay

[weirdvic]

Every so often I remember Sugar Song To Bitter Step exists. Like I just did. And now you do too!

It's really nice, so many fandoms have their own version too it's great!

The Pokemon and Homestuck ones are my favourite so far.

[weirdvic]

I put normal amounts of hours into games I like.

[weirdvic]

Are there shows like The Mentalist or House MD or Lucifer or Sherlock but instead of being centered around one man they're centered around one woman? I can't think of any but I don't know that many shows.

Been watching House MD lately. This show sucks ass.

Anyways I'm at the fourth season. I think I have a problem.

[weirdvic]

Been watching House MD lately. This show sucks ass.

Anyways I'm at the fourth season. I think I have a problem.

[weirdvic]

New ideas! Functoriality achieved :)

Have C again be a concrete category.

First new idea, I thought of a relatively obvious modification to your construction:

Have the objects be a an element of A for some A object of C, and morphisms from a in A to b in B be morphisms f from A to B such that f(a)=b.

Your construction from yesterday maps each object to a subcategory of this category.

Having shifted our focus away from endomorphisms, we are no longer looking to build f' such that F○f = f'○F (reusing notations from before), but simply F○f=f' (with f' from A to A').

This is obviously no longer an obstruction, but it is a much less interesting question! This modification is not neutral.

Second new idea, there is a cool projection functor from the associated category to C that I missed yesterday, mapping a in A to A and f from A to B such that f(a)=b to f from A to B.

This last part is really long. I'm so sorry.

You might notice I haven't adressed the functoriality of this new construction. This is still a problem, but now the struggle is with the objects, not the morphisms.

In short, if you have a functor F from C to D concrete categories and an element a in an object A, there is no natural way to associate to it an element of F(A) as is.

As is often the case in category theory, when something isn't functorial it's simply not abstract enough!

Third new idea, this can be generalized to a great extent. Take a category C and F a covariant functor from C to Sets.

You can use the forgetful functor if C is concrete, or hom(X, . ) for some X if C is locally small for example!

You can form a category whose objects are (A,a) where A is an object of C and a is an element of F(A), where the morphisms from (A,a) to (B,b) are morphisms f from A to B such that (F(f))(a) = b.

Notice that there is still a projection functor, which is neat :)

We are associating categories to functors, so in order to make this association functorial we need to associate a functor to a natural transformation!

To be precise, our objects are pairs (C, F) with C a category and F a covariant functor from C to Sets, so our morphisms from (C, F) to (D, G) should be pairs (E, eta) with E covariant functor from C to D and eta natural transformation from F to G○E.

Given an object A of C, eta gives a morphism from F(A) to G(E(A)). So we simply associate to (A,a) the pair (E(A), (eta(A))(a)).

Given a morphism f from (A,a) to (B,b), i.e a morphism from A to B such that (F(f))(a) = b, E(f) is a morphism of E(A) to E(B) such that, by definition of eta, (G(E(f))((eta(A))(a)) = (G(E(f))((eta(B))(b)).

At last, we have achieved functoriality!

So in order to make your previous association functorial, you have to

1. Associate only one category to a concrete category instead of one category for every object in a concrete category. Tragically, this takes the focus away from endomorphisms completely.

2. For morphisms between concrete categories, take functors that are compatible with the forgetful functors up to a natural transformation.

I need someone more knowledgeable than me to tell me if this is useful at all. Recently I’ve been thinking about endomorphisms in concrete categories (that is categories where we can speak of elements of objects) and I realized that for every object A in such a category can be associated with a category where objects are elements of A and for any two elements a and b hom(a,b) is the set of endomorphisms f such that f(a)=b. This association doesn’t seem to be functorial, or if it is I can’t figure out how it acts on morphisms. This construction seems interesting but I can’t figure out how. So my questions are:

1. Does what I’m describing already exist and if so where can I read about it?

2. Is this interesting at all or am I just being silly?

[weirdvic]

I admire the physicists. I could marry a physicist.

[weirdvic]

Something weird happened yesterday. I spent most of the day studying, and at some point I was really tired of it and I was looking for some fun maths for a break.

And I realized I had no such thing. Since I no longer have classes, I no longer have light subjects.

I need to pick up some fun little thing. Some of my most esoteric knowledge comes from just being bored and investigating random stuff.

[weirdvic]

I could and did in fact fuck a physicist. And I would do it again.

I admire the physicists. I could marry a physicist.

[weirdvic]

Functoriality doesn't happen as is, but it is a very interesting idea!

Let A, A' objects, F morphism from A to A', a and b in A, and f morphism from A to itself such that f(a)=b.

You associate to A (and A') categories, so in order to make this association functorial you would want to associate to F a functor from the category associated to A to that associated with A'.

Obviously to a (and b) you associate F(a) and F(b).

Now the question is whether the exists f' morphism from A' to itself such that f'(F(a)) = F(b), and whether you can choose such an f' in a functorial way.

Such a morphism simply doesn't always exist, and even if it does there isn't always a way to choose one in a functorial way.

For example you can keep only isomorphisms, then obviously f' = F ○ f ○ F^-1 works.

What you can do is simply restrict your functor! Take some subcategory of C with the same objects, but morphisms having appropriate lifting properties.

To be clear, this means restricting the choice on F, not f. It is sad, but not that sad.

You can take bigger subcategories, for example surjections that admit a unique section (just replace F^-1 with the section.)

Or better yet, take for morphisms pairs (F,S) of a surjection and a section! This isn't a subcategory but it feels like an extremely appropriate sidestep!

I need someone more knowledgeable than me to tell me if this is useful at all. Recently I’ve been thinking about endomorphisms in concrete categories (that is categories where we can speak of elements of objects) and I realized that for every object A in such a category can be associated with a category where objects are elements of A and for any two elements a and b hom(a,b) is the set of endomorphisms f such that f(a)=b. This association doesn’t seem to be functorial, or if it is I can’t figure out how it acts on morphisms. This construction seems interesting but I can’t figure out how. So my questions are:

1. Does what I’m describing already exist and if so where can I read about it?

2. Is this interesting at all or am I just being silly?

[weirdvic]

Wearing sandals outside has brought me disproportionate amounts of euphoria. Maybe life isn't so bad :)

[weirdvic]

Probably my favourite book I read this year!! If you ever indulge and want to talk about it you can do it with me :3

My copy of Fourier Analysis on Number Fields calls to me, but it would be indulgent to read it at this time. Definitely endorse that book though.

[weirdvic]

i’ll say it a hundred times because some of you need to hear it a hundred times but the trick to liking yourself again is learning new skills and hobbies or returning to ones you had. it makes you so confident learning new shit all the time.