Please explain the basics of category theory to in terms I can understand(I am a slightly gifted code monkey).
>>16540844It’s easiest to understand if you don’t try to be a smartass about it and just use concepts that you already know.A function is a certain relation between two sets. If those two sets have some additional “structure” to them, you want to “preserve” this structure. For example, a map between linear spaces that preserves structure is a linear transformation. You can generalize this to many other objects in mathematics. This general notion of a structure-preserving map is called a morphism. Sets with structure are called objects and they’re group by “category”. So you have the categories of linear space, topological spaces, groups, rings, manifolds etc.You then just start playing the game of what information can you deduce just from the net of morphisms alone. The classic example is the isomorphism theorems in algebra. Picrel is the second isomorphism theorem.
>>16540849oh i thought a morphism was any kind of function between two categories or objects within categories
>>16540854A function is a morphism in the category of sets. A morphism is a generalization of a function that ensures that the structure of whatever it is you’re mapping is preserved.
>>16540855is a morphism less rigid or more rigid than a function. it sounds like your saying it's only a morphism if it preserves some kind of structure between categories. But from my experience as a code monkey, a function is essentially any conceivable process that can be encoded in machine interpretable language. So is a morphism more general or less general than a function (from my code monkey interpretation).
>>16540855>the structure of whatever it is you’re mapping is preserved.Give a concrete example.
>>16540859It depends on what you mean by rigid. A morphism obeys more rules. A function f between linear space can be any wildest thing you can imagine, but if it doesn’t satisfy f(a+b) = f(a) + f(b), then it’s not a linear transformation aka a morphism of linear space. Another example is a homeomorphism of topological space. A function must be continuous to count as a homeomorphism. Continuity in this case is the “rule” that “preserves” the topological structure.>>16540861Already did with linear spaces. See above for topological spaces. For groups, a group morphism must satisfy f(ab) = f(a)f(b) and f(identity) = identity. And so on. You generally get a better intuition for categories the less “pure category theory” you do, ironically enough.
>>16540867oh, i seem to have been interpreting category theory wrong, i was thinking of morphisms as "processes" between abstract objects or categories, i guess that's just my code monkey brain thinking it's familiar with abstract mathematics because it uses some of the same terminology
>>16540869You can think of them however you please as long as it works in your head. I’m a math guy so I’m telling you how it works in mine. The most important thing imo is to develop an intuition for it by doing concrete examples (I’m sure CS has plenty) because pure category theory sounds like gibberish if you don’t do that.
>>16540871can category theory be generalized and applied to real world phenomena or is it mostly useful in the real of mathematical logic and abstractions
>>16540872I mean, depends on what you mean by applications. It’s like asking “what are the applications of set theory”. Technically everything and nothing because both set theory and category theory underlie all of mathematics but on a “meta” level that is not immediately apparent. I already told you about linear transformations. Those are ubiquitous.
>>16540873i'm just wondering if i can reason about what i want to have for lunch using category theory or if I'm going to study this stuff and it's going to be a waste of time
>>16540875Oh, so you’re trolling. Take care, faggot.
>>16540877see ya later nerd
it sounds like i need to invent my own category theory free from the constraints of mathematical formalism
>>16540844https://www.logicmatters.net/categories/
>>16540844>20th century>people noticed lots of different mathematical theories fell into the pattern of studying certain classes of objects and the transformations between them>many arguments relied fundamentally only on the formalism of juggling around diagrams of transformations instead of the particular theories or objects being talked about>people abstracted out these common procedures to create a very general but very boring theory that basically just deals with all this shit without having to rehash it a million times in a million different contexts>>16540875>it's going to be a waste of timeyes
>>16542311is categorical thinking useful in real life though, i've been having thought processes about categories and morphisms that seem interesting
>>16543269Anything is useful in real life if you are not braindead cattle. Imagine coming up to a car designer and going “bro how will this help me pick up chick doe?” It’s neither his job nor responsibility. If you suck at picking up chicks, no bugatteh will help you and a bugatti designer will certainly be of no help to you.
can some category theorist here explain the yoneda lemma in simple terms
It's pretty weird stuff. It started as a meme about replacing sets with drawings, then it morphed into using abbreviations for two formulas (for example: the class Grp of groups is an abbreviation for the two formulas phi(x) and psi(y) where phi(x) holds when x = (G,1,*,^-1) is a group and psi(y) holds when y is a homomorphism from one group to another, but we have to include the domain and codomain as well, so when y = (G,H,rho) where G and H are groups and rho : G -> H is a group homomorphism) and then it morphed into outright confusion where you claim "Let C be a category" could refer to sets or formulas, but you don't know.From a mathematical point of view, the actual object of study is a certain kind of partial semigroup, and that's what people should be studying, but instead they just talk nonsense and get confused. Also, if you title your work "... for the Working Mathematician" then you're a pseud.The actual mathematical objects of study are categorical semigroups. To define this, you first have to define partial semigroup as follows: a _partial semigroup_ is a pair (S,*) where * is a partial binary operation * : T -> S where T c= S x S is a subset (possibly empty) and * satisfies: for all a,b,c <- S, if (a,b) <- T, (a*b,c) <- T, (b,c) <- T, and (a,b*c) <- T, then (a*b)*c = a*(b*c). A partial semigroup is _categorical_ if- for all a,b,c <- S, if (a,b) <- T and (b,c) <- T, then (a*b,c) <- T and (a,b*c) <- T.
>>16543269Unless you're a literal mathematician or physicist, no high level math will ever be (directly) useful to you, and outside of a tiny handful of other professions nothing past elementary school is useful to anyone either.
>>16540872It's as useful to a non-mathematician as OOP is to a non-programmer.
>>16540844It's quite simple, frogshitter:1. kys2. to3. /pol/
>>16545898OOP is POO backwards
>>16546077Took you 4 days to come up with that one?
>>16540844it’s what the previous generation of tranny’s did before sudoku and before the current generation of tranny’s discovered rust
>>16544009It's worth pointing out that the category theory obsession with using pairs of classes (they even have a special term for this, the large category) a.k.a. formulas for representing the objects and morphisms in a category is utterly and totally superfluous. Taking the example of groups, for instance, we can simply say:Let A be a set, X the set of groups G such that G c= A as a set of elements, and Y the set of triples (G,G',phi) such that G,G' <- X and phi: G -> G' is a group homomorphism. Then (X,Y) is a category.If we want to prove something about direct products of groups, we can add the following assumption; for all a,a' <- A, the ordered pair (a,a') <- A. This will guarantee that for all groups G,G' <- X, their direct product G x G' is also an element of X.Let us now prove the characteristic property of direct product:Suppose H <- X and phi : H -> G and psi : H -> G' are homomorphisms. Then there is a unique homomorphism theta : H -> G x G' such that pi[G] o theta = phi and pi[G'] o theta = psi where pi[G] and pi[G'] are projections onto the factors in the subscript, namely theta(h) = (phi(h),psi(h)).We can now say: let V be the set of triples (H,phi,psi) where H <- X and phi: H -> G and psi: H -> G' are homomorphisms, and let W be the set of triples ((H,phi[H],psi[H]),(H',phi[H'],psi[H']),mu) where (H,phi[H],psi[H]),(H',phi[H'],psi[H']) <- V, mu: H -> H' is a homomorphism, phi[H] = phi[H'] o mu, and psi[H] = psi[H'] o mu. Then (V,W) is a category and G x G' is a terminal object.Similar remarks hold for free products of groups G * G', and in a vaguely similar way, we can construct a related category with an initial object.By the way, the practice of needlessly using classes and formulas when sets will do isn't limited to category theory; model theory is just as guilty of this.See pic related for details.
>>16540844You have points (objects) and you have arrows (morphisms) between them. CT is most useful in the cases where the morphisms matter more than the specific details of the objects.>>16540861In the Poset category the objects are partially ordered sets and the morphisms are order-preserving (monotone) maps.
