Category

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>image must be object
The arrows don't even have to be functions and the notion of image doesn't appear in the definition of a category. The object n is, by definition, what you may call codomain (target) of each arrow in hom(m,n).

an arrow f in hom(m,n) and an arrow g in hom(n,k) can be composed (f and g are matrices and the composition is matrix multiplication). The objects (integers in this case) don't have do be codomains of functions. In fact, there are axiomatizations of the notion of Category that avoid object altogether (you can come up with axioms that drop them, because the class of objects is always in bijection with the class of identity arrows in a category)

Categories that can be viewed as classes of sets with functions as arrows ... those are called "concrete categories"
https://en.wikipedia.org/wiki/Concrete_category

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I see your point, but how am I supposed to understand pic related definition "morphism A to B"? Am i "escaping" category of natural numbers via hom(m,n) in Mat or are there two types of objects in Mat (numbers and matrices)?

>Objects don't have to be codomains of functions
you mean: of morphisms?

Will this thing become clearer later, or should I fully understand it at the very beginning of category theory?

>>8652140
a category has two sorts: objects and arrows. There's nothing that says that the objects and arrows need to be similarly constructed (whatever that would mean), just that they need to satisfy the category axioms.

In this example, it might help to think about the objects being R^0, R^1, R^2, etc. and the arrows being linear maps between them. Then we're just abstracting out the fact that each R^n is a concrete space, and instead associate each with its dimension (n).

>>8652165
So in general, arrows (hom) do not preserve structure of category, they are more of operation on two objects in Category, and the codomain may or may not be in same category?

In you example of R^n spaces, is it healthy to think that there is no need for specific map from N to R^n in order to distinguish different spaces of same category and instead just consider hom(m,n) which is set of linear maps from R^m to R^n?

>>8652086
Not all morphisms are functions.

I think a nice example of this is the category 1Cob.

Objects are 0-dimensional manifolds.

Morphisms are 1-manifolds whose boundary is the disjoint union of the points.


For example, consider the discrete points {0} and {t}. These are 0-manifolds.

A morphism from {0} to {t} could be given be the interval [0,t].

>>8652265
But isn't your example a function from 0manifold x 0manifold -> R. f({0},{t}) = [0,t] ?

>>8652234
>structure of category
What do you mean by this?

>map from N to R^n
?

Again:
The category is a type of objects, namely {0,1,2,3,...}. Those are called objects.
And for each pair of those, e.g. 5 and 8, you have another type, the hom-classes, e.g. hom(5,8).
What's interesting in category theory is the hom-classes. The objects are almost irrelevant. The objects are just labels to distinguish different hom-classes.

E.g. you work in the cateogry of topological spaces and continuous functions. You study the category to understand the continuous functions and their concatenation. The fact that there are topological spaces at the domain and codomain of each function is secondary.
In the case of Mat, you have n x m matrices. Each type gets a hom-class. So you have
hom(1,1)
hom(1,2)
hom(2,1)
hom(2,2)
hom(1,3)
hom(2,3)
... and so on.
You study the matrix product.
And the objects happen to be the numbers.

You can now be more concrete (and thus consider a representation in terms of a "concrete cateogry") and say:
Let's look at the category where the objects are the spaces R^n and the hom-classes are the linear maps L(R^n, R^m).
But the so defined cateogry is in bijection to the one defined above (Mat, with numbers as objects and sets of matrices as hom-classes), so nothing relevant is gained.
(Except maybe for the insight that the straught forwardly defined cateogry Mat can be faithfully be embedded in Set, the category of sets and functions. And there are many theorems about Set.)

But again, the hom-classes are what's relevant. The objects are in fact redundant, you could use a more minimal but complicated language and write down all theorems without mentioning any objects (and instead expressing the theorems in a way that only speaks of the domains of the arrows that obey the identify condition, i.e. A==dom("id_A")

>>8652296
No. [0,t]:{0} --> {t} is the morphism, it is not the target of a function.

Another example of a morphism in this category:

S^1 : {1} --> {1} , the circle can be viewed as an endomorphism of an arbitrary point.

>>8652318

As structure, i thought that when we say dom(f) or cod(f) we mean that morphism gets us to some object in the same category as f. But now I understand what you are saying about irelevance of objects. I am only using numbers to express dimensions of matrices Im working with.

>>8652318
and one more thing, how come that hom(m,n) = matrix(m x n) isn't function between space of NxN and Matrices (bijection even, I think)

>>8652326

so if I have f = hom(0,t), then cod(f) = t and dom(f)=0. What is [0,t] then called?

I am sorry, Im having problems understanding this concept.

>>8652346
The set of morphisms Hom({0},{t}) would be the set of all 1-manifolds M s.t. ∂M={0}U{t}.

>>8652341
>function between space of NxN and Matrices (bijection even, I think)

There are infinitely many 2x2 matrices with rational, or real, or complex entries in hom(m,n) (i.e. matrix(m x n)) but you can count the labels in NxN (e.g. (1,1), (1,2), (2,1), ...).

I.e. you can count the hom-classes (the set of hom-classes associated with the category Mat is in bijection with NxN), but each such hom-class has an uncountable number of arrows.

>>8652086
Why are you looking at this stuff if you don't have a basic understanding of abstract algebra.

>>8652480
i have