What are...initial and terminal objects?

  Рет қаралды 1,304

VisualMath

VisualMath

Күн бұрын

Пікірлер: 10
@spogel9981
@spogel9981 Жыл бұрын
2:17 : On the left side the map is the colume-zero-vector (0,0) and the initial object is also the colume-zero-vector (0,0), is this right? But how can this map result in any vector of K2? I think for (a,b) the map should be +(a,b). What is my mistake? Many thanks for your support.😊
@VisualMath
@VisualMath Жыл бұрын
No, the initial object (and also the terminal object) is the zero vector space, not any vector. It has only one element, namely zero. The map send 0 to (0,0).
@spogel9981
@spogel9981 Жыл бұрын
​@@VisualMathI have a problem understanding how these maps look like. The map from K2 to the zerovector space should send any vector (a,b) to (0,0). The 2x2 zeromatrix would do this. But for the map from the zerovectorspace to K2, I have no idea, because this mapping is not injective.
@VisualMath
@VisualMath Жыл бұрын
@@spogel9981I would not try to think in terms of n-m matrices. The zero vector space is strange, it has dimension zero with the empty set being a basis. So any "matrix" from it is simply a formal zero.
@spogel9981
@spogel9981 Жыл бұрын
​@@VisualMaththanks for your advice. I will try to stop thinking about matrices in this case, which is not very simple, because as maths school teacher I am used to matrices.
@VisualMath
@VisualMath Жыл бұрын
@@spogel9981 Well, these are 0xn or nx0 matrices 😅
@MathForLife
@MathForLife 3 жыл бұрын
Cool video, so as I understand, there is no initial or terminal object for a cobordism category?
@VisualMath
@VisualMath 3 жыл бұрын
That is correct: 1COB has almost no objects "with universal properties" (they are called limits). In some sense I like to think about categories that have such objects (e.g. initial and terminal objects) as "linear" in the sense of "close to KVECT", and 1COB is certainly far away from being linear.
@MathForLife
@MathForLife 3 жыл бұрын
@@VisualMath that is helpful. Thank you!
@VisualMath
@VisualMath 3 жыл бұрын
@@MathForLife Welcome!
What are...products and coproducts?
10:55
VisualMath
Рет қаралды 2,8 М.
27 Unhelpful Facts About Category Theory
9:26
Oliver Lugg
Рет қаралды 428 М.
Mom Hack for Cooking Solo with a Little One! 🍳👶
00:15
5-Minute Crafts HOUSE
Рет қаралды 23 МЛН
This Is the Calculus They Won't Teach You
30:17
A Well-Rested Dog
Рет қаралды 3,5 МЛН
Category Theory For Beginners: Universal Properties
1:12:37
Richard Southwell
Рет қаралды 14 М.
Galois Theory Explained Simply
14:45
Math Visualized
Рет қаралды 484 М.
The Genius Way Computers Multiply Big Numbers
22:04
PurpleMind
Рет қаралды 232 М.
Teaching myself abstract algebra
14:41
Zach Star
Рет қаралды 283 М.
The SAT Question Everyone Got Wrong
18:25
Veritasium
Рет қаралды 14 МЛН
The Biggest Myth In Education
14:27
Veritasium
Рет қаралды 14 МЛН
Why You Can't Bring Checkerboards to Math Exams
21:45
Wrath of Math
Рет қаралды 440 М.
Mom Hack for Cooking Solo with a Little One! 🍳👶
00:15
5-Minute Crafts HOUSE
Рет қаралды 23 МЛН