What are...initial and terminal objects?

[@spogel9981]

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]

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]

​@@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]

@@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]

​@@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]

@@spogel9981 Well, these are 0xn or nx0 matrices 😅

[@MathForLife]

Cool video, so as I understand, there is no initial or terminal object for a cobordism category?

[@VisualMath]

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]

@@VisualMath that is helpful. Thank you!

[@VisualMath]

@@MathForLife Welcome!