Lawrence Breen lecture

0:00 It's exactly the same as the things we talked about before.

5:00 The point means that I let you write explicitly. This is the first one. The second one is the first one. This is the second one. And so, since this is a line, a line like this, I'm going to encode this. Z, Z, Z. In the form of Z. And it's always the same situation here, lambda in delta 1 in the direction of a, if we walk in the cable, and now the diagram which is on the table which is made in 3,

10:00 is kept by the function at source, times 3 of delta 1 which comes from this space of paths, so I have this in conjunction with delta 1, plus 1, lambda 1, 0.

12:30 The first thing we looked at was the fact that the groups were well aligned. So we went to the portal and we found one thing, the most interesting thing. So it remains to be demonstrated. I'm going to show you what it is. With all the spaces that we can use. It's an isomorph to the group phi n-1. The space is there, it's omega, it's a parameter. It's like an isomorph. There is the group law, the co-group law.

15:00 So really this one, it's well aligned. And now, the first slide will show you the isolants of physics, for the visual homozygous group of X. The advantage of this demonstration is that, as I said last time, it is not the demonstration of the visual homozygous group, but rather the demonstration of the visual homozygous group of Y. This is the real reason for this. For this, there is a theorem, which is the following basic theorem. Fibre.

17:30 Fibre. Let's take a look at the fibres. So, basically... I think I will be able to write exactly what I want to say later, so I hope it's clear. So, it starts here. Pi n, F, V. Pi n, V. Since the point P is in the fiber, Pi n, Y, star. The equation annuls by P here. Pi n of the point P. To read this, it means to calculate for each element in it, this action, to give them an image, an element in it, an element in it, and it's just an action.

20:00 And then the exactitudes below, that's the points together, pi zero, two f, and so the exactitudes below, the exactitudes, it's not the core.

22:30 If I have something beta minus one, the exactitudes of the other points, it's not written.

25:00 There was a lot of exposure since then, for those who have been, I don't know if anyone remembers, but if you know the times there were exactly the situations, not exactly the groups of homotopies at all, but we had exactly the situation, there was a situation like that, it's very characteristic of the homotopic theory, isn't it? I'm not going to go through them all because they're long but they're the subjects we're used to now. So I'm going to give you two things I'm going to do. I'm going to give you the definition of these things here. I'm going to give you the exactness at this level. The other things are easier.