Like many other people, I sometimes happen to have lucid dreams. Alas, also like many other people, I do not remember most of my dreams; I seem to be able to recall them only when I am woken up suddenly, if then. I was down with a nasty head cold earlier this week, and one of the friends I’m sharing a flat with had to wake me up to catch the morning bus. As a result, the following dream situation has been preserved for history.
At first I was having a dream that seemed unusually coherent but not truly lucid. (I could reflect on it while it was going on, but I was not fully aware that it was a dream.) Later, I became aware, of course, that it was a dream, but the odd thing is that I was still dreaming. In other words, I had a dream within which I was aware that a dream that had just concluded was in fact a dream.
What is more, within the second dream, I visualised something representing the first dream, framed by a yellow dashed line.
Has anybody had a similar experience?
(Quick meta-reflection. Given that I do not really talk about my personal life here, should I talk about a dream, or dreams in general? A lunchtime conversation has convinced me that a dream will be seen as much less personal, simply because nobody can truly figure it out. Is the presence of a dream here a sign of the waning of Freudianism?)
Not to be negative (or positive), but is there anybody else there who, in spite of not being a category theorist or even an algebraic geometer properly speaking, thinks, in retrospect, that Chapter 2 of Hartshorne’s ‘Algebraic Geometry’ would have made more sense on a first reading if the author had introduced more abstract nonsense?
What can be simpler than defining a presheaf as a contravariant functor? More to the point, isn’t the definition of the inverse image sheaf in section II.1 a little unnatural and ugly on its own? It makes more sense to define the inverse image in terms of the universal property it satisfies, and then give Hartshorne’s definition, not as a definition, but as a construction that shows that such an object in fact exists.
(For that matter, we could go back even to Atiyah and Macdonald and say the same about at least one point – why are things called “the direct limit” and “the inverse limit” ever defined as they are, the (younger) reader wondered? Now I can see that it makes sense to define direct and inverse limits in general, and then show that, for the objects handled in commutative algebra, these limits can be constructed in very concrete ways.)
Note that I am asking for a very limited amount of nonsense here. I am not asking for anybody to introduce from the start Grothendieck topologies, which may be perversely elegant, but which I truly do not need at this stage in my life. (I may need to eat my words shortly.)
What is your take on this?