Talk:Specialization (pre)order

Specialization
Should x &le; y be read "x is a specialization of y" or "y is a specialization of x"?

Hartshorne says the former Algebraic geometry (Ex 3.17 p.93). However the motivational paragraph seems to claim the latter. Actually, the paragraph appears to be self-contradictory. If first claims that y is more specific, but then uses an analogy with genus and species and generic points which indicates that x is more specific. Can anyone sort this out? -- Fropuff (talk) 05:25, 10 December 2007 (UTC)

"we say that x is a specialization of y and that y is a generization of x" Is the latter word correct, or should it be "generalization"? Also, are these preorders the 0-dimensional case of weak n-categories where the cells are points, paths, homotopies and so on? (Perhaps I am completely off-base here.) If so, a note near the end would be useful. 2001:984:9396:1:6C97:7703:246A:3C4A (talk) 22:32, 21 December 2016 (UTC)

Overemphasis on the direction of the preorder
It seems that some authors write $$x \le y$$ to mean "x is a specialisation of y" while others use it to mean "y is a specialisation of x". Fine.

The current version of the "Definition and Motivation" section seems to put a huge emphasis on this trivial point. This makes it impossible to follow, because it's constantly weighing up the pros and cons of one convention versus the other, while at the same time trying to explain what they both mean. If someone's in the mood for editing, it would be a great help just to pick one and include a brief note somewhere - just once, preferably at the end - that some authors use the other convention.

Nathaniel Virgo (talk) 20:23, 18 March 2020 (UTC)