Bartosz MilewskiI'm struggling with the definition of the category of elements--the direction of morphisms. Grothendieck worked with presheaves \(C^{op} \to \mathbf{Set}\), with a morphism \((a, x) \to (b, y)\) being an an arrow \(a \to b\) in \(C\). The question is, what is it for co-presheaves? Is it \(b \to a\)? nLab defines it as \(a \to b\) and doesn't talk about presheaves. Emily Riehl defines both as \(a \to b\), which makes one wonder what it is for (š¶įµįµ)įµįµāššš , not to mention \(C^{op}\times C \to \mathbf{Set}\).
John Carlos Baez@BartoszMilewski - it's just a convention; either convention is possible. So in what sense are you struggling with it? Are you struggling to decide what you like best? You don't really need to pick a favorite.
Martin Escardo@johncarlosbaez @BartoszMilewski
This reminds me that two different communities write the specialization order for points x and y of a topological space as x ā¤ y and y ā¤ x respectively.
It is just a matter of convention.
But it does make papers from the other community very hard to read, as we need to flip the order every time in our heads to match our intuition and previous knowledge.
@johncarlosbaez @BartoszMilewski
Another analogy would be mirroring the keyboard in a piano. Even the most skilled pianist would have a severe difficulties playing in this isomorphic keyboard.
@johncarlosbaez It's actually a serious problem. How do I define the category of elements for a profunctor: both arrows in the same direction, or twisted?
@BartoszMilewski - they both exist, neither is "right" by decree of god, so use the one that works for what you're doing... which may change next week.
@johncarlosbaez I thought that once you define a thing called the "category of elements" for \(C \to Set\), then it should work the same if I replace \(C\) with \(C^{op}\) or with \(C \times C\) or \(C^{op} \times C\) and so on. All other definition in nLab (or in math, in general) worked this way.
Omar AntolĆn@BartoszMilewski @johncarlosbaez It does work for all those other cases too, but there is also a second possible definition which also works for all those cases. š¤·š½