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.
@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