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Atrium > Mathematics, Physics & Philosophy: Projective space as a category

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    • CommentRowNumber1.
    • CommentAuthorGavinWraith
    • CommentTimeNov 27th 2009
    Anders Kock pointed this out to me a long time ago, and it may be too well known for comment. Suppose we consider a projective space over a field k. We can make a category by taking objects to be points and maps from X to Y to be triples (X,P,Y) where P is a point on the line XY. Composition is rather obvious: (X,P,Y)(Y,Q,Z) = (X,R,Z) where P,Q,R are collinear. Associativity is DesArgues' theorem. But what if X=Y? We have to interpret a map from a point to itself as given by an invertible element of k. Composition with endomaps shifts a point further down the line, so that the crossratio of the domain, codomain, point and shifted point is the invertible element of k. I omit the details. Has anyone thought about this or carried the idea further?
    • CommentRowNumber2.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 4th 2009

    I was holding off in case someone had something specific to say to your actual question, but no-one has (you could try it on MathOverflow) so I'll chip in with something else.

    Together with 'metric spaces as enriched categories', this seems to be a nice example of 'everyday object seen through the eyes of category theory'. When I get a minute, I'll put these two examples on a page in the lab and it'd be nice to add to them.

    Specifically to this example, I wonder what the functors are. Do you (Gavin) know? Did Anders say anything about that? It probably wouldn't be hard to work out and of course I'd do that on the proposed page but I wondered what was already known.

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeDec 7th 2009

    I see that Anders has written up his observation:

    0912.0822

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMay 6th 2015

    Coming back to this very old discussion, which I just happened across: it’s not clear to me that it’s really buying us very much here to think of a projective line as a category. In contrast to the case of metric spaces, where the axioms of a metric space really are literally the axioms of an enriched category, here it seems that the real data of a projective line resides in the cross-ratio axioms that are imposed on top of the category structure and essentially subsume it (in particular, he shows that if a map preserves cross-ratios then it is automatically a functor). So it seems that really we are just saying that a projective line is a set equipped with a cross-ratio operation.

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