Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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.