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 definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 nforum 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 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.
    • CommentAuthorFosco
    • CommentTimeJan 30th 2017
    • (edited Jan 30th 2017)

    Why do you call two arrows f,g with mutual lifting properties “orthogonal”?

    Is there an immediate link between v,w=0 and fg={*}, where * is the unique element that lifts g:XY on the right of f:AB in

    AXBY

    ?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 30th 2017
    • (edited Jan 30th 2017)

    That’s a good question. I don’t know an “official” answer, but here’s a thought: consider what we mean when we say an object X is orthogonal to an arrow f:AB. It means X “thinks” f is an isomorphism in the sense that hom(f,X):hom(B,X)hom(A,X) is an isomorphism (i.e., as far as X-probes or coprobes are concerned, f is an iso). If we think of an isomorphism as a notion of sameness, then f makes A and B look just the same as gauged by the instrument X.

    Now in elementary physics, say if we are measuring temperature by a thermometer X, there are certain trajectories f:AB between points A,B in space which exhibit sameness as gauged by X, namely by traveling along isothermal lines. These are orthogonal to the “direction” of X given by gradients (really the here is a relation between tangent vectors which are infinitesimal f and 1-forms dX, but hopefully you get the idea).

    When you transport all this to the slice category over Y, you get the standard notion of an object g:XY being orthogonal to an arrow f in the slice.

    • CommentRowNumber3.
    • CommentAuthorFosco
    • CommentTimeJan 31st 2017

    That’s a beautiful analogy. Mine was that lifting f against g is like pairing two vectors:

    If v0 is a vector, then vw=v(w+w//)=vw// where wv and w// is the projection of w along v. So you factor every vector as something in the direction of v and something “orthogonal”.

    If f:XY is a generic arrow in 𝒞 then it factors as f:XAY, and you can “pair it” against other arrows.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2017

    Might be worth asking at the categories list to see if anyone knows.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeJan 31st 2017

    Some time ago when I was working on a section of some monograph i asked myself exactly that question. The well known key result is:

    A pair, (,), of classes of morphisms in a category, 𝒞, is a factorisation system if, and only if, the three conditions

    • every morphism, f:XY, admits a (,)-factorisation, $XErY;$

    • the classes are closed under isomorphism;

    • .

    are satisfied. (In fact, = and =.)

    Looking at the last statement is reminiscent of the ‘orthogonal complement’ relationship between subspaces of an inner product space, and I think that that was how it was explained in some source that I was consulting. (I may even have found this on an nLab page or on Joyal’s pages.)

    • CommentRowNumber6.
    • CommentAuthorKarol Szumiło
    • CommentTimeJan 31st 2017

    I don’t know the history, but I guess I always interpreted it as Tim suggests. The analogy with orthogonal complements in a inner product space can be seen in some other ways too. For example, the factorization axiom is reminiscent of the fact the a subspace and its orthogonal complement together generate the ambient inner product space. Also, two classes of a factorization system intersect in isomorphisms, similarly to orthogonal complements in an inner product space intersecting at 0.