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 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 infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds 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 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.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010

    Over at orthogonal subcategory problem, it’s not clear to me whether or not the “objects orthogonal to Σ\Sigma” should be morphisms orthogonal to Σ\Sigma, or if it should mean objects of XX of CC such that X*X\to * is orthogonal to Σ\Sigma (where ** denotes the terminal object). (Hell, it could even mean objects that are the source of a map orthogonal to Σ\Sigma). I was in the process of changing stuff to fit the first interpretation, but I rolled it back and decided to ask here.

    If it should in fact be the second (or third) definition, I would definitely suggest changing the notation Σ \Sigma^\perp, which is extremely misleading, since that is the standard notation for the first notion.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 24th 2010

    It means objects XX such that X1X \to 1 is orthogonal to Σ\Sigma. Sorry if that was confusing to you, but I was under the impression that this is fairly standard, even if it hasn’t been standardized within the nLab. (It’s also deducible from the proof given on that page.) For example, one may define a jj-sheaf w.r.t. a topology jj to be a presheaf which is orthogonal to the class of jj-dense inclusions of presheaves.

    For those who know what I meant by “object orthogonal to Σ\Sigma”, I don’t think the usage Σ \Sigma^\perp would be “extremely misleading”, particularly since the intended usage is spelled out right on that page (and that’s true even if the notation Σ \Sigma^\perp is elsewhere used to denote a class of morphisms). I think also that the notation is standardly used for both notions (not at the same time of course!). Am I wrong about that?

    I think one can simply make a note of “object orthogonal to Σ\Sigma” at the article orthogonality, and note that this usually means f(X1)f \perp (X \to 1) for all morphisms ff in Σ\Sigma (as opposed to (0X)f(0 \to X) \perp f).

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    Sorry! I was going to actually edit the page orthogonality based on the answer to this question! =)

    I also noticed that the definition is essentially given in the proof after I posted this topic.

    As far as I’ve seen, in the 4-5 books I’ve seen the notion of orthogonality mentioned, Σ \Sigma^\perp always means “the class of morphisms orthogonal to Σ\Sigma”. I’ll take your word for it, since you know a lot more about the subject than I do.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2010

    Yeah, unfortunately the same notation is used “standardly” to mean two different things in different contexts.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    I edited orthogonal subcategory problem:

    added subsections and hyperlinks. Also linked to it from orthogonality, where it might have been a subsection otherwise.