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 comma 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 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.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.

    And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?

    I think the entry-situation here deserves to be further harmonized.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    You misread orthogonality, it does indeed say that there “exists a unique diagonal filler”.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2010

    Perhaps the category-theoretic notion should be at orthogonal morphisms or something.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    You misread orthogonality, it does indeed say that there “exists a unique diagonal filler”.

    Ah, right, sorry.

    Hm, but the notation \perp is also used for non-unique LLPs.

    Perhaps the category-theoretic notion should be at orthogonal morphisms or something.

    Yes, maybe. I won’t do it now, however.

    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    I really like Cisinski’s notation for non-unique llp (resp. rlp) (l(X) (resp. r(X)) for a class of morphisms X). It’s simple, clean, clear, and doesn’t require left exponents.

    In fact, I think that this notation should be extended to orthogonality as l (X)l_\perp(X) (resp. r (X)r_\perp(X)).

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2010

    Hm, but the notation \perp is also used for non-unique LLPs.

    It shouldn’t be. That notation for unique lifting is older, and also is the only one that matches the terminology (\perp = orthogonal = unique lifting).

    The notation I like for non-unique lifting properties is \boxslash, which is very easy to remember because it is nothing other than a picture of a lifting in a square! Unfortunately iTex doesn’t support it yet. (I am not a fan of appropriating lowercase roman letters to have a specific global meaning – I want to be able to use letters like “l” and “r” as local variables.)

    • CommentRowNumber7.
    • CommentAuthorAndrew Stacey
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    Do you mean ⧄? If so, you can get it in iTeX by using the entity syntax: ⧄. Not sure what the context would be, but you could say aba ⧄ b if you wanted. (If it’s the other way, then that’s ⧅: ⧅. I’m hoping that the unicode version of detexify will help a lot with finding symbols like this.)

    If you think it would be a useful addition to iTeX, it’s really easy to extend the symbol set so just drop a line to Jacques. Make sure you tell him what sort of object it should be (standard symbol, operator, relation etc). If there’s similar characters, it’s nice to provide the “whole set” rather than just one or two.

    • CommentRowNumber8.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    Totally serious, how about llp(X)llp(X) (resp. rlp(X)rlp(X))?

    Oh, and by the way, Urs. The \perp notation you’re thinking of is in Lurie, but for nonunique lifting, he uses it as a subscript.

    Meanwhile, the reason why I hate the \perp notation is that you have to write stupid things like (M ){}^\perp (M^\perp) or (M ){}_\perp (M_\perp) with parentheses, which look kinda silly.

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2010

    Thanks, Andrew. I’m kinda busy right now but maybe I’ll mention it to Jacques.

    llp(X) and rlp(X) wouldn’t be too bad. I don’t think (M )^\perp(M^\perp) looks silly though; I think it describes what’s going on pretty well.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2010

    Also, the nice thing about \perp and ⧄ is that they let you use the same notation for the relation between two morphism and the operation on sets of morphisms. E.g. we have fgf \perp g if and only if g{f} g \in \{f\}^\perp if and only if f {g}f \in {}^{\perp}\{g\}.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    Okay.

    I think many of our entries use LLP and RLP. Such as small object argument.

    (They do because I typed that, but after Mike made me not use \perp ;-)

    • CommentRowNumber12.
    • CommentAuthorIan_Durham
    • CommentTimeJun 24th 2010
    I'm very glad to see you guys debating good notation. Too many physicists and even some mathematicians don't put enough thought into good notation. (The majority of the rest of this discussion is over my head...). :-)
    • CommentRowNumber13.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 24th 2010
    • (edited Jun 24th 2010)

    @Mike: I don’t know. I just really dislike how (X ){}^\perp (X^\perp) looks. It’s an ugly string of symbols.

    If I had to rationalize why I don’t like it, I think it looks silly to have an operator acting on the left as a superscript/subscript with a parenthesized argument.