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.
    • CommentAuthorSridharRamesh
    • CommentTimeFeb 20th 2010
    • (edited Feb 20th 2010)
    I added some thoughts I had to the page on anafunctors, outlining another way of viewing their definition. It is quite possible that there exists standard terminology for what I called "AllButChosen(-)" in my edit, but I'm afraid I don't know what it is.
    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 21st 2010
    • (edited Feb 21st 2010)

    I can't edit the nlab at the moment, so I'll reply here. I think your construction AllButChosen(C) is just Cat_{ana}(1,C). If we assume choice in the meta-logic (this is needed to choose pullbacks of categories, which to my mind is not as strong as Choice, because there are constructions for ordered pairs of things, given certain foundations like ZF, even if they are not unique), then Cat_{ana} is a bicategory, but otherwise it is an anabicategory (=category enriched in Cat_{ana}). In the former case, AllButChosen(C) is an ordinary category, and the functor AllButChosen(-) is Cat_{ana}(1,-):Cat_{ana} \to Cat. If we do away with choice altogether (and thus have the latter case of the aboce two options), then this really lands in $Cat_{ana}$$ again. This is a good idea, though.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 21st 2010
    Sridhar, it seems to me that I have seen the terminology "clique" for all but chosen. See for example the Adv. Math. paper Braided Monoidal Categories by Joyal and Street (and I've seen it elsewhere in articles with Joyal's name attached).
    • CommentRowNumber4.
    • CommentAuthorSridharRamesh
    • CommentTimeFeb 21st 2010
    Thanks; I've updated the edit to reflect that terminology.
    • CommentRowNumber5.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 21st 2010
    • (edited Feb 21st 2010)

    You should add an entry for that terminology if possible. Or at least make a stub and post it on the n-forum so someone else can take care of it.

    • CommentRowNumber6.
    • CommentAuthorSridharRamesh
    • CommentTimeFeb 21st 2010
    I've created an entry for cliques and moved most of the material there; I'll make a separate post on the n-forum for this as well.
    • CommentRowNumber7.
    • CommentAuthorTim_Porter
    • CommentTimeFeb 22nd 2010

    It may be a good idea to have a link to a page on cliques in graph theory. there is a Wikipedia page

    http://en.wikipedia.org/wiki/Clique_%28graph_theory%29

    which may do.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2010

    I edited "clique" a bit -- Makkai calls these "anaobjects" although I think "clique" is a good word.

    I didn't notice whether this happened recently, but I don't like having the page anafunctor talk about functors being "k-surjective for all k." Right now the page is just about anafunctors between ordinary (internal) 1-categories, so it suffices to say "fully faithful and essentially surjective," which I think is more friendly to a lot of people. If we want to write about anafunctors between higher categories, great, but let's do it on a different page, or a special section at the end of that page.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2010

    I didn't notice whether this happened recently, but I don't like having the page anafunctor talk about functors being "k-surjective for all k."

    I am too lazy to check, but if this is an old leftover from an edit I made, feel free to remove it. I agree that the page should concentrate on the 1-categorical case.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 22nd 2010

    I fixed it.

    • CommentRowNumber11.
    • CommentAuthorDexter Chua
    • CommentTimeJul 29th 2016

    Added the definitions of full and faithful anafunctors and composition of anafunctors, and some examples, including the promised inverses of fullly faithful essentially surjective anafunctors.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 29th 2016

    I fixed the syntax for referring back to an earlier section (on how to do this, see here).

    • CommentRowNumber13.
    • CommentAuthorDexter Chua
    • CommentTimeJul 29th 2016

    Thanks! I somehow managed to get it the wrong way round without noticing…