Not signed in (Sign In)

Start a new discussion

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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science connection constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory 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-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory 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.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 24th 2010

    I did a little bit of rewriting and cleaning up at reflective subcategory, in an effort to make things clearer for the neophyte. Part of the cleaning-up was to remove a query initiated by Zoran under the section Characterizations (I rewrote a bit to make the question vanish altogether).

    There’s another query of Zoran at the bottom which I think was answered by Mike, but let me ask before removing it.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 25th 2010

    This were the discussions:

    Zoran: Gabriel–Zisman neglect the set theoretical issues on the EXISTENCE of localizations. Is the last conditions really equivalent or we need to make some set-theoretical assumptions ?

    Urs Schreiber: the point here is that the localization is not at any arbitrary set of morphisms, but at precisely the class that the left adjoint sends to equivalences. This is a very special class with very nice properties and is what makes the localization come out nicely. More details on this happen to be at reflective sub-(infinity,1)-category.

    About word localization:

    Zoran: this is not universally accepted. In topos theory community yes. But in the setup of abelian categories, like categories of modules, people often use word localization even if left exactness is not met. If it is it is often said flat localization in those circles (though sometimes one says flat localization only if the stronger condition is satisfied: composed endofunctor is flat). The localization of the underlying ring (in the case of module categories) is the component of adjunction at that ring, and for Gabriel localizations (where T is flat) the arget module is canonically a ring and the component of the adjunction is a ring morphism. But only if the localization is perfect this morphism of rings tell you all information about the localization functor.

    Mike Shulman: I changed it, how’s that?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 25th 2010

    Yes, those were the discussions, but I’m asking whether it’s alright by you to remove those discussions, i.e., have your queries been answered to your satisfaction?

    For example, Urs answered your first query, and I went ahead and rewrote the statement to say that the reflection r:ABr: A \to B exhibits BB as the localization with respect to the class of arrows s:aas: a \to a' for which ir(s)i r(s) is an isomorphism. I suppose it would be good to write down a formal proof, but the point is that there is no need to worry about set-theoretical assumptions since the statement just gives you the localization explicitly. (It may be clearer if I write something down.)

    In the other case, apparently Mike changed the wording in response to your comment, and asked if you were now satisfied.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJun 26th 2010

    I removed the boxes at least half an hour before you were asking me if that is OK…sorry

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 26th 2010

    Oh, sorry. You were just recording those discussions in the Forum; I get it now. Makes sense.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)